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| Meta Title | Rethinking Video Transmission: A Quantum Fourier Transform-Based Approach for Error-Prone Channels | MDPI |
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| Boilerpipe Text | Article
22 March 2026
and
Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XQ, UK
*
Author to whom correspondence should be addressed.
Abstract
Reliable video transmission over error-prone channels remains a significant challenge due to the inherent trade-off between compression efficiency and noise resilience in conventional systems. To address these issues, this paper introduces a novel quantum Fourier transform (QFT)-based framework that integrates video compression and transmission within a unified quantum frequency-domain representation. The framework converts video data into a classical bitstream and maps it onto multi-qubit quantum states with variable encoding sizes (
n
), enabling flexible control over compression levels. Through the application of the QFT, these states are transformed into the frequency domain, where only selected coefficients are transmitted to reduce bandwidth requirements. At the receiver, the transmitted components are used to reconstruct the full representation, followed by inverse transformation and decoding to recover the video sequence. The performance of the proposed framework is evaluated using peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and video multi-method assessment fusion (VMAF). The results demonstrate that increasing the number of qubits enables exponential compression, achieving ratios up to
2
n
:
1
, while maintaining high reconstruction quality under ideal transmission conditions. However, higher-qubit configurations exhibit increased sensitivity to channel noise, leading to a more rapid degradation as the signal-to-noise ratio decreases. In contrast, lower-qubit configurations provide improved robustness, maintaining more stable reconstruction quality under noisy conditions, albeit with reduced compression efficiency. Among the evaluated configurations, the two-qubit system achieves an effective trade-off, providing a compression ratio of
4
:
1
while maintaining strong visual and structural fidelity along with enhanced resilience to channel impairments.
1. Introduction
The rapid evolution of consumer electronics and intelligent multimedia systems has amplified the central role of video transmission in modern communication architectures. Applications such as ultra-high-definition (UHD) video streaming, telepresence conferencing, medical video diagnostics, augmented reality (AR) [
1
], virtual reality (VR) [
2
], autonomous navigation, cloud-based surveillance, interactive digital displays, and immersive extended-reality platforms increasingly depend on the seamless transfer of real-time video content at high fidelity. As resolutions scale from high definition (HD) to 4 K, 8 K, and beyond, alongside the growing popularity of high-frame-rate formats and multi-view video, the size and complexity of video data have grown exponentially. These advancements, while enabling more immersive user experiences, impose substantial demands on communication networks, which must handle massive data volumes under stringent requirements for latency, reliability, and robustness.
Classical video compression technologies, including advanced video coding (AVC/H.264) [
3
], high efficiency video coding (HEVC/H.265) [
4
], and the more recent versatile video coding (VVC/H.266) [
5
], achieve high compression performance by exploiting spatial and temporal redundancies through techniques such as motion-compensated prediction, transform coding, rate–distortion optimization, and entropy coding. These methods have significantly improved the efficiency of multimedia delivery over modern networks. However, the underlying framework remains constrained by the principles of classical signal processing and information theory. When aggressive compression is applied, classical codecs often produce visual distortions such as blocking artifacts, ringing patterns, motion smearing, and loss of fine texture details. Moreover, because modern video standards rely heavily on predictive coding structures such as group of pictures (GOP), errors introduced during transmission can propagate across multiple frames, causing noticeable temporal degradation.
To address these impairments, classical communication systems utilize various reliability and transmission mechanisms including adaptive bitrate streaming, channel coding [
6
], retransmission protocols, error concealment [
7
], and feature-based transmission strategies [
8
,
9
] that attempt to prioritize perceptually important visual information. While these approaches can improve robustness and maintain acceptable quality under moderate channel impairments, they do not fundamentally eliminate the sensitivity of compressed bitstreams to channel disturbances. In bandwidth-constrained, wireless, or long-distance communication environments, fluctuations in channel capacity, packet losses, and additive noise can still lead to significant degradation in video quality. With the growing demand for ultra-high-resolution visual content, real-time streaming, and highly reliable multimedia services, the performance of conventional compression and transmission frameworks is increasingly approaching the practical limits imposed by classical communication paradigms.
In response to these limitations, quantum communication [
10
] has emerged as a transformative paradigm that leverages intrinsic properties of quantum mechanics [
11
], such as quantum superposition [
12
], quantum entanglement [
13
], and unitary state evolution, to achieve new modalities of information representation and transmission. In contrast to classical communication, where information is represented as fixed binary sequences, quantum communication represents information using quantum states, which provide a substantially larger and more expressive representation space. Initial studies on quantum multimedia transmission have mainly focused on time-domain quantum encoding [
14
,
15
], where classical data are directly represented as quantum states without applying any transformation into the frequency or spectral domain. These time-domain models have demonstrated benefits in terms of enhanced state representation and improved data fidelity [
16
]. However, they remain inherently inefficient for high-dimensional video content due to the large number of qubits required to achieve high-quality reconstruction, as well as their susceptibility to accumulated noise effects, including depolarizing noise, amplitude-damping noise, phase-damping noise, and other quantum decoherence processes. Video data, with their temporal continuity and spatial variability, exhibit patterns that cannot be efficiently exploited in the time domain alone [
17
].
To respond to this, quantum Fourier transform (QFT) [
18
] offers a powerful alternative for representing quantum-encoded video data in the frequency domain. The QFT, an essential unitary transformation in quantum domain, maps quantum amplitude distributions into frequency-domain components, enabling efficient identification of redundancies and sparsity in the underlying signal structure [
19
]. By applying the QFT to quantum video data, it becomes possible to achieve intrinsic quantum-domain compression, suppress noise-sensitive spectral components, and reduce quantum resource requirements, all while maintaining the essential visual and temporal coherence necessary for high-quality video reconstruction.
Motivated by these advantages, this study introduces the first unified QFT-based framework for video compression and transmission, integrating quantum encoding, frequency-domain transformation, compression, and quantum-channel transmission into a single cohesive pipeline. Unlike prior work, which treats compression and transmission as separate processes or relies on fixed-size qubit encodings with limited adaptability, the proposed method enables flexible multi-qubit symbol encoding, where the number of qubits allocated per symbol can vary according to the desired compression ratio and available quantum resources. This design makes the framework the first to provide an adaptive quantum symbol-level compression mechanism within the frequency domain specifically for video transmission. By allowing adjustment of qubit allocation, the system can effectively balance compression efficiency, transmission robustness, and computational complexity. This makes it suitable for a wide range of video characteristics and diverse quantum channel conditions. Consequently, the proposed approach represents a significant step toward practical and high-performance quantum video communication systems, offering both high fidelity and efficient utilization of quantum resources.
The system begins by converting the input video into a classical bitstream for subsequent processing. To enhance resilience against channel-induced errors, polar coding with a code rate of 1/2 is applied to generate a channel-encoded bit sequence. After classical channel encoding, the bitstream is mapped to quantum states according to the selected qubit encoding size. This step forms a structured quantum state vector that captures both spatial and temporal characteristics of video data. The QFT is then applied to this state vector, converting it from the time domain into the quantum frequency domain. The transformation exposes sparsity, enabling selective retention of dominant frequency coefficients while discarding or compressing insignificant ones. As a result, the transmitted quantum symbol payload is significantly reduced while preserving essential visual features.
Following quantum-frequency-domain compression, the retained coefficients are transmitted over a noisy quantum communication channel. The channel introduces realistic quantum noise, modeled through quantum noise operators such as Kraus operators associated with bit-flip, phase-flip, depolarizing, amplitude-damping, and phase-damping processes. At the receiver, the system reconstructs the complete frequency-domain quantum state by appropriately restoring suppressed coefficients based on the compression ratio and encoding parameters. The inverse quantum Fourier transform (IQFT) is then applied to convert the restored frequency-domain representation back into the time domain. After measurement of the quantum state, the resulting classical bitstream is passed through the polar decoding process to correct channel-induced errors. The final step reconstructs the output video frames, restoring both the temporal continuity and structural consistency of the original sequence. Experimental evaluations demonstrate that the proposed QFT-based architecture achieves significant improvements in compression efficiency, and reconstruction fidelity compared to conventional quantum communication and classical video transmission. By evaluating multiple qubit encoding sizes, the framework reveals how quantum compression behavior and robustness vary with resource allocation, offering important insights for deploying quantum video transmission systems in future quantum networks.
1.1. Key Contributions
The key contributions of this study are summarized as follows:
A novel quantum frequency-domain approach for video compression, utilizing QFT to reduce redundancy and enhance robustness within a quantum communication framework.
A unified architecture for quantum video transmission that integrates frequency-domain compression and noisy-channel transmission to achieve a scalable, efficient, and noise-resilient system.
A comprehensive analysis of multi-qubit encoding configurations, demonstrating how qubit symbol size influences compression efficiency, spectral sparsity, and end-to-end video reconstruction performance.
Extensive experimental validation demonstrating the effectiveness of the proposed framework across multiple video datasets and channel conditions, highlighting compression efficiency, robustness to quantum noise, and improved reconstructed video quality compared to classical and conventional quantum approaches.
1.2. Organization of the Paper
The remainder of this paper is structured as follows:
Section 2
reviews prior work in quantum communication, quantum multimedia processing, and QFT.
Section 3
presents the proposed QFT–based methodology for unified video compression and transmission, detailing the multi-qubit encoding process, frequency-domain transformation, and noisy quantum channel modeling.
Section 4
provides experimental results and comparative analysis with conventional quantum methods. Finally,
Section 5
concludes the study by summarizing key findings, discussing the significance of the proposed framework, and outlining directions for future research.
3. System Model
The system illustrated in
Figure 1
is designed to provide efficient video compression and transmission while maintaining reliable delivery of video content. It works with a variety of video sequences, supporting different resolutions and frame rates, and focuses on reducing the amount of data that needs to be transmitted without compromising the overall visual quality. Video segments are represented in a compact form, allowing for easier handling and transmission over limited-bandwidth channels. The workflow is organized to ensure that all steps, from preparing the video data to reconstructing the received frames, are seamless and coordinated. By prioritizing simplicity and efficiency, the system is able to deliver videos effectively across different scenarios while preserving clarity, continuity, and smooth playback.
Figure 1.
Proposed quantum communication architecture for compressing and transmitting video data.
The performance of the system is assessed using a set of video sequences selected to cover a wide range of realistic conditions. Testing involves three videos with varying amounts of structural information (SI) and temporal information (TI). The high-motion video [
43
] features fast-moving objects and frequent scene transitions, leading to high SI and TI values, which places significant demands on both compression and transmission processes. The medium-motion video [
44
] features slower-paced actions and less frequent scene transitions, with intermediate SI and TI, representing content of average complexity. The low-motion video [
45
] has limited movement and low TI, while retaining some structural detail, making it easier to compress. These videos are uncompressed and available in three spatial resolutions:
320
×
180
,
1280
×
720
, and
1920
×
1080
, covering low-resolution to full-HD content. The videos are encoded at 20, 30, and 50 frames per second (fps) to reflect different levels of motion activity. Along with raw uncompressed sequences, the evaluation also includes videos encoded using the VVC standard with GOP sizes of 8 and 32. These encoded versions share the same resolutions and frame rates as the uncompressed set, enabling analysis under different temporal compression structures while keeping spatial and temporal properties consistent. This selection allows the framework to demonstrate its versatility and compatibility with different video formats and coding schemes. Furthermore, the design is scalable, capable of supporting various video resolutions, frame rates, and source coding methods, making it adaptable to a wide range of practical applications.
The video processing begins by converting each video into a sequential bitstream representation. These bitstreams can be optionally protected using polar codes with a rate of
1
/
2
to provide error resilience during transmission. To evaluate the inherent performance of the system independently of error correction, scenarios without channel coding (uncoded) are also considered. During the quantum encoding stage, groups of bits are mapped into multi-qubit states, where the size of each qubit encoding size (
n
) is selected between 1 and 8. This mapping determines the effective compression ratio for each segment. The resulting quantum state is then transformed into the frequency domain using a QFT-based approach, allowing the video information to be represented with fewer coefficients rather than the entire state vector. This frequency-domain compression reduces the amount of data transmitted while preserving the essential information needed for accurate video reconstruction.
After the frequency-domain compression, the encoded video coefficients are transmitted through a channel that may introduce noise and distortions. At the receiver, the system first interprets the compression parameters to reconstruct the complete frequency-domain representation of each qubit system. Next, an inverse transform is performed to map the data from the frequency domain back to the time domain. The resulting quantum states are measured and transformed into classical bitstreams. These bitstreams can subsequently be processed by a channel decoder to reconstruct the original video content.
In the following subsections, each key functional block depicted in
Figure 1
is explained in detail.
3.1. Classical Channel Encoder and Decoder
The transmission reliability of video bitstreams is improved through the integration of polar codes [
46
,
47
] operating at a 1/2 coding rate. This configuration establishes an optimal equilibrium between error correction performance and resource utilization, introducing controlled redundancy that effectively mitigates channel distortions while maintaining reasonable computational demands and spectral efficiency. The selection of polar codes over alternative error correction schemes, such as low-density parity check (LDPC) [
48
] and turbo codes [
49
], is motivated by their compelling combination of theoretical performance guarantees and practical implementation advantages. Unlike conventional QEC methodologies that primarily target time-domain perturbations, the proposed architecture specifically leverages frequency-domain processing, rendering traditional QEC approaches unsuitable for this particular framework design.
3.2. Quantum Encoder and Decoder
The proposed quantum video compression and transmission system comprises two fundamental components: the quantum encoder and quantum decoder. The encoder transforms classical video data into a quantum representation and applies compression in the frequency domain, while the decoder reconstructs the original data by reversing these operations. The following subsections present a detailed mathematical formulation of both components and their operating principles.
3.2.1. Quantum Encoder
The quantum encoder illustrated in
Figure 2
performs the conversion of classical video data from the time domain into a compact frequency-domain representation. Its operation is divided into several sequential steps. The process begins by assigning each classical bit from the video to a corresponding quantum basis state. In this initialization stage, a bit value of 0 is encoded as the state
|
0
⟩
, whereas a bit value of 1 is encoded as
|
1
⟩
, forming the qubit inputs for the subsequent quantum transformations.
Figure 2.
Proposed quantum encoder operating in the frequency domain.
In the next stage, the system selects the qubit encoding size (
n
), which may vary from 1 to 8 qubits. This parameter determines how many consecutive classical bits are grouped together and converted into a single multi-qubit state. Although there is no strict theoretical upper limit on the
n
, in this work a maximum of 8 qubits is used to maintain manageable system complexity. This choice is sufficient to directly map the pixel values into the corresponding quantum states. The encoder then constructs the complete quantum state vector for each group of bits, incorporating both the initialized qubits and the selected value of
n
. This results in a structured multi-qubit representation that is ready for frequency-domain processing.
Mathematically, a video bitstream (
V
) can be represented as in Equation (
1
).
V
=
{
v
1
,
v
2
,
…
,
v
M
}
,
v
i
∈
{
0
,
1
}
(1)
where
M
denotes the total number of bits. These bits are then processed according to the selected
n
to form the corresponding quantum states for further frequency-domain encoding.
In the case of single-qubit encoding (
n
=
1
), each individual bit from the video bitstream is directly represented by a single qubit within the two-dimensional Hilbert space
H
2
. In this mapping stage, each classical bit is directly converted into its corresponding computational basis state, as expressed in Equation (
2
).
v
i
=
0
↦
|
0
⟩
=
1
0
,
v
i
=
1
↦
|
1
⟩
=
0
1
(2)
For multi-qubit encoding, consecutive blocks of
n
bits are grouped together and encoded as an
n
-qubit register, forming a state in the
2
n
-dimensional Hilbert space
H
2
n
. This grouping process forms a composite quantum state by taking the tensor product of the individual qubits. In this way, several classical bits are combined into a single multi-qubit vector, as described in Equation (
3
).
(
v
1
,
v
2
,
…
,
v
n
)
↦
|
v
1
v
2
…
v
n
⟩
=
|
v
1
⟩
⊗
|
v
2
⟩
⊗
…
⊗
|
v
n
⟩
(3)
As an illustration, consider the case of
n
= 2. In this setting, two classical bits are grouped together, and each pair is represented as a four-dimensional quantum state obtained through a tensor product construction. The resulting states correspond to the formulations shown in Equations (
4
)–(
7
).
|
00
⟩
=
1
0
⊗
1
0
=
1
0
0
0
(4)
|
01
⟩
=
1
0
⊗
0
1
=
0
1
0
0
(5)
|
10
⟩
=
0
1
⊗
1
0
=
0
0
1
0
(6)
|
11
⟩
=
0
1
⊗
0
1
=
0
0
0
1
(7)
When
n
qubits are combined, they form a composite system whose state vector resides in a
2
n
-dimensional Hilbert space, generated by the tensor product of the constituent single-qubit states. This establishes a direct link between the number of qubits and the size of the quantum state representation.
Following the preparation of the quantum state vector, the next step is to shift from the time-domain representation to the frequency domain. The transformation requires selecting a QFT gate whose dimension matches that of the quantum state vector. For instance, a state vector of size
4
×
1
(
n
=
2
) is transformed using a
4
×
4
QFT matrix, while a vector of size
8
×
1
(
n
=
3
) requires an
8
×
8
matrix, and so forth. Accordingly, the QFT matrix, denoted by
F
N
, is constructed as shown in Equation (
8
).
F
N
=
1
N
1
1
1
…
1
1
ω
ω
2
…
ω
N
−
1
1
ω
2
ω
4
…
ω
2
(
N
−
1
)
⋮
⋮
⋮
⋱
⋮
1
ω
N
−
1
ω
2
(
N
−
1
)
…
ω
(
N
−
1
)
(
N
−
1
)
(8)
where
N
=
2
n
corresponds to the quantum state vector dimension, and
ω
represents the primitive root of unity given by Equation (
9
).
ω
=
e
2
π
i
/
N
(9)
This construction ensures that the QFT appropriately transforms the amplitudes of the time-domain quantum state into the frequency domain, analogous to the classical discrete Fourier transform but in the quantum setting.
For a qubit encoding size of
n
=
2
, the QFT is applied using a
4
×
4
unitary matrix, defined in Equation (
10
).
F
4
=
1
2
1
1
1
1
1
ω
ω
2
ω
3
1
ω
2
ω
4
ω
6
1
ω
3
ω
6
ω
9
,
ω
=
e
2
π
i
/
4
=
i
(10)
This procedure can be generalized to higher qubit numbers, with the QFT reorganizing the amplitudes of a quantum state into the frequency domain, analogous to how the discrete Fourier transform converts classical signals.
When the QFT is applied to the two-qubit computational basis states, the corresponding frequency-domain vectors are given by Equations (
11
) through (
14
).
F
4
|
00
⟩
=
1
2
1
1
1
1
(11)
F
4
|
01
⟩
=
1
2
1
ω
ω
2
ω
3
(12)
F
4
|
10
⟩
=
1
2
1
ω
2
ω
4
ω
6
(13)
F
4
|
11
⟩
=
1
2
1
ω
3
ω
6
ω
9
(14)
Equations (
11
)–(
14
) mathematically show that each time-domain basis state maps to a distinct column of the QFT matrix. Careful examination of these columns shows that the second element alone uniquely determines the entire column. Therefore, it is sufficient to transmit only this second component to retain complete knowledge of the quantum state, eliminating the need to send the full frequency-domain vector. In practical implementation, the quantum state is not obtained by extracting amplitudes from an existing superposition, but is instead directly prepared using deterministically computed coefficients via unitary operations. Specifically, for each input symbol, a representative coefficient (e.g., the second QFT coefficient) is computed classically from the bit pattern and mapped to a discrete phase on the complex unit circle. A set of
2
n
such coefficients is then used to synthesize a valid quantum state in the
2
n
-dimensional Hilbert space, ensuring normalization and preserving quantum coherence.
Importantly, the proposed framework operates under a constrained state preparation model in which the inputs to the QFT are computational basis states derived from classical bitstreams rather than arbitrary quantum states. As a result, the possible transmitted states belong to a finite and known set corresponding to the columns of the QFT matrix. Because this set is predetermined and shared by both the transmitter and receiver, the transmitted coefficient functions as a symbol that uniquely identifies the corresponding basis index within this codebook.
This process does not involve any projective measurement or partial state extraction. Instead, it defines a structured state preparation strategy in which the full quantum state is generated from a classical description using unitary operations. Although only a single representative coefficient per symbol is transmitted, the remaining components of the corresponding quantum state are implicitly determined by the predefined transform structure, allowing deterministic reconstruction of the original state at the receiver through the inverse QFT.
For instance, when the qubit register size is
n
=
2
, the full state vector contains 4 elements. By transmitting just the second element, a compression ratio of
4
:
1
is achieved. As the qubit size grows, the compression effect becomes more pronounced: with
n
=
8
, only a single element out of 256 needs to be sent, resulting in a
256
:
1
reduction in transmitted data. As the register size
n
increases, the approach allows effective compression while preserving the integrity of the quantum information. This selective transmission approach significantly reduces the quantum symbol payload while retaining essential quantum information.
In summary, to clarify the effect of the qubit encoding size on the system, the qubit encoding size
n
determines how many classical bits are grouped and represented by an
n
-qubit quantum register prior to the application of the QFT. Increasing
n
increases the dimensionality of the QFT representation, producing
2
n
frequency-domain components and enabling higher compression ratios, since a single transmitted coefficient can represent one of
2
n
possible input states. However, larger encoding sizes also reduce the angular separation between adjacent phase components, given by
Δ
θ
=
2
π
2
n
, which increases sensitivity to channel noise. Consequently, larger qubit group sizes provide higher compression efficiency but exhibit greater susceptibility to transmission errors, while smaller encoding sizes offer improved noise robustness at the cost of lower compression efficiency. Therefore, the choice of
n
introduces a trade-off between compression performance and reconstruction reliability, which is analyzed experimentally in this work for qubit sizes ranging from
n
=
1
to
n
=
8
.
The QFT operation applied in this system can be described using its general theoretical formulation, shown in Equation (
15
).
F
N
|
x
⟩
=
1
N
∑
k
=
0
N
−
1
e
2
π
i
x
k
N
|
k
⟩
(15)
Here,
F
N
represents the QFT matrix corresponding to a qubit register of size
n
. It performs a unitary transformation on an input quantum state
|
x
⟩
expressed in the computational basis. Through this operation, the input state is converted into a superposition of all frequency-domain basis states
|
k
⟩
, with each component weighted by a complex phase factor
e
2
π
i
x
k
/
N
that captures its frequency-domain characteristics. The factor
1
/
N
normalizes the transformation, ensuring that the operation preserves the overall quantum state’s norm and maintains unitarity.
3.2.2. Quantum Decoder
During the quantum decoding stage, the operations applied during encoding are effectively reversed, as illustrated in
Figure 3
. The process begins by evaluating the compression ratio and using the lightweight side information to determine the original qubit encoding size (
n
) and ensure decoder synchronization. Using this information, the full frequency-domain quantum state is reconstructed from the received compressed representation. Subsequently, the IQFT is applied to convert the frequency-domain state back into its corresponding time-domain quantum state. Mathematically, the IQFT is the Hermitian conjugate of the QFT and serves a role analogous to the inverse discrete Fourier transform in classical signal processing. Once the time-domain quantum state is restored, standard measurements are performed to map the state back into classical bits, enabling reconstruction of the original bitstream.
Figure 3.
Proposed quantum decoder.
The IQFT operation (
F
N
−
1
) can be mathematically expressed as the Hermitian conjugate of the QFT, as shown in Equation (
16
).
F
N
−
1
=
F
N
†
(16)
Being the conjugate transpose of the QFT, the IQFT reverses the transformation applied in the frequency domain. When applied to a computational basis state
|
k
⟩
, the IQFT performs the following operation, as shown in Equation (
17
).
F
N
−
1
|
k
⟩
=
1
N
∑
j
=
0
N
−
1
e
−
2
π
i
j
k
/
N
|
j
⟩
(17)
where
N
=
2
n
is the dimension of the quantum state space. The index
j
runs over all computational basis states and represents the output basis states generated by the inverse transform. The negative sign in the exponent indicates the reversal of the phase evolution compared to the forward QFT. This operation maps the frequency-domain representation of a quantum state back to its corresponding time-domain state while preserving unitarity and orthogonality.
For simulation purposes, the QFT matrix
F
∈
C
N
×
N
, where
C
denotes the set of complex numbers, is defined element-wise as shown in Equation (
18
).
F
k
,
l
=
1
N
e
2
π
i
k
l
/
N
,
k
,
l
=
0
,
…
,
N
−
1
(18)
In this expression, the indices
k
and
l
denote the row and column positions of the matrix, respectively. Each entry
F
k
,
l
corresponds to the complex phase factor associated with mapping the computational basis state
|
l
⟩
to the output component indexed by
k
. Each column of the QFT matrix, denoted by
q
j
, can therefore be represented as a vector, as shown in Equation (
19
).
q
j
=
1
N
1
e
2
π
i
j
/
N
e
2
π
i
2
j
/
N
⋮
e
2
π
i
(
N
−
1
)
j
/
N
(19)
It is important to note that the second element of each column uniquely identifies that column. This second entry of the column vector
q
j
is given by Equation (
20
).
q
j
(
2
)
=
1
N
e
2
π
i
j
/
N
(20)
The second element of each QFT column corresponds to a distinct point on the complex unit circle, scaled by
1
/
N
. This unique mapping allows each transmitted coefficient to identify its original column in the QFT matrix unambiguously.
During transmission, the received coefficient, denoted by
q
˜
(
2
)
, may be affected by channel noise
η
∈
C
, where
C
denotes the set of complex numbers and
η
can alter both the amplitude and phase of the transmitted coefficient, as expressed in Equation (
21
).
q
˜
(
2
)
=
q
j
(
2
)
+
η
(21)
The process of reconstructing the original quantum state from the received coefficient can be described as follows:
Construct the set of all ideal second elements corresponding to the noiseless QFT columns. Here,
N
=
2
n
represents the dimension of the Hilbert space for an
n
-qubit register. As shown in Equation (
22
), the set of ideal second elements for the noiseless QFT columns is defined as
Q
.
Q
=
1
N
e
2
π
i
j
/
N
|
j
=
0
,
1
,
…
,
N
−
1
(22)
Each element of
Q
corresponds to the second component of the
j
-th column
q
j
of the QFT matrix
F
.
Identify the closest match
j
*
to the received noisy coefficient
q
˜
(
2
)
affected by channel noise
η
, as shown in Equation (
23
).
j
*
=
arg
min
j
q
˜
(
2
)
−
1
N
e
2
π
i
j
/
N
(23)
Here,
j
*
indicates the index of the ideal QFT column most closely corresponding to the received signal.
Retrieve the full QFT column vector corresponding to
j
*
, as shown in Equation (
24
).
q
^
=
q
j
*
(24)
The vector
q
^
represents the retrieved frequency-domain quantum state before applying the inverse transformation.
Apply the IQFT to reconstruct the time-domain quantum state (
x
^
), as shown in Equation (
25
).
x
^
=
F
N
†
q
^
(25)
where
F
N
†
is the Hermitian conjugate of the QFT matrix
F
.
After reconstructing the time-domain quantum state, measurements are performed on each qubit in the computational (Z) basis. This produces an
n
-bit classical string corresponding to the decoded symbol, which can then be used to reconstruct the original bitstream. This approach ensures that the simulation faithfully represents the theoretical model while preserving mathematical rigor throughout the entire encoding and decoding process.
It is important to note that Equation (
23
) represents a classical symbol detection step rather than a quantum state reconstruction process. Specifically, the equation implements a nearest-neighbor decision rule that estimates the transmitted symbol index by comparing the received noisy phase value with the set of valid constellation points defined by the QFT structure. This operation is analogous to maximum-likelihood detection commonly used in classical communication systems.
The quantum operations in the proposed framework occur during the state preparation and transformation stages. At the transmitter, the computational basis state corresponding to the classical symbol index is transformed using the QFT, producing a structured quantum superposition whose phase relationships encode the transmitted symbol. After symbol detection at the receiver, the corresponding quantum state is deterministically reconstructed based on the known QFT structure, and the IQFT is applied as a unitary quantum operation to recover the original computational basis state. Therefore, the proposed receiver follows a hybrid quantum–classical processing model in which classical decision logic is used for symbol detection, while the QFT and IQFT operations constitute the quantum transformations responsible for encoding and decoding the transmitted information.
In general, transmitting a single coefficient of a quantum state would not be sufficient to reconstruct an arbitrary
n
-qubit quantum state, since a general state is described by
2
n
complex amplitudes. However, the proposed framework operates under a constrained state preparation model in which the input states to the QFT are computational basis states generated deterministically from classical bitstreams. Each block of
n
classical bits is mapped to a basis state
|
j
⟩
, where
j
∈
{
0
,
1
,
…
,
2
n
−
1
}
. When the QFT is applied to a computational basis state, the resulting frequency-domain vector corresponds to a specific column of the QFT matrix with a deterministic phase structure. Since the encoder and decoder both know this finite set of possible QFT columns, a single transmitted coefficient can uniquely identify the corresponding column index
j
. Once this index is determined at the receiver, the complete frequency-domain representation can be regenerated deterministically, and the inverse QFT can be applied to recover the original computational basis state. Therefore, the proposed approach does not attempt to reconstruct an arbitrary quantum state from a single coefficient; instead, the transmitted coefficient acts as a symbol that identifies one element from a predefined set of structured QFT states.
3.3. Quantum Communication Channel
To simulate realistic quantum transmission conditions, the proposed system considers several standard quantum noise mechanisms [
50
]. Five primary types of quantum noise are included: bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping. Each noise channel is associated with a probability parameter that determines the likelihood of an error occurring during transmission. The effect of hardware imperfections is not included, as these abstract noise models are sufficient for analyzing early-stage quantum communication performance.
Composite Noise Model
The combined influence of all quantum noise sources is represented by a composite quantum channel, denoted as
C
(
ρ
)
, where
ρ
is the density matrix of the transmitted quantum state. This channel combines the effects of bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping errors, as defined in Equation (
26
). In practical quantum communication systems, noise rarely occurs as a single isolated process. Instead, multiple decoherence mechanisms typically act simultaneously due to environmental interactions, imperfect control operations, and physical device limitations. For example, superconducting qubits may experience energy relaxation and dephasing concurrently, while photonic communication channels may be affected by loss, depolarization, and phase fluctuations. Therefore, representing the channel as a probabilistic combination of several elementary noise processes provides a more realistic abstraction of practical quantum transmission conditions. The adopted composite noise model enables the proposed framework to capture the simultaneous influence of multiple error mechanisms and to evaluate system robustness under diverse channel conditions rather than assuming a single dominant noise source.
C
(
ρ
)
=
(
1
−
p
tot
)
ρ
+
p
B
B
(
ρ
)
+
p
P
P
(
ρ
)
+
p
D
D
(
ρ
)
+
p
A
A
(
ρ
)
+
p
Φ
F
(
ρ
)
(26)
where:
p
B
,
p
P
,
p
D
,
p
A
,
p
Φ
∈
[
0
,
1
]
are the probabilities of bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping errors, respectively.
B
(
·
)
,
P
(
·
)
,
D
(
·
)
,
A
(
·
)
,
F
(
·
)
denote the corresponding quantum channels implementing these error processes.
The total noise probability
p
tot
is dynamically determined as a function of the system SNR to reflect realistic channel conditions. It is modeled as in Equation (
27
).
p
tot
=
min
1
,
1
1
+
10
SNR
/
10
(27)
where SNR is expressed in dB. This ensures that the overall error probability decreases with increasing SNR. It should be noted that the SNR used in Equation (
27
) is not intended to represent a fundamental quantum-mechanical parameter. In physical quantum systems, noise processes are typically described using quantum channels characterized by Kraus operators or completely positive trace-preserving (CPTP) maps, and their parameters depend on specific physical mechanisms rather than directly on an SNR value. In the proposed framework, the SNR parameter is used as an abstract simulation-level control variable to regulate the overall severity of channel noise. The mapping defined in Equation (
27
) therefore determines the total noise probability
p
tot
, which is subsequently distributed among several standard quantum noise channels, including bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping processes. In this way, the SNR parameter provides a convenient mechanism for evaluating system performance across different channel conditions while the underlying noise evolution remains governed by physically consistent quantum channel models.
To allocate
p
tot
among the five individual noise channels, independent random weights
w
i
are drawn from a uniform distribution, as shown in Equation (
28
).
w
1
,
w
2
,
w
3
,
w
4
,
w
5
∼
U
(
0
,
1
)
(28)
These weights are then normalized to compute the individual channel probabilities, as in Equation (
29
).
[
p
B
,
p
P
,
p
D
,
p
A
,
p
Φ
]
=
p
tot
∑
i
=
1
5
w
i
·
[
w
1
,
w
2
,
w
3
,
w
4
,
w
5
]
(29)
This normalization guarantees that the sum of the individual channel probabilities equals the total noise probability, as expressed in Equation (
30
).
p
B
+
p
P
+
p
D
+
p
A
+
p
Φ
=
p
tot
(30)
Each individual noise channel represents a physically meaningful type of quantum error, as described in the following.
Bit-flip noise (
B
): Models the process where a qubit randomly flips from
|
0
⟩
to
|
1
⟩
or vice versa, similar to a classical binary error. Mathematically, this is represented in Equation (
31
).
B
(
ρ
)
=
(
1
−
p
B
)
ρ
+
p
B
X
ρ
X
†
(31)
Here,
X
is the Pauli-X operator (bit-flip gate).
Phase-flip noise (
P
): Represents a phase error that flips the relative sign of
|
1
⟩
with respect to
|
0
⟩
, leaving populations unchanged, as shown in Equation (
32
). This is crucial in coherent superposition states.
P
(
ρ
)
=
(
1
−
p
P
)
ρ
+
p
P
Z
ρ
Z
†
(32)
Z
is the Pauli-Z operator, which inverts the phase of
|
1
⟩
.
Depolarizing noise (
D
): Simulates isotropic errors that randomly apply any Pauli operator
X
,
Y
,
Z
with equal probability, driving the qubit toward a maximally mixed state, as shown in Equation (
33
).
D
(
ρ
)
=
(
1
−
p
D
)
ρ
+
p
D
3
(
X
ρ
X
†
+
Y
ρ
Y
†
+
Z
ρ
Z
†
)
(33)
This models decoherence that affects both bit and phase simultaneously.
Amplitude damping (
A
): Represents energy loss mechanisms, such as spontaneous emission in optical or superconducting qubits. It models the relaxation from
|
1
⟩
to
|
0
⟩
, as defined in Equation (
34
).
A
(
ρ
)
=
E
0
ρ
E
0
†
+
E
1
ρ
E
1
†
,
E
0
=
1
0
0
1
−
p
A
,
E
1
=
0
p
A
0
0
(34)
Phase damping (
F
): Models pure dephasing without energy loss, which reduces off-diagonal elements of the density matrix in the computational basis, as defined in Equation (
35
).
F
(
ρ
)
=
F
0
ρ
F
0
†
+
F
1
ρ
F
1
†
,
F
0
=
1
0
0
1
−
p
Φ
,
F
1
=
0
0
0
p
Φ
(35)
By combining these channels probabilistically, the framework provides a flexible and physically meaningful model for testing quantum communication systems under realistic noisy conditions. SNR is mapped to the total noise probability
p
tot
to adjust error severity according to channel quality [
15
,
19
].
3.4. Physical Realization of QFT Using Quantum Gates
In real quantum circuit implementations, the QFT is constructed using a sequence of single-qubit Hadamard operations together with controlled phase rotation gates. To illustrate this structure, a QFT circuit consisting of three qubits is presented in
Figure 4
.
Figure 4.
Quantum circuit for a three-qubit QFT.
The Hadamard gate [
51
], represented by
H
, is a key quantum logic gate that converts computational basis states into superposition states. The matrix form of this operation is provided in Equation (
36
).
H
=
1
2
1
1
1
−
1
(36)
When applied to a single qubit, it performs the transformations given in Equations (
37
) and (
38
).
H
|
0
⟩
=
1
2
(
|
0
⟩
+
|
1
⟩
)
=
|
+
⟩
(37)
H
|
1
⟩
=
1
2
(
|
0
⟩
−
|
1
⟩
)
=
|
−
⟩
(38)
Here,
|
+
⟩
and
|
−
⟩
represent the superposition states along the
X
-axis of the Bloch sphere.
Phase gates introduce controlled rotations, which are crucial for encoding the relative phases between qubits in QFT. The general single-qubit phase gate is written as in Equation (
39
).
P
(
ϕ
)
=
1
0
0
e
i
ϕ
(39)
For QFT, the rotation angle
ϕ
is determined by the qubit positions:
ϕ
=
2
π
/
2
k
, where
k
is the distance between the control and target qubits in a controlled-phase operation.
Three-Qubit QFT Procedure
Let the input state be
|
x
1
x
2
x
3
⟩
, where
x
1
is the most significant qubit. The QFT is performed as follows:
Apply a Hadamard gate to the first qubit
x
1
.
Apply a controlled-
R
2
gate between
x
1
and
x
2
, where the
R
2
phase gate is defined in Equation (
40
).
R
2
=
1
0
0
e
i
π
/
2
(40)
Apply a controlled-
R
3
gate between
x
1
and
x
3
, where the
R
3
phase gate is defined in Equation (
41
).
R
3
=
1
0
0
e
i
π
/
4
(41)
Perform a Hadamard gate on
x
2
, followed by a controlled-
R
2
gate between
x
2
and
x
3
.
Apply a Hadamard gate to
x
3
.
Swap qubits
x
1
↔
x
3
to correct the qubit order in the computational basis.
The resulting three-qubit QFT state can be expressed as in Equation (
42
).
QFT
(
|
x
1
x
2
x
3
⟩
)
=
1
8
|
0
⟩
+
e
2
π
i
0
.
x
3
|
1
⟩
⊗
|
0
⟩
+
e
2
π
i
0
.
x
2
x
3
|
1
⟩
⊗
|
0
⟩
+
e
2
π
i
0
.
x
1
x
2
x
3
|
1
⟩
(42)
This formulation demonstrates that each qubit sequentially accumulates phase contributions from less significant qubits, effectively encoding the input into the frequency domain. The output of this circuit matches the theoretical QFT computations for all basis states, confirming the correctness of the implementation [
52
]. Similarly, this physical implementation can be generalized to accommodate arbitrary qubit sizes.
3.5. Quantum and Classical Components of the Framework
The proposed system follows a hybrid quantum–classical architecture. The genuinely quantum operations in the framework include: (i) preparation of the
n
-qubit computational basis state corresponding to each data segment, (ii) application of the QFT using quantum gate circuits composed of Hadamard and controlled phase-rotation gates, (iii) propagation of the quantum state through the quantum communication channel modeled by quantum noise processes such as bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping channels, (iv) application of the IQFT at the receiver, and (v) measurement of the qubits in the computational basis to obtain classical outcomes.
The classical components of the framework include preprocessing and post-processing stages. Specifically, the video data is first compressed using the classical VVC encoder, and the resulting bitstream is segmented into
n
-bit blocks before quantum encoding. In scenarios where uncompressed video is used, the raw pixel data is directly converted into a binary bitstream and segmented into
n
-bit blocks without applying source compression. After quantum measurement at the receiver, the recovered binary data is concatenated and decoded using the classical VVC decoder to reconstruct the video frames when compressed inputs are used. For uncompressed inputs, the recovered bitstream is directly mapped back to the corresponding pixel representation. In addition, classical detection procedures such as nearest-neighbor symbol estimation are used to determine the most likely transmitted index from the received noisy observations. Although the system is evaluated through numerical simulation, the evolution of the quantum states is modeled according to the standard formalism of quantum mechanics, including unitary QFT/IQFT operations and quantum channel noise represented by completely positive trace-preserving maps
3.6. Performance Evaluation Methodology
The performance of the proposed system is evaluated across multiple scenarios to assess compression efficiency, error resilience, and robustness under different channel conditions. The evaluation framework is described as follows:
Uncompressed video inputs: Multi-qubit configurations with
n
=
1
to 8 are tested using raw video sequences. These experiments examine how the qubit dimension affects the compression ratio and robustness to quantum channel noise. Results are reported for both cases without and with classical channel coding.
Compressed video inputs: To study transmission efficiency improvements, multi-qubit systems (
n
=
1
to 8) are applied to VVC-compressed video sequences. Two GOP structures, 8 and 32, are considered. Performance is evaluated both with and without classical channel coding.
Performance of QFT encoding without compression: The QFT-based full-vector transmission, which sends the complete quantum state without compression [
19
], is compared against alternative methods under the same bandwidth constraints. These alternatives include a time-domain Hadamard-based multi-qubit system with qubit sizes
n
=
1
to 8, and a classical system using binary phase-shift keying (BPSK). In the Hadamard-based approach, the multi-qubit encoding matrix is generated via the tensor product of the Hadamard matrix defined in Equation (
36
), and decoding is performed using the inverse Hadamard transform to recover the transmitted quantum state [
32
].
In the proposed framework, the single-qubit configuration is considered the reference transmission system. A single qubit can represent two computational basis states,
|
0
⟩
and
|
1
⟩
, which correspond to a two-point constellation. This is directly comparable to the two-symbol constellation used in a classical BPSK modulation scheme. Therefore, BPSK provides a natural classical counterpart for evaluating the transmission behavior of the QFT-based frequency-domain representation under equivalent binary symbol conditions.
To maintain equivalent bandwidth usage across all qubit configurations, the input bitstream is compressed using different quantization parameter (QP) settings in the VVC encoder. This produces progressively shorter bitstreams as the qubit grouping size increases. As a result, the total amount of transmitted information remains comparable across different configurations, allowing the impact of the QFT-based transmission process to be evaluated without introducing bandwidth bias.
3.7. Simulation Configuration
The proposed system is evaluated using a numerical simulation framework that implements the complete transmission and reconstruction pipeline described in the previous subsections. The results are obtained using a Monte Carlo simulation framework to evaluate system performance under stochastic noise conditions. For each video sequence and each SNR value, the complete transmission and reconstruction process is repeated over 1000 independent trials. In each trial, a new realization of the composite quantum noise channel is generated by drawing random weights for the individual noise processes from a uniform distribution and normalizing them to satisfy the total noise probability constraint. The transmitted states are propagated through the resulting channel, and the reconstructed video quality is evaluated using bit error rate (BER), peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM) [
53
], and video multi-method assessment fusion (VMAF) [
54
] metrics. The final curves shown in the figures correspond to the average performance obtained across these 1000 trials, which explains the smooth appearance of the plotted results despite the underlying stochastic noise model. The key parameters and implementation settings used in the experiments are summarized in
Table 1
.
Table 1.
Experimental Setup Parameters.
4. Results and Discussion
In this section, we present and analyze the performance of the proposed QFT-based video compression and transmission system under varying quantum channel conditions. The evaluation examines how the system handles videos with different spatial and temporal complexities, providing insight into its robustness and effectiveness. Quantitative results are reported using BER, PSNR, SSIM, and VMAF. Together, these metrics evaluate reconstruction fidelity, perceptual quality, and temporal consistency, offering a comprehensive view of system performance. The discussion highlights trends observed in average results across multiple test videos, emphasizing the effects of motion complexity, channel noise, and compression on video quality, while summarizing overall system behavior through aggregated performance metrics.
Each comparison scenario introduced in
Section 3.6
is explained in detail in the following subsections.
4.1. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for Uncompressed Video Inputs Using Multi-Qubit Configurations Without Channel Coding
The results of the proposed QFT-based video compression and transmission system, evaluated without channel coding, is summarized in
Figure 5
. The figure illustrates the system’s behavior across four key metrics, BER, PSNR, SSIM, and VMAF, as a function of the channel SNR for different qubit encoding sizes (F1 to F8, corresponding to n = 1 to n = 8 qubits). The BER results in
Figure 5
a reveal a critical trade-off governed by the qubit encoding size,
n
. In this system, a block of video data is mapped via the QFT into a
2
n
-dimensional frequency vector. The unitary nature of the QFT enables powerful, exponential compression; an
n
-qubit encoding achieves a theoretical compression ratio of
2
n
:
1
, as only the dominant frequency coefficient needs to be transmitted to reconstruct the original block. However, this compression gain comes at the cost of increased susceptibility to noise. The angular separation between adjacent frequency components decreases exponentially with
n
, as given by Equation (
43
). A smaller
Δ
θ
causes the constellation of frequency states to become denser, making it more difficult for the receiver to discriminate between them in the presence of phase noise induced by the channel.
Δ
θ
=
2
π
2
n
(43)
Figure 5.
Performance proposed QFT-based video compression and transmission system for the uncompressed video inputs without channel coding (
a
) bit error rate (BER), (
b
) peak signal-to-noise ratio (PSNR), (
c
) structural similarity index measure (SSIM), and (
d
) video multi-method assessment fusion (VMAF).
Consequently, while the F8 configuration (
n
=
8
) achieves a very high 256:1 compression ratio, it exhibits the highest BER across all SNR levels, as shown in
Figure 5
a. In contrast, the F2 configuration (
n
=
2
) benefits from a more robust angular separation of
π
/
2
radians and a moderate 4:1 compression ratio, resulting in superior error resilience. The F1 configuration, although offering the highest error tolerance, provides only minimal compression (2:1). Overall, these results indicate that intermediate encoding sizes (e.g., F2–F4) offer the most practical operating point, achieving a favorable trade-off between substantial compression and acceptable error rates under typical noisy channel conditions.
The video reconstruction quality metrics, PSNR, SSIM, and VMAF, shown in
Figure 5
b,
Figure 5
c, and
Figure 5
d, respectively, follow consistent trends. As expected, all metrics generally improve with increasing SNR, as a cleaner channel allows for more accurate reconstruction of the transmitted video data. The performance hierarchy across different encoding sizes mirrors the findings from the BER analysis. Systems with lower qubit counts (F1, F2) consistently achieve higher PSNR, SSIM, and VMAF scores at a given SNR due to their inherent noise resilience. The superior performance of F2, in particular, confirms its optimal balance between compression efficiency and reconstruction quality.
An interesting observation is the behavior of the F8 configuration. While its BER is significantly higher, its PSNR does not degrade as catastrophically as one might expect. This can be attributed to the nature of the QFT/IQFT process and the structure of video data. Channel noise primarily perturbs the finer, less significant details in the frequency domain. During the inverse QFT, these errors are distributed across the pixel block. Furthermore, natural video has significant spatial and temporal correlation, allowing adjacent pixels to mask these distributed errors. As a result, the overall structural integrity and perceptual quality, as captured by PSNR, SSIM and VMAF, are preserved to a greater degree than the raw BER might suggest. However, the F8 system requires a very high SNRs to achieve the highest quality levels that lower-qubit systems can achieve at moderate SNRs.
Therefore, the results in
Figure 5
highlight a fundamental trade-off in the proposed system: higher qubit encoding sizes achieve greater compression but are more susceptible to channel noise. Under realistic noisy channel conditions, smaller or intermediate qubit configurations, such as F2 to F4, provide a more favorable balance between compression and error resilience, ensuring reliable video reconstruction.
4.2. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for Uncompressed Video Inputs Using Multi-Qubit Configurations with Channel Coding
The introduction of channel coding dramatically alters the performance landscape of the proposed system, as illustrated in
Figure 6
. When compared to the uncoded results in
Figure 5
, the application of forward error correction provides a significant coding gain, effectively shifting all performance curves to the left and enabling reliable operation at much lower SNR levels. The BER characteristics, depicted in
Figure 6
a, demonstrate the substantial channel coding gain achieved through forward error correction. Although the trade-off between qubit encoding size and error susceptibility persists, the inclusion of channel coding substantially reduces the absolute BER across all SNR levels relative to the uncoded scenario. The channel coding effectively compensates for the inherent vulnerability of larger encoding sizes, enabling their practical deployment in noisy environments where they would otherwise be unusable. The error correction manifests most prominently in the intermediate SNR regime (10–25 dB), where the coding gain provides the greatest marginal benefit. In this region, the BER curves exhibit the characteristic steep descent associated with effective coding schemes, transitioning rapidly from high to low error probability as SNR increases.
Figure 6.
Performance proposed QFT video compression and transmission system for the uncompressed video inputs with channel coding (
a
) bit error rate (BER), (
b
) peak signal-to-noise ratio (PSNR), (
c
) structural similarity index measure (SSIM), and (
d
) video multi-method assessment fusion (VMAF).
The PSNR results in
Figure 6
b reveal a significant qualitative shift in system behavior compared to the uncoded scenario. Channel coding eliminates the anomalous performance previously observed with F8, establishing a consistent monotonic relationship between encoding dimension and reconstruction fidelity. The F8, which demonstrated unexpected resilience in the uncoded system due to its direct pixel mapping characteristics, now exhibits the most pronounced sensitivity to channel impairments. This normalization of behavior arises because the error correction process fundamentally alters how quantization errors propagate through the system. The coding redundancy, while essential for error protection, disrupts the spatial correlation properties that previously provided natural error masking for certain encoding configurations. Consequently, the theoretical vulnerability of high-dimensional encodings to angular separation constraints becomes fully manifested in the channel coded system.
The SSIM metrics, presented in
Figure 6
c, corroborate the PSNR findings while providing additional insights into structural preservation. Channel coding maintains superior structural similarity across all encoding sizes, particularly in the critical mid-range SNR conditions where perceptual quality is most vulnerable. However, the progressive degradation with increasing encoding size remains evident, confirming that while channel coding improves absolute performance, it does not alter the fundamental compression-robustness trade-off intrinsic to the QFT encoding approach.
The VMAF results in
Figure 6
d provide the most comprehensive assessment of perceptual video quality. This metric, which incorporates characteristics of the human visual system and accounts for temporal relationships across consecutive frames, reveals that channel coding substantially improves the viewing experience across all encoding sizes. Nonetheless, the hierarchical pattern remains: smaller encoding sizes (F1–F3) maintain excellent perceptual quality (VMAF
>
80
) at moderate SNR levels, while larger encodings require progressively higher SNR to achieve comparable viewing quality. Notably, VMAF highlights the superiority of the two-qubit encoding (F2), which achieves an optimal balance between compression efficiency and perceptual quality preservation.
The comparative analysis across all four metrics provides crucial insights for system optimization. The single-qubit configuration demonstrates the highest robustness, maintaining reliable performance under noisy conditions while achieving a 2:1 compression ratio. However, the two-qubit system emerges as the optimal compromise, delivering a favorable 4:1 compression ratio while requiring only modest SNR to maintain acceptable quality across diverse channel conditions, as consistently reflected in its superior BER, PSNR, SSIM, and VMAF performance. In contrast, higher-qubit configurations (e.g., F4–F8) offer greater compression but demand significantly higher SNR to achieve comparable quality, highlighting the trade-off between compression efficiency and error resilience.
4.3. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for VVC Encoded Video Inputs Using Multi-Qubit Configurations Without Channel Coding
The performance of the system with pre-compressed VVC input using GOP 8 exhibits a notably different behavior compared to uncompressed video, as illustrated in
Figure 7
. All three quality metrics, PSNR, SSIM, and VMAF, follow a consistent pattern across different qubit encoding sizes. The smaller qubit encodings (F1–F3) maintain superior performance throughout the SNR range, while larger encodings (F6–F8) show significant degradation, particularly at low to moderate SNR levels. The PSNR results in
Figure 7
a indicate that VVC-compressed inputs are more susceptible to quality degradation under channel noise than uncompressed content. The SSIM and VMAF metrics, shown in
Figure 7
b,c, reveal similar overall trends, albeit with different sensitivity characteristics. SSIM values for smaller encodings remain above 0.8 for SNR > 10 dB, indicating good structural preservation. However, larger encodings struggle to maintain acceptable structural similarity, with the F8 configuration performing the worst under channel noise. VMAF scores show the most pronounced separation between encoding sizes. The F1 configuration maintains excellent perceptual quality (VMAF > 80) across most of the SNR range, whereas F8 only exceeds a VMAF of 40 at around 40 dB channel SNR, indicating a poor viewing experience across nearly all channel conditions.
Figure 7.
Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP8) (
a
) peak Signal-to-Noise Ratio (PSNR), (
b
) structural similarity index measure (SSIM), and (
c
) video multi-method assessment Fusion (VMAF).
The results clearly indicate that VVC-encoded content is less tolerant to the additional compression introduced by larger qubit encodings compared to uncompressed inputs. Moreover, the combination of VVC compression artifacts and the noise sensitivity of high-dimensional QFT encodings produces a compounding effect that significantly degrades reconstruction quality. For practical systems using pre-compressed video content, smaller qubit encodings (F1–F2) are essential to maintain acceptable quality. The theoretical compression benefits of larger encodings are negated by the severe quality degradation.
Based on the results shown in
Figure 8
, which depicts the system performance with a GOP size of 32 and without channel coding, the effect of larger GOP sizes on video quality is clearly observed. The results for PSNR, SSIM, and VMAF consistently show that all configurations (F1 to F8) exhibit reduced quality compared to scenarios with GOP size 8. As the GOP size increases to 32, the number of inter-coded frames rises, leading to greater reliance on predictive coding. Errors introduced in early frames within the GOP propagate through subsequent inter-frames, amplifying quality loss. This propagation effect is evident in
Figure 8
, where even robust encodings such as F1 and F2 exhibit lower PSNR, SSIM, and VMAF scores compared to the corresponding results for GOP size 8. To mitigate this, adaptive error correction strategies tailored to the GOP structure, such as strengthening protection for key frames or using error-resilient encoding techniques, could be employed to reduce error propagation and enhance overall performance.
Figure 8.
Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP32) (
a
) peak Signal-to-Noise Ratio (PSNR), (
b
) structural similarity index measure (SSIM), and (
c
) video multi-method assessment Fusion (VMAF).
4.4. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for VVC Encoded Video Inputs Using Multi-Qubit Configurations with Channel Coding
The results for VVC-encoded video (GOP size 8, shown here as an example) with channel coding, presented in
Figure 9
, demonstrate a significant improvement compared to uncoded VVC transmission. Channel coding effectively mitigates the error propagation issues inherent in predictive coding with GOP size 8. All three quality metrics, PSNR, SSIM, and VMAF, show substantial enhancement compared to the uncoded case. The system now maintains acceptable quality levels even at moderate SNR conditions, where previously severe degradation occurred. The hierarchical relationship between different qubit encodings persists, but with improved absolute performance. Smaller encodings (F1–F3) achieve excellent reconstruction quality, with F1 and F2 maintaining PSNR above 25 dB, SSIM above 0.8, and VMAF above 80 across most of the SNR range. Even larger encodings (F4–F6) now provide usable quality at sufficient SNR levels, though F7 and F8 still show limitations due to their sensitivity to phase noise. The combination of VVC encoding, QFT-based compression, and channel coding represents a viable operational point for practical systems. The two-qubit encoding (F2) emerges as particularly advantageous, offering a favorable 4:1 compression ratio while maintaining robust performance across all quality metrics. This configuration balances compression efficiency with error resilience, making it suitable for bandwidth-constrained applications requiring reliable video transmission.
Figure 9.
Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs with channel coding (GOP8) (
a
) peak Signal-to-Noise Ratio (PSNR), (
b
) structural similarity index measure (SSIM), and (
c
) video multi-method assessment Fusion (VMAF).
4.5. Performance of Frequency-Domain Encoding Without Compression for VVC-Encoded Inputs with Channel Coding
Based on the results shown in
Figure 10
for VVC-encoded videos, a clear performance hierarchy emerges between the quantum-inspired encoding systems operating without additional compression and the classical baseline approach. Both the QFT-based frequency-domain system (F1–F8) and the time-domain Hadamard system (T1–T8) consistently outperform the bandwidth equivalent classical system (C) across all quality metrics, PSNR, SSIM, and VMAF. Notably, unlike the scenarios involving QFT-based compression where larger qubit sizes degrade performance, in this case increasing the qubit encoding size (
n
) actually improves reconstruction quality for both quantum-inspired systems. This improvement occurs because, in this scenario, no QFT-based compression is applied and the full quantum state vector is transmitted. As a result, the video does not contain the artifacts and inter-frame dependencies that typically amplify transmission errors in compressed content. As a result, the superior noise resilience and representational capacity of higher-dimensional quantum encodings become evident. Larger qubit configurations are able to capture and represent more complex quantum states, allowing them to encode greater amounts of information per transmitted symbol. This leads to improved reconstruction fidelity and robustness against channel noise, which is why larger-qubit configurations such as F7/F8 and T7/T8 consistently outperform smaller qubit systems.
Figure 10.
Performance of frequency-domain encoding without compression for VVC-encoded inputs with channel coding (GOP8) (
a
) peak Signal-to-Noise Ratio (PSNR), (
b
) structural similarity index measure (SSIM), and (
c
) video multi-method assessment Fusion (VMAF).
In particular, the QFT-based encoding system benefits from high-dimensional frequency-domain representations in Hilbert space, utilizing both amplitude and phase encoding to capture intricate correlations within the video data. In contrast, the Hadamard-based encoding operates in the time domain, relying on an orthogonal basis structure with amplitude encoding alone. The dual encoding capability of the QFT system enables it to more accurately preserve the full quantum state and resist errors, resulting in superior performance compared to Hadamard-based systems, especially in high-dimensional qubit configurations. Consequently, the combination of larger qubit sizes and QFT-based frequency-domain encoding provides the highest reconstruction quality, demonstrating the fundamental advantages of quantum-inspired encoding for high-fidelity transmission of uncompressed video. This encoding scheme differs from the proposed method in that it does not achieve compression; instead, it transmits the full quantum state vector without reducing the data volume.
4.6. Performance Assessment of the Proposed QFT-Based Compression and Transmission System Across Different Resolutions and Qubit Configurations
As shown in
Table 2
, the channel SNR gains compared to the eight-qubit encoding under the channel-coded system remain consistent across all tested video resolutions and frame rates. This indicates that the resolution and frame rate of the video do not significantly impact the performance of the QFT-based compression and transmission system. Instead, the maximum SNR gains are primarily determined by the qubit encoding size, demonstrating that the system’s error resilience and compression efficiency depend on the quantum encoding configuration rather than the specific characteristics of the video content.
Table 2.
Maximum SNR Gains (in dB) Achieved by the QFT-Based Compression and Transmission System for Each Qubit Configuration Compared to the Eight-Qubit System Across Different Video Resolutions and Frame Rates.
An illustrative example of reconstructed video frames using the proposed QFT-based system under varying channel conditions is shown in
Figure 11
. The frames span SNR levels from 38 dB down to
−
10
dB, with each subfigure labeled alphabetically for easy reference. The system employs eight-qubit encoding (F8) with a 256:1 compression ratio applied to uncompressed video inputs, demonstrating the high compression efficiency of the proposed QFT-based video compression and transmission framework. At higher SNR levels, the reconstructed frames closely preserve the original visual details, indicating effective retention of critical image information. As the SNR decreases, visual degradations such as blurring and minor artifacts become more apparent, reflecting the impact of channel noise on the transmitted quantum-encoded data. Despite these distortions, the system demonstrates graceful degradation, highlighting the robustness of the QFT-based video compression and transmission system with eight-qubit encoding against channel impairments. This illustrative example complements quantitative evaluations using PSNR, SSIM, and VMAF, providing a clear visual demonstration of the proposed method’s performance across a wide range of channel conditions.
Figure 11.
Reconstructed video frames at SNR levels from 38 dB to
−
10
dB for the QFT-based video compression and transmission system with eight-qubit encoding (F8) and 256:1 compression of uncompressed video inputs.
4.7. Performance Evaluation Compared to Classical Communication Systems
To contextualize the performance of the proposed framework in a video transmission setting, we compare it with representative classical baselines, as illustrated in
Figure 12
for the two-qubit quantum system. QPSK serves as a modulation-level reference with comparable bandwidth, while both HEVC and VVC represent widely used state-of-the-art classical video compression standards. All systems are evaluated under equivalent bitrate constraints to ensure fair bandwidth utilization.
Figure 12.
Comparison of the proposed system with classical communication systems: (
a
) Peak Signal-to-Noise Ratio (PSNR), (
b
) Structural Similarity Index Measure (SSIM).
The QFT-based system enables frequency-domain video transmission by encoding frame-level information into quantum states, allowing efficient representation of spatial content. Even without channel coding, the proposed approach achieves performance comparable to uncompressed QPSK transmission while benefiting from intrinsic compression (4:1) due to the quantum state representation. When polar coding with a rate of 1/4 is applied, the QFT-based system significantly improves robustness against channel noise, achieving substantial PSNR gains (up to 10 dB) over uncompressed QPSK transmission, as shown in
Figure 12
a, under the same bitrate.
Unlike classical video compression methods such as HEVC and VVC, which rely on predictive coding and inter-frame dependencies, the proposed QFT-based framework demonstrates a more gradual degradation in the presence of channel noise. This behavior is evident in the PSNR trends in
Figure 12
a, where the QFT-coded system maintains high reconstruction quality at lower SNR values compared to both uncoded QPSK and classical codec-based transmission schemes. Similarly, the SSIM results in
Figure 12
b confirm that the proposed method preserves structural information more effectively under channel impairments.
While HEVC and VVC provide strong rate–distortion performance under ideal channel conditions, they rely on inherently lossy compression mechanisms and predictive coding structures that make them highly sensitive to transmission errors. Even small bitstream corruptions can propagate across multiple frames due to inter-frame dependencies, leading to abrupt visual degradation. In contrast, the proposed QFT-based framework operates on independently encoded frequency-domain representations, which reduces error propagation and enables more stable reconstruction quality in noisy channels. Furthermore, since the QFT and its inverse are unitary transformations, the proposed system can theoretically achieve lossless reconstruction under ideal channel conditions.
Although classical codecs such as HEVC and VVC benefit from mature implementations and highly optimized compression efficiency in error-free environments, the proposed quantum-inspired framework offers a promising alternative for video transmission in bandwidth-limited and error-prone channels, providing improved robustness and graceful degradation characteristics.
4.8. Advantages and Practical Implications of the Proposed QFT-Based Symbol-Level Compression and Transmission Framework
It is important to emphasize that the proposed method introduces quantum domain symbol-level compression. In this framework, compression occurs during quantum encoding by selectively transmitting a reduced set of QFT-domain coefficients. This process decreases the transmission payload but does not reduce the file size of the original video for storage. Unlike classical codecs, which replace the stored representation with a compressed version, the proposed QFT-based approach requires access to the full quantum state vector for decoding. As such, the method provides compression strictly within the communication pipeline, not for persistent storage.
The mathematical foundation of the QFT ensures theoretically perfect reconstruction (lossless) under ideal noise-free conditions. Since the QFT and its inverse (IQFT) are unitary operations, a computational basis state can be perfectly recovered when no channel noise, decoherence, or measurement errors are present. Therefore, the lossless property represents the theoretical upper bound of the system rather than a guarantee of lossless performance in practical quantum communication environments. Under practical noisy channel conditions, however, the system exhibits graceful degradation: increasing the qubit dimension reduces the angular separation between encoded frequency components, making higher-dimensional states more susceptible to noise. Importantly, this controlled degradation profile offers system designers well-defined trade-off parameters between compression efficiency and noise resilience, providing a level of tunability not typically available in conventional communication systems.
Based on the comprehensive evaluation of the proposed QFT-based compression and transmission system, several key advantages emerge that highlight its potential for practical video transmission applications. The system exhibits strong adaptability, allowing the optimal qubit encoding size to be selected according to specific operational requirements and channel conditions. Smaller encoding configurations (
n
=
1
–2 qubits) excel in low-SNR scenarios, providing robust and reliable performance that is well suited for mission-critical applications where transmission reliability outweighs compression efficiency. Medium-sized encodings (
n
=
3
–5 qubits) offer an effective balance between compression gains and robustness, making them appropriate for typical multimedia transmission environments. Larger encoding sizes (
n
=
6
–8 qubits) achieve substantial compression ratios and are most beneficial in high-SNR channels or in scenarios where bandwidth conservation is a primary objective.
4.9. Channel Use, Resource Cost, and Noise Sensitivity
In the proposed framework, the encoded video bitstream is segmented into blocks containing
n
classical bits, as described in the encoding stage. Each block is mapped to the computational basis of an
n
-qubit register and processed using the QFT to obtain a frequency-domain representation of the encoded data.
A single representative coefficient from the QFT representation is selected and encoded into a transmitted quantum state. Therefore, a channel use in the proposed system is defined as the transmission of one encoded quantum state corresponding to a QFT-transformed data block. Under this definition, each segmented block requires one channel use. Increasing the qubit grouping size
n
increases the number of classical bits represented by each block and therefore reduces the number of channel uses required to transmit the entire bitstream.
From an information-theoretic perspective, the channel capacity can be related to the classical Shannon capacity formula, as shown in Equation (
44
).
C
raw
=
B
log
2
(
1
+
SNR
)
(44)
In Equation (
44
),
C
raw
denotes the raw channel capacity in bits per second,
B
represents the channel bandwidth in Hertz, and SNR is the signal-to-noise ratio.
Because the proposed QFT-based representation transmits only one coefficient out of the
2
n
possible frequency components produced by the QFT, the effective transmitted information rate can be expressed as in Equation (
45
).
C
eff
(
n
)
=
C
raw
2
n
(45)
In Equation (
45
),
C
eff
(
n
)
represents the effective channel capacity associated with the transmitted coefficient when an
n
-qubit encoding is used, and
2
n
denotes the number of spectral components generated by the QFT.
The transmitted resource cost can therefore be interpreted as the number of quantum states required to represent the encoded bitstream. If
L
denotes the total number of bits in the compressed bitstream, the number of transmitted blocks is given as in Equation (
46
).
N
b
=
L
n
(46)
In Equation (
46
),
N
b
represents the number of encoded blocks generated from the bitstream and therefore corresponds directly to the total number of channel uses required for transmission.
To evaluate the robustness of the proposed framework under different channel conditions, the transmission process is analyzed across multiple SNR levels. Channel disturbances are modeled as noise perturbations applied to the transmitted quantum state before the IQFT is performed at the receiver. By analyzing system performance across different SNR values, the sensitivity of the proposed framework to channel noise can be systematically characterized.
4.10. Computational Complexity and System Scalability
The computational complexity of the proposed QFT-based compression system is one of its key advantages. For an
n
-qubit encoding, the total gate count scales as
O
(
n
2
)
, primarily due to the use of standard Hadamard gates and controlled-phase rotations, which are commonly supported in most quantum hardware. Circuit depth, which determines the temporal length of the computation, can be reduced to
O
(
n
)
when gates acting on independent qubits are executed in parallel; however, hardware connectivity constraints, such as linear nearest-neighbor layouts, introduce additional SWAP operations that can increase depth toward
O
(
n
2
)
. The use of widely available Hadamard and phase gates, along with approximate QFT methods that truncate small-angle rotations, ensures that both gate count and depth remain manageable while maintaining high fidelity.
Moreover, the system is scalable to higher qubit counts depending on the application requirements. Increasing the qubit dimension allows for higher compression ratios, as more information can be encoded per quantum state. However, larger qubit sizes can reduce error resilience, since each qubit becomes more susceptible to noise. Therefore, qubit count selection involves a trade-off between compression efficiency and robustness, allowing the system to be tailored to specific application scenarios.
4.11. Hardware Feasibility and Resource Requirements
The proposed compression and transmission framework is compatible with practical quantum communication architectures because it relies on standard quantum operations that are widely studied and experimentally realizable. The encoding stage requires only computational basis state preparation, which can be implemented by initializing qubits in the ground state
|
0
⟩
and applying Pauli-
X
gates when the corresponding classical bit equals one. The transformation stage employs the QFT, which can be implemented using Hadamard gates and controlled phase-rotation gates with polynomial circuit complexity. For an
n
-qubit QFT circuit, the number of Hadamard gates is
H
=
n
, while the number of controlled phase gates is
n
(
n
−
1
)
2
, resulting in an approximate total gate count of
G
(
n
)
=
n
+
n
(
n
−
1
)
2
. After transmission through the quantum channel, the receiver applies the IQFT, followed by computational-basis measurement to recover the classical bitstream.
From a hardware perspective, the proposed system requires relatively small quantum registers and a moderate number of gate operations. In the experiments presented in this work, the encoding size ranges from
n
=
1
to
n
=
8
qubits, which is well within the capabilities of current noisy intermediate-scale quantum (NISQ) devices and experimental quantum communication platforms. The required gate operations scale quadratically with the number of qubits due to the structure of the QFT circuit, while the readout stage requires only standard computational-basis measurements.
Table 3
summarizes the approximate hardware resources required for different qubit encoding sizes considered in this study.
Table 3.
Estimated QFT Hardware Resources for Different Qubit Sizes.
4.12. Simulation Methodology and Assumptions
The proposed quantum-inspired compression framework is evaluated through classical simulations on conventional computing platforms, specifically an Intel Core i5-1345U processor (Intel Corporation, Santa Clara, CA, USA) with 16 GB of RAM. Classical simulation is widely used in quantum communication research to analyze algorithmic behavior and system-level performance prior to implementation on physical quantum hardware. Current quantum processors remain limited in terms of qubit count, coherence time, and circuit depth, which makes large-scale multimedia transmission experiments impractical. Therefore, numerical simulation provides a controlled and reproducible environment for modeling quantum state preparation, QFT/IQFT transformations, and channel noise processes according to the standard formalism of quantum mechanics.
These simulations allow systematic evaluation of the proposed framework under different channel conditions and encoding configurations while avoiding hardware-specific constraints that may obscure algorithmic performance. The experimental evaluation considers video sequences with different motion characteristics (low, medium, and high-motion content) across multiple resolutions and frame rates, using both uncompressed and VVC-encoded inputs. The results should therefore be interpreted as a proof-of-concept validation of the proposed QFT-based video transmission framework, demonstrating its theoretical feasibility and system-level behavior rather than claiming immediate practical deployment on existing quantum hardware.
By abstracting some hardware-related limitations, this approach focuses on evaluating the conceptual feasibility of the proposed quantum communication framework before practical deployment. The simulation results provide theoretical support for the system design and help assess its potential effectiveness. As quantum technologies continue to advance, the findings from these studies can support future prototype development, experimental validation, and hardware-level testing, facilitating the gradual transition from simulation-based analysis to real-world implementation.
5. Conclusions
This study presents and evaluates a novel QFT-based framework for efficient video compression and transmission over noisy communication channels. Through comprehensive analysis across different encoding parameters and channel conditions, several key insights emerge regarding the trade-offs inherent in quantum-inspired video transmission systems. The results indicate that the qubit encoding size plays a crucial role in balancing compression efficiency and robustness to channel noise. Larger qubit encodings enable significantly higher compression ratios, reaching up to 256:1, by representing more classical information within a single quantum state; however, these higher-dimensional encodings become increasingly sensitive to channel impairments due to the reduced angular separation between frequency-domain components. Conversely, smaller qubit group sizes provide stronger resilience to noise at the expense of lower compression efficiency. Under ideal noise-free channel conditions, the proposed framework achieves theoretically lossless reconstruction for all qubit sizes, since the QFT and its inverse are unitary operations that preserve the encoded information. In practical noisy environments, the optimal encoding size depends on the intended application—larger qubit configurations are advantageous when maximizing compression efficiency is the primary objective, whereas smaller encodings are preferable when transmission reliability is critical. Among the evaluated configurations, the two-qubit system provides a particularly effective compromise between compression and robustness, achieving a moderate compression ratio of 4:1 while maintaining stable performance across diverse channel conditions. These findings highlight the potential of QFT-based encoding as a flexible and robust approach for future multimedia transmission systems operating in bandwidth-constrained and error-prone environments.
Looking ahead, this work lays the foundation for several important research directions. Future studies should address practical challenges such as hardware imperfections, computational efficiency, and the development of adaptive qubit allocation strategies based on channel conditions or bandwidth constraints. Designing specialized error correction schemes tailored to quantum frequency-domain representations offers a promising approach to enhancing system reliability while maintaining high compression ratios. Additionally, the development of new compression algorithms capable of optimizing both transmission and storage efficiency could further improve overall system performance. Therefore, this research contributes to the growing field of quantum-inspired signal processing, demonstrating practical approaches for efficient communication systems that leverage quantum principles while remaining compatible with emerging quantum hardware.
Author Contributions
Conceptualization, U.J.; methodology, U.J.; software, U.J.; validation, U.J. and A.F.; formal analysis, A.F.; investigation, A.F.; resources, U.J.; data curation, U.J.; writing—original draft preparation, U.J.; writing—review and editing, A.F.; visualization, U.J.; supervision, A.F.; project administration, A.F. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
The original data presented in the study are openly available at
https://www.pexels.com
(accessed on 11 December 2025) under the Creative Commons Zero (CC0) license, which allows free use, distribution, and modification without attribution.
Conflicts of Interest
The authors declare no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
AVC
Advanced Video Coding
BER
Bit Error Rate
GOP
Group of Pictures
HEVC
High Efficiency Video Coding
IQFT
Inverse Quantum Fourier Transform
LDPC
Low-Density Parity Check
MIMO
Multi-Input Multi-Output
OFDM
Orthogonal Frequency-Division Multiplexing
PSNR
Peak Signal-to-Noise Ratio
QEC
Quantum Error Correction
QFT
Quantum Fourier Transform
QKD
Quantum Key Distribution
SI
Structural Information
SNR
Signal-to-Noise Ratio
SSIM
Structural Similarity Index Measure
TI
Temporal Information
VMAF
Video Multi-Method Assessment Fusion
VVC
Versatile Video Coding
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[Abstract](https://www.mdpi.com/2079-9292/15/6/1323#Abstract)[Introduction](https://www.mdpi.com/2079-9292/15/6/1323#Introduction)[Related Works](https://www.mdpi.com/2079-9292/15/6/1323#Related_Works)[System Model](https://www.mdpi.com/2079-9292/15/6/1323#System_Model)[Results and Discussion](https://www.mdpi.com/2079-9292/15/6/1323#Results_and_Discussion)[Conclusions](https://www.mdpi.com/2079-9292/15/6/1323#Conclusions)[Author Contributions](https://www.mdpi.com/2079-9292/15/6/1323#Author_Contributions)[Funding](https://www.mdpi.com/2079-9292/15/6/1323#Funding)[Data Availability Statement](https://www.mdpi.com/2079-9292/15/6/1323#Data_Availability_Statement)[Conflicts of Interest](https://www.mdpi.com/2079-9292/15/6/1323#Conflicts_of_Interest)[Abbreviations](https://www.mdpi.com/2079-9292/15/6/1323#Abbreviations)[References](https://www.mdpi.com/2079-9292/15/6/1323#References)[Article Metrics](https://www.mdpi.com/2079-9292/15/6/1323#Article_Metrics)
- Article
- 
22 March 2026
# Rethinking Video Transmission: A Quantum Fourier Transform-Based Approach for Error-Prone Channels
Udara Jayasinghe\*
and
Anil Fernando
Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XQ, UK
\*
Author to whom correspondence should be addressed.
[Electronics](https://www.mdpi.com/journal/electronics)**2026**, *15*(6), 1323;https://doi.org/10.3390/electronics15061323
(registering DOI)
This article belongs to the Special Issue [Advancements in Signal Processing: Communications, Sensing and Imaging](https://www.mdpi.com/journal/electronics/special_issues/3RWHQ97ABG)
[Version Notes](https://www.mdpi.com/2079-9292/15/6/1323/notes)
[Order Reprints](https://www.mdpi.com/2079-9292/15/6/1323/reprints)
## Abstract
Reliable video transmission over error-prone channels remains a significant challenge due to the inherent trade-off between compression efficiency and noise resilience in conventional systems. To address these issues, this paper introduces a novel quantum Fourier transform (QFT)-based framework that integrates video compression and transmission within a unified quantum frequency-domain representation. The framework converts video data into a classical bitstream and maps it onto multi-qubit quantum states with variable encoding sizes (n), enabling flexible control over compression levels. Through the application of the QFT, these states are transformed into the frequency domain, where only selected coefficients are transmitted to reduce bandwidth requirements. At the receiver, the transmitted components are used to reconstruct the full representation, followed by inverse transformation and decoding to recover the video sequence. The performance of the proposed framework is evaluated using peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and video multi-method assessment fusion (VMAF). The results demonstrate that increasing the number of qubits enables exponential compression, achieving ratios up to
2
n
:
1
, while maintaining high reconstruction quality under ideal transmission conditions. However, higher-qubit configurations exhibit increased sensitivity to channel noise, leading to a more rapid degradation as the signal-to-noise ratio decreases. In contrast, lower-qubit configurations provide improved robustness, maintaining more stable reconstruction quality under noisy conditions, albeit with reduced compression efficiency. Among the evaluated configurations, the two-qubit system achieves an effective trade-off, providing a compression ratio of
4
:
1
while maintaining strong visual and structural fidelity along with enhanced resilience to channel impairments.
Keywords:
[quantum Fourier transform](https://www.mdpi.com/search?q=quantum+Fourier+transform); [quantum frequency domain](https://www.mdpi.com/search?q=quantum+frequency+domain); [quantum superposition](https://www.mdpi.com/search?q=quantum+superposition); [video transmission](https://www.mdpi.com/search?q=video+transmission)
## 1\. Introduction
The rapid evolution of consumer electronics and intelligent multimedia systems has amplified the central role of video transmission in modern communication architectures. Applications such as ultra-high-definition (UHD) video streaming, telepresence conferencing, medical video diagnostics, augmented reality (AR) \[1\], virtual reality (VR) \[2\], autonomous navigation, cloud-based surveillance, interactive digital displays, and immersive extended-reality platforms increasingly depend on the seamless transfer of real-time video content at high fidelity. As resolutions scale from high definition (HD) to 4 K, 8 K, and beyond, alongside the growing popularity of high-frame-rate formats and multi-view video, the size and complexity of video data have grown exponentially. These advancements, while enabling more immersive user experiences, impose substantial demands on communication networks, which must handle massive data volumes under stringent requirements for latency, reliability, and robustness.
Classical video compression technologies, including advanced video coding (AVC/H.264) \[3\], high efficiency video coding (HEVC/H.265) \[4\], and the more recent versatile video coding (VVC/H.266) \[5\], achieve high compression performance by exploiting spatial and temporal redundancies through techniques such as motion-compensated prediction, transform coding, rate–distortion optimization, and entropy coding. These methods have significantly improved the efficiency of multimedia delivery over modern networks. However, the underlying framework remains constrained by the principles of classical signal processing and information theory. When aggressive compression is applied, classical codecs often produce visual distortions such as blocking artifacts, ringing patterns, motion smearing, and loss of fine texture details. Moreover, because modern video standards rely heavily on predictive coding structures such as group of pictures (GOP), errors introduced during transmission can propagate across multiple frames, causing noticeable temporal degradation.
To address these impairments, classical communication systems utilize various reliability and transmission mechanisms including adaptive bitrate streaming, channel coding \[6\], retransmission protocols, error concealment \[7\], and feature-based transmission strategies \[8,9\] that attempt to prioritize perceptually important visual information. While these approaches can improve robustness and maintain acceptable quality under moderate channel impairments, they do not fundamentally eliminate the sensitivity of compressed bitstreams to channel disturbances. In bandwidth-constrained, wireless, or long-distance communication environments, fluctuations in channel capacity, packet losses, and additive noise can still lead to significant degradation in video quality. With the growing demand for ultra-high-resolution visual content, real-time streaming, and highly reliable multimedia services, the performance of conventional compression and transmission frameworks is increasingly approaching the practical limits imposed by classical communication paradigms.
In response to these limitations, quantum communication \[10\] has emerged as a transformative paradigm that leverages intrinsic properties of quantum mechanics \[11\], such as quantum superposition \[12\], quantum entanglement \[13\], and unitary state evolution, to achieve new modalities of information representation and transmission. In contrast to classical communication, where information is represented as fixed binary sequences, quantum communication represents information using quantum states, which provide a substantially larger and more expressive representation space. Initial studies on quantum multimedia transmission have mainly focused on time-domain quantum encoding \[14,15\], where classical data are directly represented as quantum states without applying any transformation into the frequency or spectral domain. These time-domain models have demonstrated benefits in terms of enhanced state representation and improved data fidelity \[16\]. However, they remain inherently inefficient for high-dimensional video content due to the large number of qubits required to achieve high-quality reconstruction, as well as their susceptibility to accumulated noise effects, including depolarizing noise, amplitude-damping noise, phase-damping noise, and other quantum decoherence processes. Video data, with their temporal continuity and spatial variability, exhibit patterns that cannot be efficiently exploited in the time domain alone \[17\].
To respond to this, quantum Fourier transform (QFT) \[18\] offers a powerful alternative for representing quantum-encoded video data in the frequency domain. The QFT, an essential unitary transformation in quantum domain, maps quantum amplitude distributions into frequency-domain components, enabling efficient identification of redundancies and sparsity in the underlying signal structure \[19\]. By applying the QFT to quantum video data, it becomes possible to achieve intrinsic quantum-domain compression, suppress noise-sensitive spectral components, and reduce quantum resource requirements, all while maintaining the essential visual and temporal coherence necessary for high-quality video reconstruction.
Motivated by these advantages, this study introduces the first unified QFT-based framework for video compression and transmission, integrating quantum encoding, frequency-domain transformation, compression, and quantum-channel transmission into a single cohesive pipeline. Unlike prior work, which treats compression and transmission as separate processes or relies on fixed-size qubit encodings with limited adaptability, the proposed method enables flexible multi-qubit symbol encoding, where the number of qubits allocated per symbol can vary according to the desired compression ratio and available quantum resources. This design makes the framework the first to provide an adaptive quantum symbol-level compression mechanism within the frequency domain specifically for video transmission. By allowing adjustment of qubit allocation, the system can effectively balance compression efficiency, transmission robustness, and computational complexity. This makes it suitable for a wide range of video characteristics and diverse quantum channel conditions. Consequently, the proposed approach represents a significant step toward practical and high-performance quantum video communication systems, offering both high fidelity and efficient utilization of quantum resources.
The system begins by converting the input video into a classical bitstream for subsequent processing. To enhance resilience against channel-induced errors, polar coding with a code rate of 1/2 is applied to generate a channel-encoded bit sequence. After classical channel encoding, the bitstream is mapped to quantum states according to the selected qubit encoding size. This step forms a structured quantum state vector that captures both spatial and temporal characteristics of video data. The QFT is then applied to this state vector, converting it from the time domain into the quantum frequency domain. The transformation exposes sparsity, enabling selective retention of dominant frequency coefficients while discarding or compressing insignificant ones. As a result, the transmitted quantum symbol payload is significantly reduced while preserving essential visual features.
Following quantum-frequency-domain compression, the retained coefficients are transmitted over a noisy quantum communication channel. The channel introduces realistic quantum noise, modeled through quantum noise operators such as Kraus operators associated with bit-flip, phase-flip, depolarizing, amplitude-damping, and phase-damping processes. At the receiver, the system reconstructs the complete frequency-domain quantum state by appropriately restoring suppressed coefficients based on the compression ratio and encoding parameters. The inverse quantum Fourier transform (IQFT) is then applied to convert the restored frequency-domain representation back into the time domain. After measurement of the quantum state, the resulting classical bitstream is passed through the polar decoding process to correct channel-induced errors. The final step reconstructs the output video frames, restoring both the temporal continuity and structural consistency of the original sequence. Experimental evaluations demonstrate that the proposed QFT-based architecture achieves significant improvements in compression efficiency, and reconstruction fidelity compared to conventional quantum communication and classical video transmission. By evaluating multiple qubit encoding sizes, the framework reveals how quantum compression behavior and robustness vary with resource allocation, offering important insights for deploying quantum video transmission systems in future quantum networks.
### 1\.1. Key Contributions
The key contributions of this study are summarized as follows:
- A novel quantum frequency-domain approach for video compression, utilizing QFT to reduce redundancy and enhance robustness within a quantum communication framework.
- A unified architecture for quantum video transmission that integrates frequency-domain compression and noisy-channel transmission to achieve a scalable, efficient, and noise-resilient system.
- A comprehensive analysis of multi-qubit encoding configurations, demonstrating how qubit symbol size influences compression efficiency, spectral sparsity, and end-to-end video reconstruction performance.
- Extensive experimental validation demonstrating the effectiveness of the proposed framework across multiple video datasets and channel conditions, highlighting compression efficiency, robustness to quantum noise, and improved reconstructed video quality compared to classical and conventional quantum approaches.
### 1\.2. Organization of the Paper
The remainder of this paper is structured as follows: [Section 2](https://www.mdpi.com/2079-9292/15/6/1323#sec2-electronics-15-01323) reviews prior work in quantum communication, quantum multimedia processing, and QFT. [Section 3](https://www.mdpi.com/2079-9292/15/6/1323#sec3-electronics-15-01323) presents the proposed QFT–based methodology for unified video compression and transmission, detailing the multi-qubit encoding process, frequency-domain transformation, and noisy quantum channel modeling. [Section 4](https://www.mdpi.com/2079-9292/15/6/1323#sec4-electronics-15-01323) provides experimental results and comparative analysis with conventional quantum methods. Finally, [Section 5](https://www.mdpi.com/2079-9292/15/6/1323#sec5-electronics-15-01323) concludes the study by summarizing key findings, discussing the significance of the proposed framework, and outlining directions for future research.
## 2\. Related Works
Quantum communication \[20\] is a revolutionary approach that uses the fundamental principles of quantum superposition and quantum entanglement \[21,22,23\] to transmit information in ways that classical communication cannot achieve. Early research in this field focused mainly on time-domain quantum communication, where information is encoded and transmitted directly over quantum states in time. Well-known applications of this approach include quantum key distribution (QKD) \[24,25,26\], which allows two parties to securely share cryptographic keys, and quantum teleportation \[27,28\], which enables the transfer of quantum states between distant locations. Building on these methods, researchers have also explored transmitting images and videos over quantum channels \[29,30,31\], showing improved security and reliability. However, scaling these techniques to handle high-resolution, high-dimensional media remains a significant challenge due to the large quantum resources and noise sensitivity involved.
Several studies have investigated the use of single-qubit superposition encoding for transmitting images and videos, demonstrating its feasibility for high-fidelity media applications \[14,15\]. In this approach, each qubit encodes a portion of the media information as a quantum superposition state. While these methods highlight the potential of quantum-based media transmission, they exhibit several key limitations. First, they are resource-intensive, as high-resolution or high-dimensional media require an exponentially growing number of qubits. Second, their compression capability is limited, because single-qubit encoding cannot effectively exploit the inherent spatial and temporal redundancies in the media. Third, these systems are highly sensitive to quantum noise, including depolarizing, amplitude-damping, and phase-damping effects, which can significantly degrade transmission fidelity. Moreover, time-domain encoding methods fail to fully utilize the structural and spectral correlations present in high-dimensional media, reducing their efficiency for representation, compression, and accurate reconstruction \[16\].
To address the limitations of single-qubit encoding, multi-qubit quantum encoding has been proposed \[32\], in which multiple classical bits are represented within a single quantum state. This approach significantly increases the data capacity and enhances transmission efficiency for high-dimensional media. By encoding more information per qubit, multi-qubit schemes can reduce the total number of qubits required for large media files and better exploit correlations within the data. However, most existing research focuses on time-domain implementations, which offer limited resilience to realistic quantum noise, such as depolarizing, amplitude-damping, and phase-damping effects \[19\]. Additionally, there has been no systematic study evaluating how different qubit sizes impact compression efficiency, noise robustness, and the quality of reconstructed media in a unified quantum communication framework. Understanding these trade-offs is critical for designing scalable and efficient quantum media transmission systems.
The QFT \[33\] has emerged as a powerful tool to overcome these limitations. By converting quantum states from the time domain into the frequency domain, QFT exposes sparsity in the underlying signal, enabling more efficient compression of high-dimensional media. Several studies have explored QFT exclusively for compression, without considering transmission \[34\], and have shown that frequency-domain representation can reduce the number of qubits required to store and process quantum images or videos \[35,36,37,38,39\]. However, compression achieved solely through QFT is limited, as it does not account for channel noise or the need for error-resilient transmission. Conversely, QFT has also been applied purely for transmission purposes, without compression, where frequency-domain transformations improve noise resilience and maintain fidelity during quantum communication \[19\]. While these studies demonstrate the benefits of QFT in isolation in various contexts, there is currently no unified framework that simultaneously exploits the frequency-domain representation for both efficient compression and robust transmission of high-dimensional media.
In addition, the QFT has also been explored in the context of quantum error correction (QEC) \[40\] and quantum orthogonal frequency-division multiplexing (OFDM) systems \[41\]. QFT-based QEC techniques leverage the frequency-domain representation of quantum states to detect and correct errors more efficiently, mitigating the effects of quantum noise \[42\]. Similarly, QFT-enabled quantum OFDM systems have been proposed to support parallel transmission of multiple quantum subcarriers, improving spectral efficiency and channel utilization. These approaches have shown promising theoretical and simulation results in enhancing the reliability and robustness of quantum communications. However, despite their effectiveness in general quantum channels, these QFT-based QEC and OFDM frameworks have not been specifically designed or evaluated for high-dimensional video transmission. As a result, they do not address the unique challenges posed by large-scale, temporally correlated video data, including the need for compression, efficient qubit utilization, and preservation of temporal and spatial correlations.
### Research Gaps and Rationale for a Unified Framework
Despite these advances, a significant gap remains in the literature: no study has systematically combined multi-qubit quantum encoding with QFT-based compression and transmission in a unified framework. Existing research typically treats QFT either as a tool for compression, ignoring the effects of noisy channels, or as a tool for transmission, without leveraging frequency-domain sparsity to optimize qubit usage and resource efficiency. Furthermore, there is no comprehensive analysis of how varying qubit sizes affect compression efficiency, noise resilience, and the quality of reconstructed high-dimensional media. Addressing these gaps is critical for developing scalable and efficient quantum communication systems capable of transmitting videos with both high fidelity and robust error tolerance.
In summary, prior research highlights several key trends and gaps:
- Classical compression methods are effective under ideal channels but degrade under noisy or error-prone conditions.
- Time-domain quantum encoding provides security and reliability but struggles with high-dimensional media due to resource and noise limitations.
- Time-domain single-qubit superposition demonstrates feasibility for small-scale media but has limited scalability and compression capability.
- Time-domain multi-qubit encoding can improve data capacity and transmission efficiency, but it suffers from limited noise resilience in high-noise environments and inefficient compression, reducing its effectiveness for high-dimensional media transmission.
- Using the QFT purely for compression can reduce the number of qubits required by exploiting spectral sparsity; however, this approach only addresses storage and does not account for channel noise or provide resilience against transmission errors, thereby limiting its practical effectiveness in real-world quantum communication systems.
- When the QFT is used solely for transmission, it can enhance fidelity over noisy quantum channels, but it does not provide compression or optimize the use of quantum resources, which limits efficiency when transmitting high-dimensional media such as images or videos.
These observations indicate that existing QFT-based compression and transmission approaches cannot be directly combined within a single framework, as they are designed for different objectives. This motivates the development of a novel unified quantum framework that leverages multi-qubit encoding and QFT to jointly optimize compression, transmission efficiency, and robustness against quantum noise. The present study addresses these gaps by proposing a QFT-based video compression and transmission system, systematically evaluating its performance across varying qubit sizes, and providing insights for practical deployment in future quantum networks.
## 3\. System Model
The system illustrated in [Figure 1](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f001) is designed to provide efficient video compression and transmission while maintaining reliable delivery of video content. It works with a variety of video sequences, supporting different resolutions and frame rates, and focuses on reducing the amount of data that needs to be transmitted without compromising the overall visual quality. Video segments are represented in a compact form, allowing for easier handling and transmission over limited-bandwidth channels. The workflow is organized to ensure that all steps, from preparing the video data to reconstructing the received frames, are seamless and coordinated. By prioritizing simplicity and efficiency, the system is able to deliver videos effectively across different scenarios while preserving clarity, continuity, and smooth playback.
**Figure 1.** Proposed quantum communication architecture for compressing and transmitting video data.
The performance of the system is assessed using a set of video sequences selected to cover a wide range of realistic conditions. Testing involves three videos with varying amounts of structural information (SI) and temporal information (TI). The high-motion video \[43\] features fast-moving objects and frequent scene transitions, leading to high SI and TI values, which places significant demands on both compression and transmission processes. The medium-motion video \[44\] features slower-paced actions and less frequent scene transitions, with intermediate SI and TI, representing content of average complexity. The low-motion video \[45\] has limited movement and low TI, while retaining some structural detail, making it easier to compress. These videos are uncompressed and available in three spatial resolutions:
320
×
180
,
1280
×
720
, and
1920
×
1080
, covering low-resolution to full-HD content. The videos are encoded at 20, 30, and 50 frames per second (fps) to reflect different levels of motion activity. Along with raw uncompressed sequences, the evaluation also includes videos encoded using the VVC standard with GOP sizes of 8 and 32. These encoded versions share the same resolutions and frame rates as the uncompressed set, enabling analysis under different temporal compression structures while keeping spatial and temporal properties consistent. This selection allows the framework to demonstrate its versatility and compatibility with different video formats and coding schemes. Furthermore, the design is scalable, capable of supporting various video resolutions, frame rates, and source coding methods, making it adaptable to a wide range of practical applications.
The video processing begins by converting each video into a sequential bitstream representation. These bitstreams can be optionally protected using polar codes with a rate of
1
/
2
to provide error resilience during transmission. To evaluate the inherent performance of the system independently of error correction, scenarios without channel coding (uncoded) are also considered. During the quantum encoding stage, groups of bits are mapped into multi-qubit states, where the size of each qubit encoding size (n) is selected between 1 and 8. This mapping determines the effective compression ratio for each segment. The resulting quantum state is then transformed into the frequency domain using a QFT-based approach, allowing the video information to be represented with fewer coefficients rather than the entire state vector. This frequency-domain compression reduces the amount of data transmitted while preserving the essential information needed for accurate video reconstruction.
After the frequency-domain compression, the encoded video coefficients are transmitted through a channel that may introduce noise and distortions. At the receiver, the system first interprets the compression parameters to reconstruct the complete frequency-domain representation of each qubit system. Next, an inverse transform is performed to map the data from the frequency domain back to the time domain. The resulting quantum states are measured and transformed into classical bitstreams. These bitstreams can subsequently be processed by a channel decoder to reconstruct the original video content.
In the following subsections, each key functional block depicted in [Figure 1](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f001) is explained in detail.
### 3\.1. Classical Channel Encoder and Decoder
The transmission reliability of video bitstreams is improved through the integration of polar codes \[46,47\] operating at a 1/2 coding rate. This configuration establishes an optimal equilibrium between error correction performance and resource utilization, introducing controlled redundancy that effectively mitigates channel distortions while maintaining reasonable computational demands and spectral efficiency. The selection of polar codes over alternative error correction schemes, such as low-density parity check (LDPC) \[48\] and turbo codes \[49\], is motivated by their compelling combination of theoretical performance guarantees and practical implementation advantages. Unlike conventional QEC methodologies that primarily target time-domain perturbations, the proposed architecture specifically leverages frequency-domain processing, rendering traditional QEC approaches unsuitable for this particular framework design.
### 3\.2. Quantum Encoder and Decoder
The proposed quantum video compression and transmission system comprises two fundamental components: the quantum encoder and quantum decoder. The encoder transforms classical video data into a quantum representation and applies compression in the frequency domain, while the decoder reconstructs the original data by reversing these operations. The following subsections present a detailed mathematical formulation of both components and their operating principles.
#### 3\.2.1. Quantum Encoder
The quantum encoder illustrated in [Figure 2](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f002) performs the conversion of classical video data from the time domain into a compact frequency-domain representation. Its operation is divided into several sequential steps. The process begins by assigning each classical bit from the video to a corresponding quantum basis state. In this initialization stage, a bit value of 0 is encoded as the state
\|
0
⟩
, whereas a bit value of 1 is encoded as
\|
1
⟩
, forming the qubit inputs for the subsequent quantum transformations.
**Figure 2.** Proposed quantum encoder operating in the frequency domain.
In the next stage, the system selects the qubit encoding size (n), which may vary from 1 to 8 qubits. This parameter determines how many consecutive classical bits are grouped together and converted into a single multi-qubit state. Although there is no strict theoretical upper limit on the n, in this work a maximum of 8 qubits is used to maintain manageable system complexity. This choice is sufficient to directly map the pixel values into the corresponding quantum states. The encoder then constructs the complete quantum state vector for each group of bits, incorporating both the initialized qubits and the selected value of n. This results in a structured multi-qubit representation that is ready for frequency-domain processing.
Mathematically, a video bitstream (V) can be represented as in Equation ([1](https://www.mdpi.com/2079-9292/15/6/1323#FD1-electronics-15-01323)).
V
\=
{
v
1
,
v
2
,
…
,
v
M
}
,
v
i
∈
{
0
,
1
}
(1)
where M denotes the total number of bits. These bits are then processed according to the selected n to form the corresponding quantum states for further frequency-domain encoding.
In the case of single-qubit encoding (
n
\=
1
), each individual bit from the video bitstream is directly represented by a single qubit within the two-dimensional Hilbert space
H
2
. In this mapping stage, each classical bit is directly converted into its corresponding computational basis state, as expressed in Equation ([2](https://www.mdpi.com/2079-9292/15/6/1323#FD2-electronics-15-01323)).
v
i
\=
0
↦
\|
0
⟩
\=
1
0
,
v
i
\=
1
↦
\|
1
⟩
\=
0
1
(2)
For multi-qubit encoding, consecutive blocks of n bits are grouped together and encoded as an n\-qubit register, forming a state in the
2
n
\-dimensional Hilbert space
H
2
n
. This grouping process forms a composite quantum state by taking the tensor product of the individual qubits. In this way, several classical bits are combined into a single multi-qubit vector, as described in Equation ([3](https://www.mdpi.com/2079-9292/15/6/1323#FD3-electronics-15-01323)).
(
v
1
,
v
2
,
…
,
v
n
)
↦
\|
v
1
v
2
…
v
n
⟩
\=
\|
v
1
⟩
⊗
\|
v
2
⟩
⊗
…
⊗
\|
v
n
⟩
(3)
As an illustration, consider the case of n = 2. In this setting, two classical bits are grouped together, and each pair is represented as a four-dimensional quantum state obtained through a tensor product construction. The resulting states correspond to the formulations shown in Equations ([4](https://www.mdpi.com/2079-9292/15/6/1323#FD4-electronics-15-01323))–([7](https://www.mdpi.com/2079-9292/15/6/1323#FD7-electronics-15-01323)).
\|
00
⟩
\=
1
0
⊗
1
0
\=
1
0
0
0
(4)
\|
01
⟩
\=
1
0
⊗
0
1
\=
0
1
0
0
(5)
\|
10
⟩
\=
0
1
⊗
1
0
\=
0
0
1
0
(6)
\|
11
⟩
\=
0
1
⊗
0
1
\=
0
0
0
1
(7)
When n qubits are combined, they form a composite system whose state vector resides in a
2
n
\-dimensional Hilbert space, generated by the tensor product of the constituent single-qubit states. This establishes a direct link between the number of qubits and the size of the quantum state representation.
Following the preparation of the quantum state vector, the next step is to shift from the time-domain representation to the frequency domain. The transformation requires selecting a QFT gate whose dimension matches that of the quantum state vector. For instance, a state vector of size
4
×
1
(
n
\=
2
) is transformed using a
4
×
4
QFT matrix, while a vector of size
8
×
1
(
n
\=
3
) requires an
8
×
8
matrix, and so forth. Accordingly, the QFT matrix, denoted by
F
N
, is constructed as shown in Equation ([8](https://www.mdpi.com/2079-9292/15/6/1323#FD8-electronics-15-01323)).
F
N
\=
1
N
1
1
1
…
1
1
ω
ω
2
…
ω
N
−
1
1
ω
2
ω
4
…
ω
2
(
N
−
1
)
⋮
⋮
⋮
⋱
⋮
1
ω
N
−
1
ω
2
(
N
−
1
)
…
ω
(
N
−
1
)
(
N
−
1
)
(8)
where
N
\=
2
n
corresponds to the quantum state vector dimension, and
ω
represents the primitive root of unity given by Equation ([9](https://www.mdpi.com/2079-9292/15/6/1323#FD9-electronics-15-01323)).
ω
\=
e
2
π
i
/
N
(9)
This construction ensures that the QFT appropriately transforms the amplitudes of the time-domain quantum state into the frequency domain, analogous to the classical discrete Fourier transform but in the quantum setting.
For a qubit encoding size of
n
\=
2
, the QFT is applied using a
4
×
4
unitary matrix, defined in Equation ([10](https://www.mdpi.com/2079-9292/15/6/1323#FD10-electronics-15-01323)).
F
4
\=
1
2
1
1
1
1
1
ω
ω
2
ω
3
1
ω
2
ω
4
ω
6
1
ω
3
ω
6
ω
9
,
ω
\=
e
2
π
i
/
4
\=
i
(10)
This procedure can be generalized to higher qubit numbers, with the QFT reorganizing the amplitudes of a quantum state into the frequency domain, analogous to how the discrete Fourier transform converts classical signals.
When the QFT is applied to the two-qubit computational basis states, the corresponding frequency-domain vectors are given by Equations ([11](https://www.mdpi.com/2079-9292/15/6/1323#FD11-electronics-15-01323)) through ([14](https://www.mdpi.com/2079-9292/15/6/1323#FD14-electronics-15-01323)).
F
4
\|
00
⟩
\=
1
2
1
1
1
1
(11)
F
4
\|
01
⟩
\=
1
2
1
ω
ω
2
ω
3
(12)
F
4
\|
10
⟩
\=
1
2
1
ω
2
ω
4
ω
6
(13)
F
4
\|
11
⟩
\=
1
2
1
ω
3
ω
6
ω
9
(14)
Equations ([11](https://www.mdpi.com/2079-9292/15/6/1323#FD11-electronics-15-01323))–([14](https://www.mdpi.com/2079-9292/15/6/1323#FD14-electronics-15-01323)) mathematically show that each time-domain basis state maps to a distinct column of the QFT matrix. Careful examination of these columns shows that the second element alone uniquely determines the entire column. Therefore, it is sufficient to transmit only this second component to retain complete knowledge of the quantum state, eliminating the need to send the full frequency-domain vector. In practical implementation, the quantum state is not obtained by extracting amplitudes from an existing superposition, but is instead directly prepared using deterministically computed coefficients via unitary operations. Specifically, for each input symbol, a representative coefficient (e.g., the second QFT coefficient) is computed classically from the bit pattern and mapped to a discrete phase on the complex unit circle. A set of
2
n
such coefficients is then used to synthesize a valid quantum state in the
2
n
\-dimensional Hilbert space, ensuring normalization and preserving quantum coherence.
Importantly, the proposed framework operates under a constrained state preparation model in which the inputs to the QFT are computational basis states derived from classical bitstreams rather than arbitrary quantum states. As a result, the possible transmitted states belong to a finite and known set corresponding to the columns of the QFT matrix. Because this set is predetermined and shared by both the transmitter and receiver, the transmitted coefficient functions as a symbol that uniquely identifies the corresponding basis index within this codebook.
This process does not involve any projective measurement or partial state extraction. Instead, it defines a structured state preparation strategy in which the full quantum state is generated from a classical description using unitary operations. Although only a single representative coefficient per symbol is transmitted, the remaining components of the corresponding quantum state are implicitly determined by the predefined transform structure, allowing deterministic reconstruction of the original state at the receiver through the inverse QFT.
For instance, when the qubit register size is
n
\=
2
, the full state vector contains 4 elements. By transmitting just the second element, a compression ratio of
4
:
1
is achieved. As the qubit size grows, the compression effect becomes more pronounced: with
n
\=
8
, only a single element out of 256 needs to be sent, resulting in a
256
:
1
reduction in transmitted data. As the register size n increases, the approach allows effective compression while preserving the integrity of the quantum information. This selective transmission approach significantly reduces the quantum symbol payload while retaining essential quantum information.
In summary, to clarify the effect of the qubit encoding size on the system, the qubit encoding size n determines how many classical bits are grouped and represented by an n\-qubit quantum register prior to the application of the QFT. Increasing n increases the dimensionality of the QFT representation, producing
2
n
frequency-domain components and enabling higher compression ratios, since a single transmitted coefficient can represent one of
2
n
possible input states. However, larger encoding sizes also reduce the angular separation between adjacent phase components, given by
Δ
θ
\=
2
π
2
n
, which increases sensitivity to channel noise. Consequently, larger qubit group sizes provide higher compression efficiency but exhibit greater susceptibility to transmission errors, while smaller encoding sizes offer improved noise robustness at the cost of lower compression efficiency. Therefore, the choice of n introduces a trade-off between compression performance and reconstruction reliability, which is analyzed experimentally in this work for qubit sizes ranging from
n
\=
1
to
n
\=
8
.
The QFT operation applied in this system can be described using its general theoretical formulation, shown in Equation ([15](https://www.mdpi.com/2079-9292/15/6/1323#FD15-electronics-15-01323)).
F
N
\|
x
⟩
\=
1
N
∑
k
\=
0
N
−
1
e
2
π
i
x
k
N
\|
k
⟩
(15)
Here,
F
N
represents the QFT matrix corresponding to a qubit register of size n. It performs a unitary transformation on an input quantum state
\|
x
⟩
expressed in the computational basis. Through this operation, the input state is converted into a superposition of all frequency-domain basis states
\|
k
⟩
, with each component weighted by a complex phase factor
e
2
π
i
x
k
/
N
that captures its frequency-domain characteristics. The factor
1
/
N
normalizes the transformation, ensuring that the operation preserves the overall quantum state’s norm and maintains unitarity.
#### 3\.2.2. Quantum Decoder
During the quantum decoding stage, the operations applied during encoding are effectively reversed, as illustrated in [Figure 3](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f003). The process begins by evaluating the compression ratio and using the lightweight side information to determine the original qubit encoding size (n) and ensure decoder synchronization. Using this information, the full frequency-domain quantum state is reconstructed from the received compressed representation. Subsequently, the IQFT is applied to convert the frequency-domain state back into its corresponding time-domain quantum state. Mathematically, the IQFT is the Hermitian conjugate of the QFT and serves a role analogous to the inverse discrete Fourier transform in classical signal processing. Once the time-domain quantum state is restored, standard measurements are performed to map the state back into classical bits, enabling reconstruction of the original bitstream.
**Figure 3.** Proposed quantum decoder.
The IQFT operation (
F
N
−
1
) can be mathematically expressed as the Hermitian conjugate of the QFT, as shown in Equation ([16](https://www.mdpi.com/2079-9292/15/6/1323#FD16-electronics-15-01323)).
F
N
−
1
\=
F
N
†
(16)
Being the conjugate transpose of the QFT, the IQFT reverses the transformation applied in the frequency domain. When applied to a computational basis state
\|
k
⟩
, the IQFT performs the following operation, as shown in Equation ([17](https://www.mdpi.com/2079-9292/15/6/1323#FD17-electronics-15-01323)).
F
N
−
1
\|
k
⟩
\=
1
N
∑
j
\=
0
N
−
1
e
−
2
π
i
j
k
/
N
\|
j
⟩
(17)
where
N
\=
2
n
is the dimension of the quantum state space. The index j runs over all computational basis states and represents the output basis states generated by the inverse transform. The negative sign in the exponent indicates the reversal of the phase evolution compared to the forward QFT. This operation maps the frequency-domain representation of a quantum state back to its corresponding time-domain state while preserving unitarity and orthogonality.
For simulation purposes, the QFT matrix
F
∈
C
N
×
N
, where
C
denotes the set of complex numbers, is defined element-wise as shown in Equation ([18](https://www.mdpi.com/2079-9292/15/6/1323#FD18-electronics-15-01323)).
F
k
,
l
\=
1
N
e
2
π
i
k
l
/
N
,
k
,
l
\=
0
,
…
,
N
−
1
(18)
In this expression, the indices k and l denote the row and column positions of the matrix, respectively. Each entry
F
k
,
l
corresponds to the complex phase factor associated with mapping the computational basis state
\|
l
⟩
to the output component indexed by k. Each column of the QFT matrix, denoted by
q
j
, can therefore be represented as a vector, as shown in Equation ([19](https://www.mdpi.com/2079-9292/15/6/1323#FD19-electronics-15-01323)).
q
j
\=
1
N
1
e
2
π
i
j
/
N
e
2
π
i
2
j
/
N
⋮
e
2
π
i
(
N
−
1
)
j
/
N
(19)
It is important to note that the second element of each column uniquely identifies that column. This second entry of the column vector
q
j
is given by Equation ([20](https://www.mdpi.com/2079-9292/15/6/1323#FD20-electronics-15-01323)).
q
j
(
2
)
\=
1
N
e
2
π
i
j
/
N
(20)
The second element of each QFT column corresponds to a distinct point on the complex unit circle, scaled by
1
/
N
. This unique mapping allows each transmitted coefficient to identify its original column in the QFT matrix unambiguously.
During transmission, the received coefficient, denoted by
q
˜
(
2
)
, may be affected by channel noise
η
∈
C
, where
C
denotes the set of complex numbers and
η
can alter both the amplitude and phase of the transmitted coefficient, as expressed in Equation ([21](https://www.mdpi.com/2079-9292/15/6/1323#FD21-electronics-15-01323)).
q
˜
(
2
)
\=
q
j
(
2
)
\+
η
(21)
The process of reconstructing the original quantum state from the received coefficient can be described as follows:
- Construct the set of all ideal second elements corresponding to the noiseless QFT columns. Here,
N
\=
2
n
represents the dimension of the Hilbert space for an n\-qubit register. As shown in Equation ([22](https://www.mdpi.com/2079-9292/15/6/1323#FD22-electronics-15-01323)), the set of ideal second elements for the noiseless QFT columns is defined as
Q
.
Q
\=
1
N
e
2
π
i
j
/
N
\|
j
\=
0
,
1
,
…
,
N
−
1
(22)
Each element of
Q
corresponds to the second component of the j\-th column
q
j
of the QFT matrix F.
- Identify the closest match
j
\*
to the received noisy coefficient
q
˜
(
2
)
affected by channel noise
η
, as shown in Equation ([23](https://www.mdpi.com/2079-9292/15/6/1323#FD23-electronics-15-01323)).
j
\*
\=
arg
min
j
q
˜
(
2
)
−
1
N
e
2
π
i
j
/
N
(23)
Here,
j
\*
indicates the index of the ideal QFT column most closely corresponding to the received signal.
- Retrieve the full QFT column vector corresponding to
j
\*
, as shown in Equation ([24](https://www.mdpi.com/2079-9292/15/6/1323#FD24-electronics-15-01323)).
q
^
\=
q
j
\*
(24)
The vector
q
^
represents the retrieved frequency-domain quantum state before applying the inverse transformation.
- Apply the IQFT to reconstruct the time-domain quantum state (
x
^
), as shown in Equation ([25](https://www.mdpi.com/2079-9292/15/6/1323#FD25-electronics-15-01323)).
x
^
\=
F
N
†
q
^
(25)
where
F
N
†
is the Hermitian conjugate of the QFT matrix F.
After reconstructing the time-domain quantum state, measurements are performed on each qubit in the computational (Z) basis. This produces an n\-bit classical string corresponding to the decoded symbol, which can then be used to reconstruct the original bitstream. This approach ensures that the simulation faithfully represents the theoretical model while preserving mathematical rigor throughout the entire encoding and decoding process.
It is important to note that Equation ([23](https://www.mdpi.com/2079-9292/15/6/1323#FD23-electronics-15-01323)) represents a classical symbol detection step rather than a quantum state reconstruction process. Specifically, the equation implements a nearest-neighbor decision rule that estimates the transmitted symbol index by comparing the received noisy phase value with the set of valid constellation points defined by the QFT structure. This operation is analogous to maximum-likelihood detection commonly used in classical communication systems.
The quantum operations in the proposed framework occur during the state preparation and transformation stages. At the transmitter, the computational basis state corresponding to the classical symbol index is transformed using the QFT, producing a structured quantum superposition whose phase relationships encode the transmitted symbol. After symbol detection at the receiver, the corresponding quantum state is deterministically reconstructed based on the known QFT structure, and the IQFT is applied as a unitary quantum operation to recover the original computational basis state. Therefore, the proposed receiver follows a hybrid quantum–classical processing model in which classical decision logic is used for symbol detection, while the QFT and IQFT operations constitute the quantum transformations responsible for encoding and decoding the transmitted information.
In general, transmitting a single coefficient of a quantum state would not be sufficient to reconstruct an arbitrary n\-qubit quantum state, since a general state is described by
2
n
complex amplitudes. However, the proposed framework operates under a constrained state preparation model in which the input states to the QFT are computational basis states generated deterministically from classical bitstreams. Each block of n classical bits is mapped to a basis state
\|
j
⟩
, where
j
∈
{
0
,
1
,
…
,
2
n
−
1
}
. When the QFT is applied to a computational basis state, the resulting frequency-domain vector corresponds to a specific column of the QFT matrix with a deterministic phase structure. Since the encoder and decoder both know this finite set of possible QFT columns, a single transmitted coefficient can uniquely identify the corresponding column index j. Once this index is determined at the receiver, the complete frequency-domain representation can be regenerated deterministically, and the inverse QFT can be applied to recover the original computational basis state. Therefore, the proposed approach does not attempt to reconstruct an arbitrary quantum state from a single coefficient; instead, the transmitted coefficient acts as a symbol that identifies one element from a predefined set of structured QFT states.
### 3\.3. Quantum Communication Channel
To simulate realistic quantum transmission conditions, the proposed system considers several standard quantum noise mechanisms \[50\]. Five primary types of quantum noise are included: bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping. Each noise channel is associated with a probability parameter that determines the likelihood of an error occurring during transmission. The effect of hardware imperfections is not included, as these abstract noise models are sufficient for analyzing early-stage quantum communication performance.
#### Composite Noise Model
The combined influence of all quantum noise sources is represented by a composite quantum channel, denoted as
C
(
ρ
)
, where
ρ
is the density matrix of the transmitted quantum state. This channel combines the effects of bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping errors, as defined in Equation ([26](https://www.mdpi.com/2079-9292/15/6/1323#FD26-electronics-15-01323)). In practical quantum communication systems, noise rarely occurs as a single isolated process. Instead, multiple decoherence mechanisms typically act simultaneously due to environmental interactions, imperfect control operations, and physical device limitations. For example, superconducting qubits may experience energy relaxation and dephasing concurrently, while photonic communication channels may be affected by loss, depolarization, and phase fluctuations. Therefore, representing the channel as a probabilistic combination of several elementary noise processes provides a more realistic abstraction of practical quantum transmission conditions. The adopted composite noise model enables the proposed framework to capture the simultaneous influence of multiple error mechanisms and to evaluate system robustness under diverse channel conditions rather than assuming a single dominant noise source.
C
(
ρ
)
\=
(
1
−
p
tot
)
ρ
\+
p
B
B
(
ρ
)
\+
p
P
P
(
ρ
)
\+
p
D
D
(
ρ
)
\+
p
A
A
(
ρ
)
\+
p
Φ
F
(
ρ
)
(26)
where:
- p
B
,
p
P
,
p
D
,
p
A
,
p
Φ
∈
\[
0
,
1
\]
are the probabilities of bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping errors, respectively.
- B
(
·
)
,
P
(
·
)
,
D
(
·
)
,
A
(
·
)
,
F
(
·
)
denote the corresponding quantum channels implementing these error processes.
The total noise probability
p
tot
is dynamically determined as a function of the system SNR to reflect realistic channel conditions. It is modeled as in Equation ([27](https://www.mdpi.com/2079-9292/15/6/1323#FD27-electronics-15-01323)).
p
tot
\=
min
1
,
1
1
\+
10
SNR
/
10
(27)
where SNR is expressed in dB. This ensures that the overall error probability decreases with increasing SNR. It should be noted that the SNR used in Equation ([27](https://www.mdpi.com/2079-9292/15/6/1323#FD27-electronics-15-01323)) is not intended to represent a fundamental quantum-mechanical parameter. In physical quantum systems, noise processes are typically described using quantum channels characterized by Kraus operators or completely positive trace-preserving (CPTP) maps, and their parameters depend on specific physical mechanisms rather than directly on an SNR value. In the proposed framework, the SNR parameter is used as an abstract simulation-level control variable to regulate the overall severity of channel noise. The mapping defined in Equation ([27](https://www.mdpi.com/2079-9292/15/6/1323#FD27-electronics-15-01323)) therefore determines the total noise probability
p
tot
, which is subsequently distributed among several standard quantum noise channels, including bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping processes. In this way, the SNR parameter provides a convenient mechanism for evaluating system performance across different channel conditions while the underlying noise evolution remains governed by physically consistent quantum channel models.
To allocate
p
tot
among the five individual noise channels, independent random weights
w
i
are drawn from a uniform distribution, as shown in Equation ([28](https://www.mdpi.com/2079-9292/15/6/1323#FD28-electronics-15-01323)).
w
1
,
w
2
,
w
3
,
w
4
,
w
5
∼
U
(
0
,
1
)
(28)
These weights are then normalized to compute the individual channel probabilities, as in Equation ([29](https://www.mdpi.com/2079-9292/15/6/1323#FD29-electronics-15-01323)).
\[
p
B
,
p
P
,
p
D
,
p
A
,
p
Φ
\]
\=
p
tot
∑
i
\=
1
5
w
i
·
\[
w
1
,
w
2
,
w
3
,
w
4
,
w
5
\]
(29)
This normalization guarantees that the sum of the individual channel probabilities equals the total noise probability, as expressed in Equation ([30](https://www.mdpi.com/2079-9292/15/6/1323#FD30-electronics-15-01323)).
p
B
\+
p
P
\+
p
D
\+
p
A
\+
p
Φ
\=
p
tot
(30)
Each individual noise channel represents a physically meaningful type of quantum error, as described in the following.
- Bit-flip noise (
B
): Models the process where a qubit randomly flips from
\|
0
⟩
to
\|
1
⟩
or vice versa, similar to a classical binary error. Mathematically, this is represented in Equation ([31](https://www.mdpi.com/2079-9292/15/6/1323#FD31-electronics-15-01323)).
B
(
ρ
)
\=
(
1
−
p
B
)
ρ
\+
p
B
X
ρ
X
†
(31)
Here, X is the Pauli-X operator (bit-flip gate).
- Phase-flip noise (
P
): Represents a phase error that flips the relative sign of
\|
1
⟩
with respect to
\|
0
⟩
, leaving populations unchanged, as shown in Equation ([32](https://www.mdpi.com/2079-9292/15/6/1323#FD32-electronics-15-01323)). This is crucial in coherent superposition states.
P
(
ρ
)
\=
(
1
−
p
P
)
ρ
\+
p
P
Z
ρ
Z
†
(32)
Z is the Pauli-Z operator, which inverts the phase of
\|
1
⟩
.
- Depolarizing noise (
D
): Simulates isotropic errors that randomly apply any Pauli operator
X
,
Y
,
Z
with equal probability, driving the qubit toward a maximally mixed state, as shown in Equation ([33](https://www.mdpi.com/2079-9292/15/6/1323#FD33-electronics-15-01323)).
D
(
ρ
)
\=
(
1
−
p
D
)
ρ
\+
p
D
3
(
X
ρ
X
†
\+
Y
ρ
Y
†
\+
Z
ρ
Z
†
)
(33)
This models decoherence that affects both bit and phase simultaneously.
- Amplitude damping (
A
): Represents energy loss mechanisms, such as spontaneous emission in optical or superconducting qubits. It models the relaxation from
\|
1
⟩
to
\|
0
⟩
, as defined in Equation ([34](https://www.mdpi.com/2079-9292/15/6/1323#FD34-electronics-15-01323)).
A
(
ρ
)
\=
E
0
ρ
E
0
†
\+
E
1
ρ
E
1
†
,
E
0
\=
1
0
0
1
−
p
A
,
E
1
\=
0
p
A
0
0
(34)
- Phase damping (
F
): Models pure dephasing without energy loss, which reduces off-diagonal elements of the density matrix in the computational basis, as defined in Equation ([35](https://www.mdpi.com/2079-9292/15/6/1323#FD35-electronics-15-01323)).
F
(
ρ
)
\=
F
0
ρ
F
0
†
\+
F
1
ρ
F
1
†
,
F
0
\=
1
0
0
1
−
p
Φ
,
F
1
\=
0
0
0
p
Φ
(35)
By combining these channels probabilistically, the framework provides a flexible and physically meaningful model for testing quantum communication systems under realistic noisy conditions. SNR is mapped to the total noise probability
p
tot
to adjust error severity according to channel quality \[15,19\].
### 3\.4. Physical Realization of QFT Using Quantum Gates
In real quantum circuit implementations, the QFT is constructed using a sequence of single-qubit Hadamard operations together with controlled phase rotation gates. To illustrate this structure, a QFT circuit consisting of three qubits is presented in [Figure 4](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f004).
**Figure 4.** Quantum circuit for a three-qubit QFT.
The Hadamard gate \[51\], represented by H, is a key quantum logic gate that converts computational basis states into superposition states. The matrix form of this operation is provided in Equation ([36](https://www.mdpi.com/2079-9292/15/6/1323#FD36-electronics-15-01323)).
H
\=
1
2
1
1
1
−
1
(36)
When applied to a single qubit, it performs the transformations given in Equations ([37](https://www.mdpi.com/2079-9292/15/6/1323#FD37-electronics-15-01323)) and ([38](https://www.mdpi.com/2079-9292/15/6/1323#FD38-electronics-15-01323)).
H
\|
0
⟩
\=
1
2
(
\|
0
⟩
\+
\|
1
⟩
)
\=
\|
\+
⟩
(37)
H
\|
1
⟩
\=
1
2
(
\|
0
⟩
−
\|
1
⟩
)
\=
\|
−
⟩
(38)
Here,
\|
\+
⟩
and
\|
−
⟩
represent the superposition states along the X\-axis of the Bloch sphere.
Phase gates introduce controlled rotations, which are crucial for encoding the relative phases between qubits in QFT. The general single-qubit phase gate is written as in Equation ([39](https://www.mdpi.com/2079-9292/15/6/1323#FD39-electronics-15-01323)).
P
(
ϕ
)
\=
1
0
0
e
i
ϕ
(39)
For QFT, the rotation angle
ϕ
is determined by the qubit positions:
ϕ
\=
2
π
/
2
k
, where k is the distance between the control and target qubits in a controlled-phase operation.
#### Three-Qubit QFT Procedure
Let the input state be
\|
x
1
x
2
x
3
⟩
, where
x
1
is the most significant qubit. The QFT is performed as follows:
- Apply a Hadamard gate to the first qubit
x
1
.
- Apply a controlled-
R
2
gate between
x
1
and
x
2
, where the
R
2
phase gate is defined in Equation ([40](https://www.mdpi.com/2079-9292/15/6/1323#FD40-electronics-15-01323)).
R
2
\=
1
0
0
e
i
π
/
2
(40)
- Apply a controlled-
R
3
gate between
x
1
and
x
3
, where the
R
3
phase gate is defined in Equation ([41](https://www.mdpi.com/2079-9292/15/6/1323#FD41-electronics-15-01323)).
R
3
\=
1
0
0
e
i
π
/
4
(41)
- Perform a Hadamard gate on
x
2
, followed by a controlled-
R
2
gate between
x
2
and
x
3
.
- Apply a Hadamard gate to
x
3
.
- Swap qubits
x
1
↔
x
3
to correct the qubit order in the computational basis.
The resulting three-qubit QFT state can be expressed as in Equation ([42](https://www.mdpi.com/2079-9292/15/6/1323#FD42-electronics-15-01323)).
QFT
(
\|
x
1
x
2
x
3
⟩
)
\=
1
8
\|
0
⟩
\+
e
2
π
i
0
.
x
3
\|
1
⟩
⊗
\|
0
⟩
\+
e
2
π
i
0
.
x
2
x
3
\|
1
⟩
⊗
\|
0
⟩
\+
e
2
π
i
0
.
x
1
x
2
x
3
\|
1
⟩
(42)
This formulation demonstrates that each qubit sequentially accumulates phase contributions from less significant qubits, effectively encoding the input into the frequency domain. The output of this circuit matches the theoretical QFT computations for all basis states, confirming the correctness of the implementation \[52\]. Similarly, this physical implementation can be generalized to accommodate arbitrary qubit sizes.
### 3\.5. Quantum and Classical Components of the Framework
The proposed system follows a hybrid quantum–classical architecture. The genuinely quantum operations in the framework include: (i) preparation of the n\-qubit computational basis state corresponding to each data segment, (ii) application of the QFT using quantum gate circuits composed of Hadamard and controlled phase-rotation gates, (iii) propagation of the quantum state through the quantum communication channel modeled by quantum noise processes such as bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping channels, (iv) application of the IQFT at the receiver, and (v) measurement of the qubits in the computational basis to obtain classical outcomes.
The classical components of the framework include preprocessing and post-processing stages. Specifically, the video data is first compressed using the classical VVC encoder, and the resulting bitstream is segmented into n\-bit blocks before quantum encoding. In scenarios where uncompressed video is used, the raw pixel data is directly converted into a binary bitstream and segmented into n\-bit blocks without applying source compression. After quantum measurement at the receiver, the recovered binary data is concatenated and decoded using the classical VVC decoder to reconstruct the video frames when compressed inputs are used. For uncompressed inputs, the recovered bitstream is directly mapped back to the corresponding pixel representation. In addition, classical detection procedures such as nearest-neighbor symbol estimation are used to determine the most likely transmitted index from the received noisy observations. Although the system is evaluated through numerical simulation, the evolution of the quantum states is modeled according to the standard formalism of quantum mechanics, including unitary QFT/IQFT operations and quantum channel noise represented by completely positive trace-preserving maps
### 3\.6. Performance Evaluation Methodology
The performance of the proposed system is evaluated across multiple scenarios to assess compression efficiency, error resilience, and robustness under different channel conditions. The evaluation framework is described as follows:
- Uncompressed video inputs: Multi-qubit configurations with
n
\=
1
to 8 are tested using raw video sequences. These experiments examine how the qubit dimension affects the compression ratio and robustness to quantum channel noise. Results are reported for both cases without and with classical channel coding.
- Compressed video inputs: To study transmission efficiency improvements, multi-qubit systems (
n
\=
1
to 8) are applied to VVC-compressed video sequences. Two GOP structures, 8 and 32, are considered. Performance is evaluated both with and without classical channel coding.
- Performance of QFT encoding without compression: The QFT-based full-vector transmission, which sends the complete quantum state without compression \[19\], is compared against alternative methods under the same bandwidth constraints. These alternatives include a time-domain Hadamard-based multi-qubit system with qubit sizes
n
\=
1
to 8, and a classical system using binary phase-shift keying (BPSK). In the Hadamard-based approach, the multi-qubit encoding matrix is generated via the tensor product of the Hadamard matrix defined in Equation ([36](https://www.mdpi.com/2079-9292/15/6/1323#FD36-electronics-15-01323)), and decoding is performed using the inverse Hadamard transform to recover the transmitted quantum state \[32\].
In the proposed framework, the single-qubit configuration is considered the reference transmission system. A single qubit can represent two computational basis states,
\|
0
⟩
and
\|
1
⟩
, which correspond to a two-point constellation. This is directly comparable to the two-symbol constellation used in a classical BPSK modulation scheme. Therefore, BPSK provides a natural classical counterpart for evaluating the transmission behavior of the QFT-based frequency-domain representation under equivalent binary symbol conditions.
To maintain equivalent bandwidth usage across all qubit configurations, the input bitstream is compressed using different quantization parameter (QP) settings in the VVC encoder. This produces progressively shorter bitstreams as the qubit grouping size increases. As a result, the total amount of transmitted information remains comparable across different configurations, allowing the impact of the QFT-based transmission process to be evaluated without introducing bandwidth bias.
### 3\.7. Simulation Configuration
The proposed system is evaluated using a numerical simulation framework that implements the complete transmission and reconstruction pipeline described in the previous subsections. The results are obtained using a Monte Carlo simulation framework to evaluate system performance under stochastic noise conditions. For each video sequence and each SNR value, the complete transmission and reconstruction process is repeated over 1000 independent trials. In each trial, a new realization of the composite quantum noise channel is generated by drawing random weights for the individual noise processes from a uniform distribution and normalizing them to satisfy the total noise probability constraint. The transmitted states are propagated through the resulting channel, and the reconstructed video quality is evaluated using bit error rate (BER), peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM) \[53\], and video multi-method assessment fusion (VMAF) \[54\] metrics. The final curves shown in the figures correspond to the average performance obtained across these 1000 trials, which explains the smooth appearance of the plotted results despite the underlying stochastic noise model. The key parameters and implementation settings used in the experiments are summarized in [Table 1](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-t001).
**Table 1.** Experimental Setup Parameters.
## 4\. Results and Discussion
In this section, we present and analyze the performance of the proposed QFT-based video compression and transmission system under varying quantum channel conditions. The evaluation examines how the system handles videos with different spatial and temporal complexities, providing insight into its robustness and effectiveness. Quantitative results are reported using BER, PSNR, SSIM, and VMAF. Together, these metrics evaluate reconstruction fidelity, perceptual quality, and temporal consistency, offering a comprehensive view of system performance. The discussion highlights trends observed in average results across multiple test videos, emphasizing the effects of motion complexity, channel noise, and compression on video quality, while summarizing overall system behavior through aggregated performance metrics.
Each comparison scenario introduced in [Section 3.6](https://www.mdpi.com/2079-9292/15/6/1323#sec3dot6-electronics-15-01323) is explained in detail in the following subsections.
### 4\.1. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for Uncompressed Video Inputs Using Multi-Qubit Configurations Without Channel Coding
The results of the proposed QFT-based video compression and transmission system, evaluated without channel coding, is summarized in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005). The figure illustrates the system’s behavior across four key metrics, BER, PSNR, SSIM, and VMAF, as a function of the channel SNR for different qubit encoding sizes (F1 to F8, corresponding to n = 1 to n = 8 qubits). The BER results in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)a reveal a critical trade-off governed by the qubit encoding size, n. In this system, a block of video data is mapped via the QFT into a
2
n
\-dimensional frequency vector. The unitary nature of the QFT enables powerful, exponential compression; an n\-qubit encoding achieves a theoretical compression ratio of
2
n
:
1
, as only the dominant frequency coefficient needs to be transmitted to reconstruct the original block. However, this compression gain comes at the cost of increased susceptibility to noise. The angular separation between adjacent frequency components decreases exponentially with n, as given by Equation ([43](https://www.mdpi.com/2079-9292/15/6/1323#FD43-electronics-15-01323)). A smaller
Δ
θ
causes the constellation of frequency states to become denser, making it more difficult for the receiver to discriminate between them in the presence of phase noise induced by the channel.
Δ
θ
\=
2
π
2
n
(43)
**Figure 5.** Performance proposed QFT-based video compression and transmission system for the uncompressed video inputs without channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
Consequently, while the F8 configuration (
n
\=
8
) achieves a very high 256:1 compression ratio, it exhibits the highest BER across all SNR levels, as shown in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)a. In contrast, the F2 configuration (
n
\=
2
) benefits from a more robust angular separation of
π
/
2
radians and a moderate 4:1 compression ratio, resulting in superior error resilience. The F1 configuration, although offering the highest error tolerance, provides only minimal compression (2:1). Overall, these results indicate that intermediate encoding sizes (e.g., F2–F4) offer the most practical operating point, achieving a favorable trade-off between substantial compression and acceptable error rates under typical noisy channel conditions.
The video reconstruction quality metrics, PSNR, SSIM, and VMAF, shown in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)b, [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)c, and [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)d, respectively, follow consistent trends. As expected, all metrics generally improve with increasing SNR, as a cleaner channel allows for more accurate reconstruction of the transmitted video data. The performance hierarchy across different encoding sizes mirrors the findings from the BER analysis. Systems with lower qubit counts (F1, F2) consistently achieve higher PSNR, SSIM, and VMAF scores at a given SNR due to their inherent noise resilience. The superior performance of F2, in particular, confirms its optimal balance between compression efficiency and reconstruction quality.
An interesting observation is the behavior of the F8 configuration. While its BER is significantly higher, its PSNR does not degrade as catastrophically as one might expect. This can be attributed to the nature of the QFT/IQFT process and the structure of video data. Channel noise primarily perturbs the finer, less significant details in the frequency domain. During the inverse QFT, these errors are distributed across the pixel block. Furthermore, natural video has significant spatial and temporal correlation, allowing adjacent pixels to mask these distributed errors. As a result, the overall structural integrity and perceptual quality, as captured by PSNR, SSIM and VMAF, are preserved to a greater degree than the raw BER might suggest. However, the F8 system requires a very high SNRs to achieve the highest quality levels that lower-qubit systems can achieve at moderate SNRs.
Therefore, the results in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005) highlight a fundamental trade-off in the proposed system: higher qubit encoding sizes achieve greater compression but are more susceptible to channel noise. Under realistic noisy channel conditions, smaller or intermediate qubit configurations, such as F2 to F4, provide a more favorable balance between compression and error resilience, ensuring reliable video reconstruction.
### 4\.2. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for Uncompressed Video Inputs Using Multi-Qubit Configurations with Channel Coding
The introduction of channel coding dramatically alters the performance landscape of the proposed system, as illustrated in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006). When compared to the uncoded results in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005), the application of forward error correction provides a significant coding gain, effectively shifting all performance curves to the left and enabling reliable operation at much lower SNR levels. The BER characteristics, depicted in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)a, demonstrate the substantial channel coding gain achieved through forward error correction. Although the trade-off between qubit encoding size and error susceptibility persists, the inclusion of channel coding substantially reduces the absolute BER across all SNR levels relative to the uncoded scenario. The channel coding effectively compensates for the inherent vulnerability of larger encoding sizes, enabling their practical deployment in noisy environments where they would otherwise be unusable. The error correction manifests most prominently in the intermediate SNR regime (10–25 dB), where the coding gain provides the greatest marginal benefit. In this region, the BER curves exhibit the characteristic steep descent associated with effective coding schemes, transitioning rapidly from high to low error probability as SNR increases.
**Figure 6.** Performance proposed QFT video compression and transmission system for the uncompressed video inputs with channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
The PSNR results in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)b reveal a significant qualitative shift in system behavior compared to the uncoded scenario. Channel coding eliminates the anomalous performance previously observed with F8, establishing a consistent monotonic relationship between encoding dimension and reconstruction fidelity. The F8, which demonstrated unexpected resilience in the uncoded system due to its direct pixel mapping characteristics, now exhibits the most pronounced sensitivity to channel impairments. This normalization of behavior arises because the error correction process fundamentally alters how quantization errors propagate through the system. The coding redundancy, while essential for error protection, disrupts the spatial correlation properties that previously provided natural error masking for certain encoding configurations. Consequently, the theoretical vulnerability of high-dimensional encodings to angular separation constraints becomes fully manifested in the channel coded system.
The SSIM metrics, presented in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)c, corroborate the PSNR findings while providing additional insights into structural preservation. Channel coding maintains superior structural similarity across all encoding sizes, particularly in the critical mid-range SNR conditions where perceptual quality is most vulnerable. However, the progressive degradation with increasing encoding size remains evident, confirming that while channel coding improves absolute performance, it does not alter the fundamental compression-robustness trade-off intrinsic to the QFT encoding approach.
The VMAF results in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)d provide the most comprehensive assessment of perceptual video quality. This metric, which incorporates characteristics of the human visual system and accounts for temporal relationships across consecutive frames, reveals that channel coding substantially improves the viewing experience across all encoding sizes. Nonetheless, the hierarchical pattern remains: smaller encoding sizes (F1–F3) maintain excellent perceptual quality (VMAF
\>
80
) at moderate SNR levels, while larger encodings require progressively higher SNR to achieve comparable viewing quality. Notably, VMAF highlights the superiority of the two-qubit encoding (F2), which achieves an optimal balance between compression efficiency and perceptual quality preservation.
The comparative analysis across all four metrics provides crucial insights for system optimization. The single-qubit configuration demonstrates the highest robustness, maintaining reliable performance under noisy conditions while achieving a 2:1 compression ratio. However, the two-qubit system emerges as the optimal compromise, delivering a favorable 4:1 compression ratio while requiring only modest SNR to maintain acceptable quality across diverse channel conditions, as consistently reflected in its superior BER, PSNR, SSIM, and VMAF performance. In contrast, higher-qubit configurations (e.g., F4–F8) offer greater compression but demand significantly higher SNR to achieve comparable quality, highlighting the trade-off between compression efficiency and error resilience.
### 4\.3. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for VVC Encoded Video Inputs Using Multi-Qubit Configurations Without Channel Coding
The performance of the system with pre-compressed VVC input using GOP 8 exhibits a notably different behavior compared to uncompressed video, as illustrated in [Figure 7](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f007). All three quality metrics, PSNR, SSIM, and VMAF, follow a consistent pattern across different qubit encoding sizes. The smaller qubit encodings (F1–F3) maintain superior performance throughout the SNR range, while larger encodings (F6–F8) show significant degradation, particularly at low to moderate SNR levels. The PSNR results in [Figure 7](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f007)a indicate that VVC-compressed inputs are more susceptible to quality degradation under channel noise than uncompressed content. The SSIM and VMAF metrics, shown in [Figure 7](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f007)b,c, reveal similar overall trends, albeit with different sensitivity characteristics. SSIM values for smaller encodings remain above 0.8 for SNR \> 10 dB, indicating good structural preservation. However, larger encodings struggle to maintain acceptable structural similarity, with the F8 configuration performing the worst under channel noise. VMAF scores show the most pronounced separation between encoding sizes. The F1 configuration maintains excellent perceptual quality (VMAF \> 80) across most of the SNR range, whereas F8 only exceeds a VMAF of 40 at around 40 dB channel SNR, indicating a poor viewing experience across nearly all channel conditions.
**Figure 7.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
The results clearly indicate that VVC-encoded content is less tolerant to the additional compression introduced by larger qubit encodings compared to uncompressed inputs. Moreover, the combination of VVC compression artifacts and the noise sensitivity of high-dimensional QFT encodings produces a compounding effect that significantly degrades reconstruction quality. For practical systems using pre-compressed video content, smaller qubit encodings (F1–F2) are essential to maintain acceptable quality. The theoretical compression benefits of larger encodings are negated by the severe quality degradation.
Based on the results shown in [Figure 8](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f008), which depicts the system performance with a GOP size of 32 and without channel coding, the effect of larger GOP sizes on video quality is clearly observed. The results for PSNR, SSIM, and VMAF consistently show that all configurations (F1 to F8) exhibit reduced quality compared to scenarios with GOP size 8. As the GOP size increases to 32, the number of inter-coded frames rises, leading to greater reliance on predictive coding. Errors introduced in early frames within the GOP propagate through subsequent inter-frames, amplifying quality loss. This propagation effect is evident in [Figure 8](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f008), where even robust encodings such as F1 and F2 exhibit lower PSNR, SSIM, and VMAF scores compared to the corresponding results for GOP size 8. To mitigate this, adaptive error correction strategies tailored to the GOP structure, such as strengthening protection for key frames or using error-resilient encoding techniques, could be employed to reduce error propagation and enhance overall performance.
**Figure 8.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP32) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
### 4\.4. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for VVC Encoded Video Inputs Using Multi-Qubit Configurations with Channel Coding
The results for VVC-encoded video (GOP size 8, shown here as an example) with channel coding, presented in [Figure 9](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f009), demonstrate a significant improvement compared to uncoded VVC transmission. Channel coding effectively mitigates the error propagation issues inherent in predictive coding with GOP size 8. All three quality metrics, PSNR, SSIM, and VMAF, show substantial enhancement compared to the uncoded case. The system now maintains acceptable quality levels even at moderate SNR conditions, where previously severe degradation occurred. The hierarchical relationship between different qubit encodings persists, but with improved absolute performance. Smaller encodings (F1–F3) achieve excellent reconstruction quality, with F1 and F2 maintaining PSNR above 25 dB, SSIM above 0.8, and VMAF above 80 across most of the SNR range. Even larger encodings (F4–F6) now provide usable quality at sufficient SNR levels, though F7 and F8 still show limitations due to their sensitivity to phase noise. The combination of VVC encoding, QFT-based compression, and channel coding represents a viable operational point for practical systems. The two-qubit encoding (F2) emerges as particularly advantageous, offering a favorable 4:1 compression ratio while maintaining robust performance across all quality metrics. This configuration balances compression efficiency with error resilience, making it suitable for bandwidth-constrained applications requiring reliable video transmission.
**Figure 9.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
### 4\.5. Performance of Frequency-Domain Encoding Without Compression for VVC-Encoded Inputs with Channel Coding
Based on the results shown in [Figure 10](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f010) for VVC-encoded videos, a clear performance hierarchy emerges between the quantum-inspired encoding systems operating without additional compression and the classical baseline approach. Both the QFT-based frequency-domain system (F1–F8) and the time-domain Hadamard system (T1–T8) consistently outperform the bandwidth equivalent classical system (C) across all quality metrics, PSNR, SSIM, and VMAF. Notably, unlike the scenarios involving QFT-based compression where larger qubit sizes degrade performance, in this case increasing the qubit encoding size (n) actually improves reconstruction quality for both quantum-inspired systems. This improvement occurs because, in this scenario, no QFT-based compression is applied and the full quantum state vector is transmitted. As a result, the video does not contain the artifacts and inter-frame dependencies that typically amplify transmission errors in compressed content. As a result, the superior noise resilience and representational capacity of higher-dimensional quantum encodings become evident. Larger qubit configurations are able to capture and represent more complex quantum states, allowing them to encode greater amounts of information per transmitted symbol. This leads to improved reconstruction fidelity and robustness against channel noise, which is why larger-qubit configurations such as F7/F8 and T7/T8 consistently outperform smaller qubit systems.
**Figure 10.** Performance of frequency-domain encoding without compression for VVC-encoded inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
In particular, the QFT-based encoding system benefits from high-dimensional frequency-domain representations in Hilbert space, utilizing both amplitude and phase encoding to capture intricate correlations within the video data. In contrast, the Hadamard-based encoding operates in the time domain, relying on an orthogonal basis structure with amplitude encoding alone. The dual encoding capability of the QFT system enables it to more accurately preserve the full quantum state and resist errors, resulting in superior performance compared to Hadamard-based systems, especially in high-dimensional qubit configurations. Consequently, the combination of larger qubit sizes and QFT-based frequency-domain encoding provides the highest reconstruction quality, demonstrating the fundamental advantages of quantum-inspired encoding for high-fidelity transmission of uncompressed video. This encoding scheme differs from the proposed method in that it does not achieve compression; instead, it transmits the full quantum state vector without reducing the data volume.
### 4\.6. Performance Assessment of the Proposed QFT-Based Compression and Transmission System Across Different Resolutions and Qubit Configurations
As shown in [Table 2](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-t002), the channel SNR gains compared to the eight-qubit encoding under the channel-coded system remain consistent across all tested video resolutions and frame rates. This indicates that the resolution and frame rate of the video do not significantly impact the performance of the QFT-based compression and transmission system. Instead, the maximum SNR gains are primarily determined by the qubit encoding size, demonstrating that the system’s error resilience and compression efficiency depend on the quantum encoding configuration rather than the specific characteristics of the video content.
**Table 2.** Maximum SNR Gains (in dB) Achieved by the QFT-Based Compression and Transmission System for Each Qubit Configuration Compared to the Eight-Qubit System Across Different Video Resolutions and Frame Rates.
An illustrative example of reconstructed video frames using the proposed QFT-based system under varying channel conditions is shown in [Figure 11](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f011). The frames span SNR levels from 38 dB down to
−
10
dB, with each subfigure labeled alphabetically for easy reference. The system employs eight-qubit encoding (F8) with a 256:1 compression ratio applied to uncompressed video inputs, demonstrating the high compression efficiency of the proposed QFT-based video compression and transmission framework. At higher SNR levels, the reconstructed frames closely preserve the original visual details, indicating effective retention of critical image information. As the SNR decreases, visual degradations such as blurring and minor artifacts become more apparent, reflecting the impact of channel noise on the transmitted quantum-encoded data. Despite these distortions, the system demonstrates graceful degradation, highlighting the robustness of the QFT-based video compression and transmission system with eight-qubit encoding against channel impairments. This illustrative example complements quantitative evaluations using PSNR, SSIM, and VMAF, providing a clear visual demonstration of the proposed method’s performance across a wide range of channel conditions.
**Figure 11.** Reconstructed video frames at SNR levels from 38 dB to
−
10
dB for the QFT-based video compression and transmission system with eight-qubit encoding (F8) and 256:1 compression of uncompressed video inputs.
### 4\.7. Performance Evaluation Compared to Classical Communication Systems
To contextualize the performance of the proposed framework in a video transmission setting, we compare it with representative classical baselines, as illustrated in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012) for the two-qubit quantum system. QPSK serves as a modulation-level reference with comparable bandwidth, while both HEVC and VVC represent widely used state-of-the-art classical video compression standards. All systems are evaluated under equivalent bitrate constraints to ensure fair bandwidth utilization.
**Figure 12.** Comparison of the proposed system with classical communication systems: (**a**) Peak Signal-to-Noise Ratio (PSNR), (**b**) Structural Similarity Index Measure (SSIM).
The QFT-based system enables frequency-domain video transmission by encoding frame-level information into quantum states, allowing efficient representation of spatial content. Even without channel coding, the proposed approach achieves performance comparable to uncompressed QPSK transmission while benefiting from intrinsic compression (4:1) due to the quantum state representation. When polar coding with a rate of 1/4 is applied, the QFT-based system significantly improves robustness against channel noise, achieving substantial PSNR gains (up to 10 dB) over uncompressed QPSK transmission, as shown in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012)a, under the same bitrate.
Unlike classical video compression methods such as HEVC and VVC, which rely on predictive coding and inter-frame dependencies, the proposed QFT-based framework demonstrates a more gradual degradation in the presence of channel noise. This behavior is evident in the PSNR trends in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012)a, where the QFT-coded system maintains high reconstruction quality at lower SNR values compared to both uncoded QPSK and classical codec-based transmission schemes. Similarly, the SSIM results in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012)b confirm that the proposed method preserves structural information more effectively under channel impairments.
While HEVC and VVC provide strong rate–distortion performance under ideal channel conditions, they rely on inherently lossy compression mechanisms and predictive coding structures that make them highly sensitive to transmission errors. Even small bitstream corruptions can propagate across multiple frames due to inter-frame dependencies, leading to abrupt visual degradation. In contrast, the proposed QFT-based framework operates on independently encoded frequency-domain representations, which reduces error propagation and enables more stable reconstruction quality in noisy channels. Furthermore, since the QFT and its inverse are unitary transformations, the proposed system can theoretically achieve lossless reconstruction under ideal channel conditions.
Although classical codecs such as HEVC and VVC benefit from mature implementations and highly optimized compression efficiency in error-free environments, the proposed quantum-inspired framework offers a promising alternative for video transmission in bandwidth-limited and error-prone channels, providing improved robustness and graceful degradation characteristics.
### 4\.8. Advantages and Practical Implications of the Proposed QFT-Based Symbol-Level Compression and Transmission Framework
It is important to emphasize that the proposed method introduces quantum domain symbol-level compression. In this framework, compression occurs during quantum encoding by selectively transmitting a reduced set of QFT-domain coefficients. This process decreases the transmission payload but does not reduce the file size of the original video for storage. Unlike classical codecs, which replace the stored representation with a compressed version, the proposed QFT-based approach requires access to the full quantum state vector for decoding. As such, the method provides compression strictly within the communication pipeline, not for persistent storage.
The mathematical foundation of the QFT ensures theoretically perfect reconstruction (lossless) under ideal noise-free conditions. Since the QFT and its inverse (IQFT) are unitary operations, a computational basis state can be perfectly recovered when no channel noise, decoherence, or measurement errors are present. Therefore, the lossless property represents the theoretical upper bound of the system rather than a guarantee of lossless performance in practical quantum communication environments. Under practical noisy channel conditions, however, the system exhibits graceful degradation: increasing the qubit dimension reduces the angular separation between encoded frequency components, making higher-dimensional states more susceptible to noise. Importantly, this controlled degradation profile offers system designers well-defined trade-off parameters between compression efficiency and noise resilience, providing a level of tunability not typically available in conventional communication systems.
Based on the comprehensive evaluation of the proposed QFT-based compression and transmission system, several key advantages emerge that highlight its potential for practical video transmission applications. The system exhibits strong adaptability, allowing the optimal qubit encoding size to be selected according to specific operational requirements and channel conditions. Smaller encoding configurations (
n
\=
1
–2 qubits) excel in low-SNR scenarios, providing robust and reliable performance that is well suited for mission-critical applications where transmission reliability outweighs compression efficiency. Medium-sized encodings (
n
\=
3
–5 qubits) offer an effective balance between compression gains and robustness, making them appropriate for typical multimedia transmission environments. Larger encoding sizes (
n
\=
6
–8 qubits) achieve substantial compression ratios and are most beneficial in high-SNR channels or in scenarios where bandwidth conservation is a primary objective.
### 4\.9. Channel Use, Resource Cost, and Noise Sensitivity
In the proposed framework, the encoded video bitstream is segmented into blocks containing n classical bits, as described in the encoding stage. Each block is mapped to the computational basis of an n\-qubit register and processed using the QFT to obtain a frequency-domain representation of the encoded data.
A single representative coefficient from the QFT representation is selected and encoded into a transmitted quantum state. Therefore, a channel use in the proposed system is defined as the transmission of one encoded quantum state corresponding to a QFT-transformed data block. Under this definition, each segmented block requires one channel use. Increasing the qubit grouping size n increases the number of classical bits represented by each block and therefore reduces the number of channel uses required to transmit the entire bitstream.
From an information-theoretic perspective, the channel capacity can be related to the classical Shannon capacity formula, as shown in Equation ([44](https://www.mdpi.com/2079-9292/15/6/1323#FD44-electronics-15-01323)).
C
raw
\=
B
log
2
(
1
\+
SNR
)
(44)
In Equation ([44](https://www.mdpi.com/2079-9292/15/6/1323#FD44-electronics-15-01323)),
C
raw
denotes the raw channel capacity in bits per second, B represents the channel bandwidth in Hertz, and SNR is the signal-to-noise ratio.
Because the proposed QFT-based representation transmits only one coefficient out of the
2
n
possible frequency components produced by the QFT, the effective transmitted information rate can be expressed as in Equation ([45](https://www.mdpi.com/2079-9292/15/6/1323#FD45-electronics-15-01323)).
C
eff
(
n
)
\=
C
raw
2
n
(45)
In Equation ([45](https://www.mdpi.com/2079-9292/15/6/1323#FD45-electronics-15-01323)),
C
eff
(
n
)
represents the effective channel capacity associated with the transmitted coefficient when an n\-qubit encoding is used, and
2
n
denotes the number of spectral components generated by the QFT.
The transmitted resource cost can therefore be interpreted as the number of quantum states required to represent the encoded bitstream. If L denotes the total number of bits in the compressed bitstream, the number of transmitted blocks is given as in Equation ([46](https://www.mdpi.com/2079-9292/15/6/1323#FD46-electronics-15-01323)).
N
b
\=
L
n
(46)
In Equation ([46](https://www.mdpi.com/2079-9292/15/6/1323#FD46-electronics-15-01323)),
N
b
represents the number of encoded blocks generated from the bitstream and therefore corresponds directly to the total number of channel uses required for transmission.
To evaluate the robustness of the proposed framework under different channel conditions, the transmission process is analyzed across multiple SNR levels. Channel disturbances are modeled as noise perturbations applied to the transmitted quantum state before the IQFT is performed at the receiver. By analyzing system performance across different SNR values, the sensitivity of the proposed framework to channel noise can be systematically characterized.
### 4\.10. Computational Complexity and System Scalability
The computational complexity of the proposed QFT-based compression system is one of its key advantages. For an n\-qubit encoding, the total gate count scales as
O
(
n
2
)
, primarily due to the use of standard Hadamard gates and controlled-phase rotations, which are commonly supported in most quantum hardware. Circuit depth, which determines the temporal length of the computation, can be reduced to
O
(
n
)
when gates acting on independent qubits are executed in parallel; however, hardware connectivity constraints, such as linear nearest-neighbor layouts, introduce additional SWAP operations that can increase depth toward
O
(
n
2
)
. The use of widely available Hadamard and phase gates, along with approximate QFT methods that truncate small-angle rotations, ensures that both gate count and depth remain manageable while maintaining high fidelity.
Moreover, the system is scalable to higher qubit counts depending on the application requirements. Increasing the qubit dimension allows for higher compression ratios, as more information can be encoded per quantum state. However, larger qubit sizes can reduce error resilience, since each qubit becomes more susceptible to noise. Therefore, qubit count selection involves a trade-off between compression efficiency and robustness, allowing the system to be tailored to specific application scenarios.
### 4\.11. Hardware Feasibility and Resource Requirements
The proposed compression and transmission framework is compatible with practical quantum communication architectures because it relies on standard quantum operations that are widely studied and experimentally realizable. The encoding stage requires only computational basis state preparation, which can be implemented by initializing qubits in the ground state
\|
0
⟩
and applying Pauli-X gates when the corresponding classical bit equals one. The transformation stage employs the QFT, which can be implemented using Hadamard gates and controlled phase-rotation gates with polynomial circuit complexity. For an n\-qubit QFT circuit, the number of Hadamard gates is
H
\=
n
, while the number of controlled phase gates is
n
(
n
−
1
)
2
, resulting in an approximate total gate count of
G
(
n
)
\=
n
\+
n
(
n
−
1
)
2
. After transmission through the quantum channel, the receiver applies the IQFT, followed by computational-basis measurement to recover the classical bitstream.
From a hardware perspective, the proposed system requires relatively small quantum registers and a moderate number of gate operations. In the experiments presented in this work, the encoding size ranges from
n
\=
1
to
n
\=
8
qubits, which is well within the capabilities of current noisy intermediate-scale quantum (NISQ) devices and experimental quantum communication platforms. The required gate operations scale quadratically with the number of qubits due to the structure of the QFT circuit, while the readout stage requires only standard computational-basis measurements.
[Table 3](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-t003) summarizes the approximate hardware resources required for different qubit encoding sizes considered in this study.
**Table 3.** Estimated QFT Hardware Resources for Different Qubit Sizes.
### 4\.12. Simulation Methodology and Assumptions
The proposed quantum-inspired compression framework is evaluated through classical simulations on conventional computing platforms, specifically an Intel Core i5-1345U processor (Intel Corporation, Santa Clara, CA, USA) with 16 GB of RAM. Classical simulation is widely used in quantum communication research to analyze algorithmic behavior and system-level performance prior to implementation on physical quantum hardware. Current quantum processors remain limited in terms of qubit count, coherence time, and circuit depth, which makes large-scale multimedia transmission experiments impractical. Therefore, numerical simulation provides a controlled and reproducible environment for modeling quantum state preparation, QFT/IQFT transformations, and channel noise processes according to the standard formalism of quantum mechanics.
These simulations allow systematic evaluation of the proposed framework under different channel conditions and encoding configurations while avoiding hardware-specific constraints that may obscure algorithmic performance. The experimental evaluation considers video sequences with different motion characteristics (low, medium, and high-motion content) across multiple resolutions and frame rates, using both uncompressed and VVC-encoded inputs. The results should therefore be interpreted as a proof-of-concept validation of the proposed QFT-based video transmission framework, demonstrating its theoretical feasibility and system-level behavior rather than claiming immediate practical deployment on existing quantum hardware.
By abstracting some hardware-related limitations, this approach focuses on evaluating the conceptual feasibility of the proposed quantum communication framework before practical deployment. The simulation results provide theoretical support for the system design and help assess its potential effectiveness. As quantum technologies continue to advance, the findings from these studies can support future prototype development, experimental validation, and hardware-level testing, facilitating the gradual transition from simulation-based analysis to real-world implementation.
## 5\. Conclusions
This study presents and evaluates a novel QFT-based framework for efficient video compression and transmission over noisy communication channels. Through comprehensive analysis across different encoding parameters and channel conditions, several key insights emerge regarding the trade-offs inherent in quantum-inspired video transmission systems. The results indicate that the qubit encoding size plays a crucial role in balancing compression efficiency and robustness to channel noise. Larger qubit encodings enable significantly higher compression ratios, reaching up to 256:1, by representing more classical information within a single quantum state; however, these higher-dimensional encodings become increasingly sensitive to channel impairments due to the reduced angular separation between frequency-domain components. Conversely, smaller qubit group sizes provide stronger resilience to noise at the expense of lower compression efficiency. Under ideal noise-free channel conditions, the proposed framework achieves theoretically lossless reconstruction for all qubit sizes, since the QFT and its inverse are unitary operations that preserve the encoded information. In practical noisy environments, the optimal encoding size depends on the intended application—larger qubit configurations are advantageous when maximizing compression efficiency is the primary objective, whereas smaller encodings are preferable when transmission reliability is critical. Among the evaluated configurations, the two-qubit system provides a particularly effective compromise between compression and robustness, achieving a moderate compression ratio of 4:1 while maintaining stable performance across diverse channel conditions. These findings highlight the potential of QFT-based encoding as a flexible and robust approach for future multimedia transmission systems operating in bandwidth-constrained and error-prone environments.
Looking ahead, this work lays the foundation for several important research directions. Future studies should address practical challenges such as hardware imperfections, computational efficiency, and the development of adaptive qubit allocation strategies based on channel conditions or bandwidth constraints. Designing specialized error correction schemes tailored to quantum frequency-domain representations offers a promising approach to enhancing system reliability while maintaining high compression ratios. Additionally, the development of new compression algorithms capable of optimizing both transmission and storage efficiency could further improve overall system performance. Therefore, this research contributes to the growing field of quantum-inspired signal processing, demonstrating practical approaches for efficient communication systems that leverage quantum principles while remaining compatible with emerging quantum hardware.
## Author Contributions
Conceptualization, U.J.; methodology, U.J.; software, U.J.; validation, U.J. and A.F.; formal analysis, A.F.; investigation, A.F.; resources, U.J.; data curation, U.J.; writing—original draft preparation, U.J.; writing—review and editing, A.F.; visualization, U.J.; supervision, A.F.; project administration, A.F. All authors have read and agreed to the published version of the manuscript.
## Funding
This research received no external funding.
## Data Availability Statement
The original data presented in the study are openly available at <https://www.pexels.com> (accessed on 11 December 2025) under the Creative Commons Zero (CC0) license, which allows free use, distribution, and modification without attribution.
## Conflicts of Interest
The authors declare no conflicts of interest.
## Abbreviations
The following abbreviations are used in this manuscript:
| | |
|---|---|
| AVC | Advanced Video Coding |
| BER | Bit Error Rate |
| GOP | Group of Pictures |
| HEVC | High Efficiency Video Coding |
| IQFT | Inverse Quantum Fourier Transform |
| LDPC | Low-Density Parity Check |
| MIMO | Multi-Input Multi-Output |
| OFDM | Orthogonal Frequency-Division Multiplexing |
| PSNR | Peak Signal-to-Noise Ratio |
| QEC | Quantum Error Correction |
| QFT | Quantum Fourier Transform |
| QKD | Quantum Key Distribution |
| SI | Structural Information |
| SNR | Signal-to-Noise Ratio |
| SSIM | Structural Similarity Index Measure |
| TI | Temporal Information |
| VMAF | Video Multi-Method Assessment Fusion |
| VVC | Versatile Video Coding |
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**Figure 1.** Proposed quantum communication architecture for compressing and transmitting video data.
**Figure 1.** Proposed quantum communication architecture for compressing and transmitting video data.
**Figure 2.** Proposed quantum encoder operating in the frequency domain.
**Figure 2.** Proposed quantum encoder operating in the frequency domain.
**Figure 3.** Proposed quantum decoder.
**Figure 3.** Proposed quantum decoder.
**Figure 4.** Quantum circuit for a three-qubit QFT.
**Figure 4.** Quantum circuit for a three-qubit QFT.
**Figure 5.** Performance proposed QFT-based video compression and transmission system for the uncompressed video inputs without channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
**Figure 5.** Performance proposed QFT-based video compression and transmission system for the uncompressed video inputs without channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
**Figure 6.** Performance proposed QFT video compression and transmission system for the uncompressed video inputs with channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
**Figure 6.** Performance proposed QFT video compression and transmission system for the uncompressed video inputs with channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
**Figure 7.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 7.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 8.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP32) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 8.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP32) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 9.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 9.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 10.** Performance of frequency-domain encoding without compression for VVC-encoded inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 10.** Performance of frequency-domain encoding without compression for VVC-encoded inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
**Figure 11.** Reconstructed video frames at SNR levels from 38 dB to
−
10
dB for the QFT-based video compression and transmission system with eight-qubit encoding (F8) and 256:1 compression of uncompressed video inputs.
**Figure 11.** Reconstructed video frames at SNR levels from 38 dB to
−
10
dB for the QFT-based video compression and transmission system with eight-qubit encoding (F8) and 256:1 compression of uncompressed video inputs.
**Figure 12.** Comparison of the proposed system with classical communication systems: (**a**) Peak Signal-to-Noise Ratio (PSNR), (**b**) Structural Similarity Index Measure (SSIM).
**Figure 12.** Comparison of the proposed system with classical communication systems: (**a**) Peak Signal-to-Noise Ratio (PSNR), (**b**) Structural Similarity Index Measure (SSIM).
**Table 1.** Experimental Setup Parameters.
**Table 1.** Experimental Setup Parameters.
| Parameter | Value / Description |
|---|---|
| Video resolutions | 320 × 180, 1280 × 720, 1920 × 1080 |
| Frame rates | 20 fps, 30 fps, 50 fps |
| Video motion types | Low-motion, Medium-motion, High-motion sequences |
| Input formats | Uncompressed video and VVC-encoded bitstreams |
| Source encoder | VVC (Versatile Video Coding) |
| Qubit grouping size (n) | 1–8 qubits |
| Compression ratio | 2 n : 1 (2:1 to 256:1) |
| Quantum transform | Quantum Fourier Transform (QFT) |
| Receiver transform | Inverse Quantum Fourier Transform (IQFT) |
| Channel model | Composite quantum noise channel |
| Noise weighting | Random weighting in noise process (w i ∼ U ( 0 , 1 )) |
| SNR range | 50 dB to − 10 dB |
| SNR step size | 2 dB |
| Monte Carlo runs | 1000 runs per SNR value |
| Channel coding | Polar coding (rate 1/2) and no coding scenarios |
| Evaluation metrics | BER, PSNR, SSIM, VMAF |
| Simulation platform | MATLAB (R2023b ) and Python (v3.10)3 |
**Table 2.** Maximum SNR Gains (in dB) Achieved by the QFT-Based Compression and Transmission System for Each Qubit Configuration Compared to the Eight-Qubit System Across Different Video Resolutions and Frame Rates.
**Table 2.** Maximum SNR Gains (in dB) Achieved by the QFT-Based Compression and Transmission System for Each Qubit Configuration Compared to the Eight-Qubit System Across Different Video Resolutions and Frame Rates.
| Resolution | Frame Rate | Maximum Channel SNR Gains | | | | | | |
|---|---|---|---|---|---|---|---|---|
| | | 1 (2:1) | 2 (4:1) | 3 (8:1) | 4 (16:1) | 5 (32:1) | 6 (64:1) | 7 (128:1) |
| 320 × 180 | 20 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 320 × 180 | 30 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 320 × 180 | 50 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 1280 × 720 | 20 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 1280 × 720 | 30 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 1280 × 720 | 50 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 1920 × 1080 | 20 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 1920 × 1080 | 30 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
| 1920 × 1080 | 50 fps | 38 | 36 | 32 | 24 | 20 | 12 | 8 |
**Table 3.** Estimated QFT Hardware Resources for Different Qubit Sizes.
**Table 3.** Estimated QFT Hardware Resources for Different Qubit Sizes.
| Qubits (n) | Register Size (Qubits) | Hadamard Gates H \= n | Phase Gates n ( n − 1 ) 2 | Total Gates n \+ n ( n − 1 ) 2 |
|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 1 |
| 2 | 2 | 2 | 1 | 3 |
| 3 | 3 | 3 | 3 | 6 |
| 4 | 4 | 4 | 6 | 10 |
| 5 | 5 | 5 | 10 | 15 |
| 6 | 6 | 6 | 15 | 21 |
| 7 | 7 | 7 | 21 | 28 |
| 8 | 8 | 8 | 28 | 36 |
| | |
|---|---|
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| Readable Markdown | - Article
- 
22 March 2026
and
Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XQ, UK
\*
Author to whom correspondence should be addressed.
## Abstract
Reliable video transmission over error-prone channels remains a significant challenge due to the inherent trade-off between compression efficiency and noise resilience in conventional systems. To address these issues, this paper introduces a novel quantum Fourier transform (QFT)-based framework that integrates video compression and transmission within a unified quantum frequency-domain representation. The framework converts video data into a classical bitstream and maps it onto multi-qubit quantum states with variable encoding sizes (n), enabling flexible control over compression levels. Through the application of the QFT, these states are transformed into the frequency domain, where only selected coefficients are transmitted to reduce bandwidth requirements. At the receiver, the transmitted components are used to reconstruct the full representation, followed by inverse transformation and decoding to recover the video sequence. The performance of the proposed framework is evaluated using peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and video multi-method assessment fusion (VMAF). The results demonstrate that increasing the number of qubits enables exponential compression, achieving ratios up to 2 n : 1, while maintaining high reconstruction quality under ideal transmission conditions. However, higher-qubit configurations exhibit increased sensitivity to channel noise, leading to a more rapid degradation as the signal-to-noise ratio decreases. In contrast, lower-qubit configurations provide improved robustness, maintaining more stable reconstruction quality under noisy conditions, albeit with reduced compression efficiency. Among the evaluated configurations, the two-qubit system achieves an effective trade-off, providing a compression ratio of 4 : 1 while maintaining strong visual and structural fidelity along with enhanced resilience to channel impairments.
## 1\. Introduction
The rapid evolution of consumer electronics and intelligent multimedia systems has amplified the central role of video transmission in modern communication architectures. Applications such as ultra-high-definition (UHD) video streaming, telepresence conferencing, medical video diagnostics, augmented reality (AR) \[1\], virtual reality (VR) \[2\], autonomous navigation, cloud-based surveillance, interactive digital displays, and immersive extended-reality platforms increasingly depend on the seamless transfer of real-time video content at high fidelity. As resolutions scale from high definition (HD) to 4 K, 8 K, and beyond, alongside the growing popularity of high-frame-rate formats and multi-view video, the size and complexity of video data have grown exponentially. These advancements, while enabling more immersive user experiences, impose substantial demands on communication networks, which must handle massive data volumes under stringent requirements for latency, reliability, and robustness.
Classical video compression technologies, including advanced video coding (AVC/H.264) \[3\], high efficiency video coding (HEVC/H.265) \[4\], and the more recent versatile video coding (VVC/H.266) \[5\], achieve high compression performance by exploiting spatial and temporal redundancies through techniques such as motion-compensated prediction, transform coding, rate–distortion optimization, and entropy coding. These methods have significantly improved the efficiency of multimedia delivery over modern networks. However, the underlying framework remains constrained by the principles of classical signal processing and information theory. When aggressive compression is applied, classical codecs often produce visual distortions such as blocking artifacts, ringing patterns, motion smearing, and loss of fine texture details. Moreover, because modern video standards rely heavily on predictive coding structures such as group of pictures (GOP), errors introduced during transmission can propagate across multiple frames, causing noticeable temporal degradation.
To address these impairments, classical communication systems utilize various reliability and transmission mechanisms including adaptive bitrate streaming, channel coding \[6\], retransmission protocols, error concealment \[7\], and feature-based transmission strategies \[8,9\] that attempt to prioritize perceptually important visual information. While these approaches can improve robustness and maintain acceptable quality under moderate channel impairments, they do not fundamentally eliminate the sensitivity of compressed bitstreams to channel disturbances. In bandwidth-constrained, wireless, or long-distance communication environments, fluctuations in channel capacity, packet losses, and additive noise can still lead to significant degradation in video quality. With the growing demand for ultra-high-resolution visual content, real-time streaming, and highly reliable multimedia services, the performance of conventional compression and transmission frameworks is increasingly approaching the practical limits imposed by classical communication paradigms.
In response to these limitations, quantum communication \[10\] has emerged as a transformative paradigm that leverages intrinsic properties of quantum mechanics \[11\], such as quantum superposition \[12\], quantum entanglement \[13\], and unitary state evolution, to achieve new modalities of information representation and transmission. In contrast to classical communication, where information is represented as fixed binary sequences, quantum communication represents information using quantum states, which provide a substantially larger and more expressive representation space. Initial studies on quantum multimedia transmission have mainly focused on time-domain quantum encoding \[14,15\], where classical data are directly represented as quantum states without applying any transformation into the frequency or spectral domain. These time-domain models have demonstrated benefits in terms of enhanced state representation and improved data fidelity \[16\]. However, they remain inherently inefficient for high-dimensional video content due to the large number of qubits required to achieve high-quality reconstruction, as well as their susceptibility to accumulated noise effects, including depolarizing noise, amplitude-damping noise, phase-damping noise, and other quantum decoherence processes. Video data, with their temporal continuity and spatial variability, exhibit patterns that cannot be efficiently exploited in the time domain alone \[17\].
To respond to this, quantum Fourier transform (QFT) \[18\] offers a powerful alternative for representing quantum-encoded video data in the frequency domain. The QFT, an essential unitary transformation in quantum domain, maps quantum amplitude distributions into frequency-domain components, enabling efficient identification of redundancies and sparsity in the underlying signal structure \[19\]. By applying the QFT to quantum video data, it becomes possible to achieve intrinsic quantum-domain compression, suppress noise-sensitive spectral components, and reduce quantum resource requirements, all while maintaining the essential visual and temporal coherence necessary for high-quality video reconstruction.
Motivated by these advantages, this study introduces the first unified QFT-based framework for video compression and transmission, integrating quantum encoding, frequency-domain transformation, compression, and quantum-channel transmission into a single cohesive pipeline. Unlike prior work, which treats compression and transmission as separate processes or relies on fixed-size qubit encodings with limited adaptability, the proposed method enables flexible multi-qubit symbol encoding, where the number of qubits allocated per symbol can vary according to the desired compression ratio and available quantum resources. This design makes the framework the first to provide an adaptive quantum symbol-level compression mechanism within the frequency domain specifically for video transmission. By allowing adjustment of qubit allocation, the system can effectively balance compression efficiency, transmission robustness, and computational complexity. This makes it suitable for a wide range of video characteristics and diverse quantum channel conditions. Consequently, the proposed approach represents a significant step toward practical and high-performance quantum video communication systems, offering both high fidelity and efficient utilization of quantum resources.
The system begins by converting the input video into a classical bitstream for subsequent processing. To enhance resilience against channel-induced errors, polar coding with a code rate of 1/2 is applied to generate a channel-encoded bit sequence. After classical channel encoding, the bitstream is mapped to quantum states according to the selected qubit encoding size. This step forms a structured quantum state vector that captures both spatial and temporal characteristics of video data. The QFT is then applied to this state vector, converting it from the time domain into the quantum frequency domain. The transformation exposes sparsity, enabling selective retention of dominant frequency coefficients while discarding or compressing insignificant ones. As a result, the transmitted quantum symbol payload is significantly reduced while preserving essential visual features.
Following quantum-frequency-domain compression, the retained coefficients are transmitted over a noisy quantum communication channel. The channel introduces realistic quantum noise, modeled through quantum noise operators such as Kraus operators associated with bit-flip, phase-flip, depolarizing, amplitude-damping, and phase-damping processes. At the receiver, the system reconstructs the complete frequency-domain quantum state by appropriately restoring suppressed coefficients based on the compression ratio and encoding parameters. The inverse quantum Fourier transform (IQFT) is then applied to convert the restored frequency-domain representation back into the time domain. After measurement of the quantum state, the resulting classical bitstream is passed through the polar decoding process to correct channel-induced errors. The final step reconstructs the output video frames, restoring both the temporal continuity and structural consistency of the original sequence. Experimental evaluations demonstrate that the proposed QFT-based architecture achieves significant improvements in compression efficiency, and reconstruction fidelity compared to conventional quantum communication and classical video transmission. By evaluating multiple qubit encoding sizes, the framework reveals how quantum compression behavior and robustness vary with resource allocation, offering important insights for deploying quantum video transmission systems in future quantum networks.
### 1\.1. Key Contributions
The key contributions of this study are summarized as follows:
- A novel quantum frequency-domain approach for video compression, utilizing QFT to reduce redundancy and enhance robustness within a quantum communication framework.
- A unified architecture for quantum video transmission that integrates frequency-domain compression and noisy-channel transmission to achieve a scalable, efficient, and noise-resilient system.
- A comprehensive analysis of multi-qubit encoding configurations, demonstrating how qubit symbol size influences compression efficiency, spectral sparsity, and end-to-end video reconstruction performance.
- Extensive experimental validation demonstrating the effectiveness of the proposed framework across multiple video datasets and channel conditions, highlighting compression efficiency, robustness to quantum noise, and improved reconstructed video quality compared to classical and conventional quantum approaches.
### 1\.2. Organization of the Paper
The remainder of this paper is structured as follows: [Section 2](https://www.mdpi.com/2079-9292/15/6/1323#sec2-electronics-15-01323) reviews prior work in quantum communication, quantum multimedia processing, and QFT. [Section 3](https://www.mdpi.com/2079-9292/15/6/1323#sec3-electronics-15-01323) presents the proposed QFT–based methodology for unified video compression and transmission, detailing the multi-qubit encoding process, frequency-domain transformation, and noisy quantum channel modeling. [Section 4](https://www.mdpi.com/2079-9292/15/6/1323#sec4-electronics-15-01323) provides experimental results and comparative analysis with conventional quantum methods. Finally, [Section 5](https://www.mdpi.com/2079-9292/15/6/1323#sec5-electronics-15-01323) concludes the study by summarizing key findings, discussing the significance of the proposed framework, and outlining directions for future research.
## 3\. System Model
The system illustrated in [Figure 1](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f001) is designed to provide efficient video compression and transmission while maintaining reliable delivery of video content. It works with a variety of video sequences, supporting different resolutions and frame rates, and focuses on reducing the amount of data that needs to be transmitted without compromising the overall visual quality. Video segments are represented in a compact form, allowing for easier handling and transmission over limited-bandwidth channels. The workflow is organized to ensure that all steps, from preparing the video data to reconstructing the received frames, are seamless and coordinated. By prioritizing simplicity and efficiency, the system is able to deliver videos effectively across different scenarios while preserving clarity, continuity, and smooth playback.
**Figure 1.** Proposed quantum communication architecture for compressing and transmitting video data.
The performance of the system is assessed using a set of video sequences selected to cover a wide range of realistic conditions. Testing involves three videos with varying amounts of structural information (SI) and temporal information (TI). The high-motion video \[43\] features fast-moving objects and frequent scene transitions, leading to high SI and TI values, which places significant demands on both compression and transmission processes. The medium-motion video \[44\] features slower-paced actions and less frequent scene transitions, with intermediate SI and TI, representing content of average complexity. The low-motion video \[45\] has limited movement and low TI, while retaining some structural detail, making it easier to compress. These videos are uncompressed and available in three spatial resolutions: 320 × 180, 1280 × 720, and 1920 × 1080, covering low-resolution to full-HD content. The videos are encoded at 20, 30, and 50 frames per second (fps) to reflect different levels of motion activity. Along with raw uncompressed sequences, the evaluation also includes videos encoded using the VVC standard with GOP sizes of 8 and 32. These encoded versions share the same resolutions and frame rates as the uncompressed set, enabling analysis under different temporal compression structures while keeping spatial and temporal properties consistent. This selection allows the framework to demonstrate its versatility and compatibility with different video formats and coding schemes. Furthermore, the design is scalable, capable of supporting various video resolutions, frame rates, and source coding methods, making it adaptable to a wide range of practical applications.
The video processing begins by converting each video into a sequential bitstream representation. These bitstreams can be optionally protected using polar codes with a rate of 1 / 2 to provide error resilience during transmission. To evaluate the inherent performance of the system independently of error correction, scenarios without channel coding (uncoded) are also considered. During the quantum encoding stage, groups of bits are mapped into multi-qubit states, where the size of each qubit encoding size (n) is selected between 1 and 8. This mapping determines the effective compression ratio for each segment. The resulting quantum state is then transformed into the frequency domain using a QFT-based approach, allowing the video information to be represented with fewer coefficients rather than the entire state vector. This frequency-domain compression reduces the amount of data transmitted while preserving the essential information needed for accurate video reconstruction.
After the frequency-domain compression, the encoded video coefficients are transmitted through a channel that may introduce noise and distortions. At the receiver, the system first interprets the compression parameters to reconstruct the complete frequency-domain representation of each qubit system. Next, an inverse transform is performed to map the data from the frequency domain back to the time domain. The resulting quantum states are measured and transformed into classical bitstreams. These bitstreams can subsequently be processed by a channel decoder to reconstruct the original video content.
In the following subsections, each key functional block depicted in [Figure 1](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f001) is explained in detail.
### 3\.1. Classical Channel Encoder and Decoder
The transmission reliability of video bitstreams is improved through the integration of polar codes \[46,47\] operating at a 1/2 coding rate. This configuration establishes an optimal equilibrium between error correction performance and resource utilization, introducing controlled redundancy that effectively mitigates channel distortions while maintaining reasonable computational demands and spectral efficiency. The selection of polar codes over alternative error correction schemes, such as low-density parity check (LDPC) \[48\] and turbo codes \[49\], is motivated by their compelling combination of theoretical performance guarantees and practical implementation advantages. Unlike conventional QEC methodologies that primarily target time-domain perturbations, the proposed architecture specifically leverages frequency-domain processing, rendering traditional QEC approaches unsuitable for this particular framework design.
### 3\.2. Quantum Encoder and Decoder
The proposed quantum video compression and transmission system comprises two fundamental components: the quantum encoder and quantum decoder. The encoder transforms classical video data into a quantum representation and applies compression in the frequency domain, while the decoder reconstructs the original data by reversing these operations. The following subsections present a detailed mathematical formulation of both components and their operating principles.
#### 3\.2.1. Quantum Encoder
The quantum encoder illustrated in [Figure 2](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f002) performs the conversion of classical video data from the time domain into a compact frequency-domain representation. Its operation is divided into several sequential steps. The process begins by assigning each classical bit from the video to a corresponding quantum basis state. In this initialization stage, a bit value of 0 is encoded as the state \| 0 ⟩, whereas a bit value of 1 is encoded as \| 1 ⟩, forming the qubit inputs for the subsequent quantum transformations.
**Figure 2.** Proposed quantum encoder operating in the frequency domain.
In the next stage, the system selects the qubit encoding size (n), which may vary from 1 to 8 qubits. This parameter determines how many consecutive classical bits are grouped together and converted into a single multi-qubit state. Although there is no strict theoretical upper limit on the n, in this work a maximum of 8 qubits is used to maintain manageable system complexity. This choice is sufficient to directly map the pixel values into the corresponding quantum states. The encoder then constructs the complete quantum state vector for each group of bits, incorporating both the initialized qubits and the selected value of n. This results in a structured multi-qubit representation that is ready for frequency-domain processing.
Mathematically, a video bitstream (V) can be represented as in Equation ([1](https://www.mdpi.com/2079-9292/15/6/1323#FD1-electronics-15-01323)).
V \= { v 1 , v 2 , … , v M } , v i ∈ { 0 , 1 }
(1)
where M denotes the total number of bits. These bits are then processed according to the selected n to form the corresponding quantum states for further frequency-domain encoding.
In the case of single-qubit encoding (n \= 1), each individual bit from the video bitstream is directly represented by a single qubit within the two-dimensional Hilbert space H 2. In this mapping stage, each classical bit is directly converted into its corresponding computational basis state, as expressed in Equation ([2](https://www.mdpi.com/2079-9292/15/6/1323#FD2-electronics-15-01323)).
v i \= 0 ↦ \| 0 ⟩ \= 1 0 , v i \= 1 ↦ \| 1 ⟩ \= 0 1
(2)
For multi-qubit encoding, consecutive blocks of n bits are grouped together and encoded as an n\-qubit register, forming a state in the 2 n\-dimensional Hilbert space H 2 n. This grouping process forms a composite quantum state by taking the tensor product of the individual qubits. In this way, several classical bits are combined into a single multi-qubit vector, as described in Equation ([3](https://www.mdpi.com/2079-9292/15/6/1323#FD3-electronics-15-01323)).
( v 1 , v 2 , … , v n ) ↦ \| v 1 v 2 … v n ⟩ \= \| v 1 ⟩ ⊗ \| v 2 ⟩ ⊗ … ⊗ \| v n ⟩
(3)
As an illustration, consider the case of n = 2. In this setting, two classical bits are grouped together, and each pair is represented as a four-dimensional quantum state obtained through a tensor product construction. The resulting states correspond to the formulations shown in Equations ([4](https://www.mdpi.com/2079-9292/15/6/1323#FD4-electronics-15-01323))–([7](https://www.mdpi.com/2079-9292/15/6/1323#FD7-electronics-15-01323)).
\| 00 ⟩ \= 1 0 ⊗ 1 0 \= 1 0 0 0
(4)
\| 01 ⟩ \= 1 0 ⊗ 0 1 \= 0 1 0 0
(5)
\| 10 ⟩ \= 0 1 ⊗ 1 0 \= 0 0 1 0
(6)
\| 11 ⟩ \= 0 1 ⊗ 0 1 \= 0 0 0 1
(7)
When n qubits are combined, they form a composite system whose state vector resides in a 2 n\-dimensional Hilbert space, generated by the tensor product of the constituent single-qubit states. This establishes a direct link between the number of qubits and the size of the quantum state representation.
Following the preparation of the quantum state vector, the next step is to shift from the time-domain representation to the frequency domain. The transformation requires selecting a QFT gate whose dimension matches that of the quantum state vector. For instance, a state vector of size 4 × 1 (n \= 2) is transformed using a 4 × 4 QFT matrix, while a vector of size 8 × 1 (n \= 3) requires an 8 × 8 matrix, and so forth. Accordingly, the QFT matrix, denoted by F N, is constructed as shown in Equation ([8](https://www.mdpi.com/2079-9292/15/6/1323#FD8-electronics-15-01323)).
F N \= 1 N 1 1 1 … 1 1 ω ω 2 … ω N − 1 1 ω 2 ω 4 … ω 2 ( N − 1 ) ⋮ ⋮ ⋮ ⋱ ⋮ 1 ω N − 1 ω 2 ( N − 1 ) … ω ( N − 1 ) ( N − 1 )
(8)
where N \= 2 n corresponds to the quantum state vector dimension, and ω represents the primitive root of unity given by Equation ([9](https://www.mdpi.com/2079-9292/15/6/1323#FD9-electronics-15-01323)).
ω \= e 2 π i / N
(9)
This construction ensures that the QFT appropriately transforms the amplitudes of the time-domain quantum state into the frequency domain, analogous to the classical discrete Fourier transform but in the quantum setting.
For a qubit encoding size of n \= 2, the QFT is applied using a 4 × 4 unitary matrix, defined in Equation ([10](https://www.mdpi.com/2079-9292/15/6/1323#FD10-electronics-15-01323)).
F 4 \= 1 2 1 1 1 1 1 ω ω 2 ω 3 1 ω 2 ω 4 ω 6 1 ω 3 ω 6 ω 9 , ω \= e 2 π i / 4 \= i
(10)
This procedure can be generalized to higher qubit numbers, with the QFT reorganizing the amplitudes of a quantum state into the frequency domain, analogous to how the discrete Fourier transform converts classical signals.
When the QFT is applied to the two-qubit computational basis states, the corresponding frequency-domain vectors are given by Equations ([11](https://www.mdpi.com/2079-9292/15/6/1323#FD11-electronics-15-01323)) through ([14](https://www.mdpi.com/2079-9292/15/6/1323#FD14-electronics-15-01323)).
F 4 \| 00 ⟩ \= 1 2 1 1 1 1
(11)
F 4 \| 01 ⟩ \= 1 2 1 ω ω 2 ω 3
(12)
F 4 \| 10 ⟩ \= 1 2 1 ω 2 ω 4 ω 6
(13)
F 4 \| 11 ⟩ \= 1 2 1 ω 3 ω 6 ω 9
(14)
Equations ([11](https://www.mdpi.com/2079-9292/15/6/1323#FD11-electronics-15-01323))–([14](https://www.mdpi.com/2079-9292/15/6/1323#FD14-electronics-15-01323)) mathematically show that each time-domain basis state maps to a distinct column of the QFT matrix. Careful examination of these columns shows that the second element alone uniquely determines the entire column. Therefore, it is sufficient to transmit only this second component to retain complete knowledge of the quantum state, eliminating the need to send the full frequency-domain vector. In practical implementation, the quantum state is not obtained by extracting amplitudes from an existing superposition, but is instead directly prepared using deterministically computed coefficients via unitary operations. Specifically, for each input symbol, a representative coefficient (e.g., the second QFT coefficient) is computed classically from the bit pattern and mapped to a discrete phase on the complex unit circle. A set of 2 n such coefficients is then used to synthesize a valid quantum state in the 2 n\-dimensional Hilbert space, ensuring normalization and preserving quantum coherence.
Importantly, the proposed framework operates under a constrained state preparation model in which the inputs to the QFT are computational basis states derived from classical bitstreams rather than arbitrary quantum states. As a result, the possible transmitted states belong to a finite and known set corresponding to the columns of the QFT matrix. Because this set is predetermined and shared by both the transmitter and receiver, the transmitted coefficient functions as a symbol that uniquely identifies the corresponding basis index within this codebook.
This process does not involve any projective measurement or partial state extraction. Instead, it defines a structured state preparation strategy in which the full quantum state is generated from a classical description using unitary operations. Although only a single representative coefficient per symbol is transmitted, the remaining components of the corresponding quantum state are implicitly determined by the predefined transform structure, allowing deterministic reconstruction of the original state at the receiver through the inverse QFT.
For instance, when the qubit register size is n \= 2, the full state vector contains 4 elements. By transmitting just the second element, a compression ratio of 4 : 1 is achieved. As the qubit size grows, the compression effect becomes more pronounced: with n \= 8, only a single element out of 256 needs to be sent, resulting in a 256 : 1 reduction in transmitted data. As the register size n increases, the approach allows effective compression while preserving the integrity of the quantum information. This selective transmission approach significantly reduces the quantum symbol payload while retaining essential quantum information.
In summary, to clarify the effect of the qubit encoding size on the system, the qubit encoding size n determines how many classical bits are grouped and represented by an n\-qubit quantum register prior to the application of the QFT. Increasing n increases the dimensionality of the QFT representation, producing 2 n frequency-domain components and enabling higher compression ratios, since a single transmitted coefficient can represent one of 2 n possible input states. However, larger encoding sizes also reduce the angular separation between adjacent phase components, given by Δ θ \= 2 π 2 n, which increases sensitivity to channel noise. Consequently, larger qubit group sizes provide higher compression efficiency but exhibit greater susceptibility to transmission errors, while smaller encoding sizes offer improved noise robustness at the cost of lower compression efficiency. Therefore, the choice of n introduces a trade-off between compression performance and reconstruction reliability, which is analyzed experimentally in this work for qubit sizes ranging from n \= 1 to n \= 8.
The QFT operation applied in this system can be described using its general theoretical formulation, shown in Equation ([15](https://www.mdpi.com/2079-9292/15/6/1323#FD15-electronics-15-01323)).
F N \| x ⟩ \= 1 N ∑ k \= 0 N − 1 e 2 π i x k N \| k ⟩
(15)
Here, F N represents the QFT matrix corresponding to a qubit register of size n. It performs a unitary transformation on an input quantum state \| x ⟩ expressed in the computational basis. Through this operation, the input state is converted into a superposition of all frequency-domain basis states \| k ⟩, with each component weighted by a complex phase factor e 2 π i x k / N that captures its frequency-domain characteristics. The factor 1 / N normalizes the transformation, ensuring that the operation preserves the overall quantum state’s norm and maintains unitarity.
#### 3\.2.2. Quantum Decoder
During the quantum decoding stage, the operations applied during encoding are effectively reversed, as illustrated in [Figure 3](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f003). The process begins by evaluating the compression ratio and using the lightweight side information to determine the original qubit encoding size (n) and ensure decoder synchronization. Using this information, the full frequency-domain quantum state is reconstructed from the received compressed representation. Subsequently, the IQFT is applied to convert the frequency-domain state back into its corresponding time-domain quantum state. Mathematically, the IQFT is the Hermitian conjugate of the QFT and serves a role analogous to the inverse discrete Fourier transform in classical signal processing. Once the time-domain quantum state is restored, standard measurements are performed to map the state back into classical bits, enabling reconstruction of the original bitstream.
**Figure 3.** Proposed quantum decoder.
The IQFT operation (F N − 1) can be mathematically expressed as the Hermitian conjugate of the QFT, as shown in Equation ([16](https://www.mdpi.com/2079-9292/15/6/1323#FD16-electronics-15-01323)).
F N − 1 \= F N †
(16)
Being the conjugate transpose of the QFT, the IQFT reverses the transformation applied in the frequency domain. When applied to a computational basis state \| k ⟩, the IQFT performs the following operation, as shown in Equation ([17](https://www.mdpi.com/2079-9292/15/6/1323#FD17-electronics-15-01323)).
F N − 1 \| k ⟩ \= 1 N ∑ j \= 0 N − 1 e − 2 π i j k / N \| j ⟩
(17)
where N \= 2 n is the dimension of the quantum state space. The index j runs over all computational basis states and represents the output basis states generated by the inverse transform. The negative sign in the exponent indicates the reversal of the phase evolution compared to the forward QFT. This operation maps the frequency-domain representation of a quantum state back to its corresponding time-domain state while preserving unitarity and orthogonality.
For simulation purposes, the QFT matrix F ∈ C N × N, where C denotes the set of complex numbers, is defined element-wise as shown in Equation ([18](https://www.mdpi.com/2079-9292/15/6/1323#FD18-electronics-15-01323)).
F k , l \= 1 N e 2 π i k l / N , k , l \= 0 , … , N − 1
(18)
In this expression, the indices k and l denote the row and column positions of the matrix, respectively. Each entry F k , l corresponds to the complex phase factor associated with mapping the computational basis state \| l ⟩ to the output component indexed by k. Each column of the QFT matrix, denoted by q j, can therefore be represented as a vector, as shown in Equation ([19](https://www.mdpi.com/2079-9292/15/6/1323#FD19-electronics-15-01323)).
q j \= 1 N 1 e 2 π i j / N e 2 π i 2 j / N ⋮ e 2 π i ( N − 1 ) j / N
(19)
It is important to note that the second element of each column uniquely identifies that column. This second entry of the column vector q j is given by Equation ([20](https://www.mdpi.com/2079-9292/15/6/1323#FD20-electronics-15-01323)).
q j ( 2 ) \= 1 N e 2 π i j / N
(20)
The second element of each QFT column corresponds to a distinct point on the complex unit circle, scaled by 1 / N. This unique mapping allows each transmitted coefficient to identify its original column in the QFT matrix unambiguously.
During transmission, the received coefficient, denoted by q ˜ ( 2 ), may be affected by channel noise η ∈ C, where C denotes the set of complex numbers and η can alter both the amplitude and phase of the transmitted coefficient, as expressed in Equation ([21](https://www.mdpi.com/2079-9292/15/6/1323#FD21-electronics-15-01323)).
q ˜ ( 2 ) \= q j ( 2 ) \+ η
(21)
The process of reconstructing the original quantum state from the received coefficient can be described as follows:
- Construct the set of all ideal second elements corresponding to the noiseless QFT columns. Here, N \= 2 n represents the dimension of the Hilbert space for an n\-qubit register. As shown in Equation ([22](https://www.mdpi.com/2079-9292/15/6/1323#FD22-electronics-15-01323)), the set of ideal second elements for the noiseless QFT columns is defined as Q.
Q \= 1 N e 2 π i j / N \| j \= 0 , 1 , … , N − 1
(22)
Each element of Q corresponds to the second component of the j\-th column q j of the QFT matrix F.
- Identify the closest match j \* to the received noisy coefficient q ˜ ( 2 ) affected by channel noise η, as shown in Equation ([23](https://www.mdpi.com/2079-9292/15/6/1323#FD23-electronics-15-01323)).
j \* \= arg min j q ˜ ( 2 ) − 1 N e 2 π i j / N
(23)
Here, j \* indicates the index of the ideal QFT column most closely corresponding to the received signal.
- Retrieve the full QFT column vector corresponding to j \*, as shown in Equation ([24](https://www.mdpi.com/2079-9292/15/6/1323#FD24-electronics-15-01323)).
q ^ \= q j \*
(24)
The vector q ^ represents the retrieved frequency-domain quantum state before applying the inverse transformation.
- Apply the IQFT to reconstruct the time-domain quantum state (x ^), as shown in Equation ([25](https://www.mdpi.com/2079-9292/15/6/1323#FD25-electronics-15-01323)).
x ^ \= F N † q ^
(25)
where F N † is the Hermitian conjugate of the QFT matrix F.
After reconstructing the time-domain quantum state, measurements are performed on each qubit in the computational (Z) basis. This produces an n\-bit classical string corresponding to the decoded symbol, which can then be used to reconstruct the original bitstream. This approach ensures that the simulation faithfully represents the theoretical model while preserving mathematical rigor throughout the entire encoding and decoding process.
It is important to note that Equation ([23](https://www.mdpi.com/2079-9292/15/6/1323#FD23-electronics-15-01323)) represents a classical symbol detection step rather than a quantum state reconstruction process. Specifically, the equation implements a nearest-neighbor decision rule that estimates the transmitted symbol index by comparing the received noisy phase value with the set of valid constellation points defined by the QFT structure. This operation is analogous to maximum-likelihood detection commonly used in classical communication systems.
The quantum operations in the proposed framework occur during the state preparation and transformation stages. At the transmitter, the computational basis state corresponding to the classical symbol index is transformed using the QFT, producing a structured quantum superposition whose phase relationships encode the transmitted symbol. After symbol detection at the receiver, the corresponding quantum state is deterministically reconstructed based on the known QFT structure, and the IQFT is applied as a unitary quantum operation to recover the original computational basis state. Therefore, the proposed receiver follows a hybrid quantum–classical processing model in which classical decision logic is used for symbol detection, while the QFT and IQFT operations constitute the quantum transformations responsible for encoding and decoding the transmitted information.
In general, transmitting a single coefficient of a quantum state would not be sufficient to reconstruct an arbitrary n\-qubit quantum state, since a general state is described by 2 n complex amplitudes. However, the proposed framework operates under a constrained state preparation model in which the input states to the QFT are computational basis states generated deterministically from classical bitstreams. Each block of n classical bits is mapped to a basis state \| j ⟩, where j ∈ { 0 , 1 , … , 2 n − 1 }. When the QFT is applied to a computational basis state, the resulting frequency-domain vector corresponds to a specific column of the QFT matrix with a deterministic phase structure. Since the encoder and decoder both know this finite set of possible QFT columns, a single transmitted coefficient can uniquely identify the corresponding column index j. Once this index is determined at the receiver, the complete frequency-domain representation can be regenerated deterministically, and the inverse QFT can be applied to recover the original computational basis state. Therefore, the proposed approach does not attempt to reconstruct an arbitrary quantum state from a single coefficient; instead, the transmitted coefficient acts as a symbol that identifies one element from a predefined set of structured QFT states.
### 3\.3. Quantum Communication Channel
To simulate realistic quantum transmission conditions, the proposed system considers several standard quantum noise mechanisms \[50\]. Five primary types of quantum noise are included: bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping. Each noise channel is associated with a probability parameter that determines the likelihood of an error occurring during transmission. The effect of hardware imperfections is not included, as these abstract noise models are sufficient for analyzing early-stage quantum communication performance.
#### Composite Noise Model
The combined influence of all quantum noise sources is represented by a composite quantum channel, denoted as C ( ρ ), where ρ is the density matrix of the transmitted quantum state. This channel combines the effects of bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping errors, as defined in Equation ([26](https://www.mdpi.com/2079-9292/15/6/1323#FD26-electronics-15-01323)). In practical quantum communication systems, noise rarely occurs as a single isolated process. Instead, multiple decoherence mechanisms typically act simultaneously due to environmental interactions, imperfect control operations, and physical device limitations. For example, superconducting qubits may experience energy relaxation and dephasing concurrently, while photonic communication channels may be affected by loss, depolarization, and phase fluctuations. Therefore, representing the channel as a probabilistic combination of several elementary noise processes provides a more realistic abstraction of practical quantum transmission conditions. The adopted composite noise model enables the proposed framework to capture the simultaneous influence of multiple error mechanisms and to evaluate system robustness under diverse channel conditions rather than assuming a single dominant noise source.
C ( ρ ) \= ( 1 − p tot ) ρ \+ p B B ( ρ ) \+ p P P ( ρ ) \+ p D D ( ρ ) \+ p A A ( ρ ) \+ p Φ F ( ρ )
(26)
where:
- p B , p P , p D , p A , p Φ ∈ \[ 0 , 1 \] are the probabilities of bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping errors, respectively.
- B ( · ) , P ( · ) , D ( · ) , A ( · ) , F ( · ) denote the corresponding quantum channels implementing these error processes.
The total noise probability p tot is dynamically determined as a function of the system SNR to reflect realistic channel conditions. It is modeled as in Equation ([27](https://www.mdpi.com/2079-9292/15/6/1323#FD27-electronics-15-01323)).
p tot \= min 1 , 1 1 \+ 10 SNR / 10
(27)
where SNR is expressed in dB. This ensures that the overall error probability decreases with increasing SNR. It should be noted that the SNR used in Equation ([27](https://www.mdpi.com/2079-9292/15/6/1323#FD27-electronics-15-01323)) is not intended to represent a fundamental quantum-mechanical parameter. In physical quantum systems, noise processes are typically described using quantum channels characterized by Kraus operators or completely positive trace-preserving (CPTP) maps, and their parameters depend on specific physical mechanisms rather than directly on an SNR value. In the proposed framework, the SNR parameter is used as an abstract simulation-level control variable to regulate the overall severity of channel noise. The mapping defined in Equation ([27](https://www.mdpi.com/2079-9292/15/6/1323#FD27-electronics-15-01323)) therefore determines the total noise probability p tot, which is subsequently distributed among several standard quantum noise channels, including bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping processes. In this way, the SNR parameter provides a convenient mechanism for evaluating system performance across different channel conditions while the underlying noise evolution remains governed by physically consistent quantum channel models.
To allocate p tot among the five individual noise channels, independent random weights w i are drawn from a uniform distribution, as shown in Equation ([28](https://www.mdpi.com/2079-9292/15/6/1323#FD28-electronics-15-01323)).
w 1 , w 2 , w 3 , w 4 , w 5 ∼ U ( 0 , 1 )
(28)
These weights are then normalized to compute the individual channel probabilities, as in Equation ([29](https://www.mdpi.com/2079-9292/15/6/1323#FD29-electronics-15-01323)).
\[ p B , p P , p D , p A , p Φ \] \= p tot ∑ i \= 1 5 w i · \[ w 1 , w 2 , w 3 , w 4 , w 5 \]
(29)
This normalization guarantees that the sum of the individual channel probabilities equals the total noise probability, as expressed in Equation ([30](https://www.mdpi.com/2079-9292/15/6/1323#FD30-electronics-15-01323)).
p B \+ p P \+ p D \+ p A \+ p Φ \= p tot
(30)
Each individual noise channel represents a physically meaningful type of quantum error, as described in the following.
- Bit-flip noise (B): Models the process where a qubit randomly flips from \| 0 ⟩ to \| 1 ⟩ or vice versa, similar to a classical binary error. Mathematically, this is represented in Equation ([31](https://www.mdpi.com/2079-9292/15/6/1323#FD31-electronics-15-01323)).
B ( ρ ) \= ( 1 − p B ) ρ \+ p B X ρ X †
(31)
Here, X is the Pauli-X operator (bit-flip gate).
- Phase-flip noise (P): Represents a phase error that flips the relative sign of \| 1 ⟩ with respect to \| 0 ⟩, leaving populations unchanged, as shown in Equation ([32](https://www.mdpi.com/2079-9292/15/6/1323#FD32-electronics-15-01323)). This is crucial in coherent superposition states.
P ( ρ ) \= ( 1 − p P ) ρ \+ p P Z ρ Z †
(32)
Z is the Pauli-Z operator, which inverts the phase of \| 1 ⟩.
- Depolarizing noise (D): Simulates isotropic errors that randomly apply any Pauli operator X , Y , Z with equal probability, driving the qubit toward a maximally mixed state, as shown in Equation ([33](https://www.mdpi.com/2079-9292/15/6/1323#FD33-electronics-15-01323)).
D ( ρ ) \= ( 1 − p D ) ρ \+ p D 3 ( X ρ X † \+ Y ρ Y † \+ Z ρ Z † )
(33)
This models decoherence that affects both bit and phase simultaneously.
- Amplitude damping (A): Represents energy loss mechanisms, such as spontaneous emission in optical or superconducting qubits. It models the relaxation from \| 1 ⟩ to \| 0 ⟩, as defined in Equation ([34](https://www.mdpi.com/2079-9292/15/6/1323#FD34-electronics-15-01323)).
A ( ρ ) \= E 0 ρ E 0 † \+ E 1 ρ E 1 † , E 0 \= 1 0 0 1 − p A , E 1 \= 0 p A 0 0
(34)
- Phase damping (F): Models pure dephasing without energy loss, which reduces off-diagonal elements of the density matrix in the computational basis, as defined in Equation ([35](https://www.mdpi.com/2079-9292/15/6/1323#FD35-electronics-15-01323)).
F ( ρ ) \= F 0 ρ F 0 † \+ F 1 ρ F 1 † , F 0 \= 1 0 0 1 − p Φ , F 1 \= 0 0 0 p Φ
(35)
By combining these channels probabilistically, the framework provides a flexible and physically meaningful model for testing quantum communication systems under realistic noisy conditions. SNR is mapped to the total noise probability p tot to adjust error severity according to channel quality \[15,19\].
### 3\.4. Physical Realization of QFT Using Quantum Gates
In real quantum circuit implementations, the QFT is constructed using a sequence of single-qubit Hadamard operations together with controlled phase rotation gates. To illustrate this structure, a QFT circuit consisting of three qubits is presented in [Figure 4](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f004).
**Figure 4.** Quantum circuit for a three-qubit QFT.
The Hadamard gate \[51\], represented by H, is a key quantum logic gate that converts computational basis states into superposition states. The matrix form of this operation is provided in Equation ([36](https://www.mdpi.com/2079-9292/15/6/1323#FD36-electronics-15-01323)).
H \= 1 2 1 1 1 − 1
(36)
When applied to a single qubit, it performs the transformations given in Equations ([37](https://www.mdpi.com/2079-9292/15/6/1323#FD37-electronics-15-01323)) and ([38](https://www.mdpi.com/2079-9292/15/6/1323#FD38-electronics-15-01323)).
H \| 0 ⟩ \= 1 2 ( \| 0 ⟩ \+ \| 1 ⟩ ) \= \| \+ ⟩
(37)
H \| 1 ⟩ \= 1 2 ( \| 0 ⟩ − \| 1 ⟩ ) \= \| − ⟩
(38)
Here, \| \+ ⟩ and \| − ⟩ represent the superposition states along the X\-axis of the Bloch sphere.
Phase gates introduce controlled rotations, which are crucial for encoding the relative phases between qubits in QFT. The general single-qubit phase gate is written as in Equation ([39](https://www.mdpi.com/2079-9292/15/6/1323#FD39-electronics-15-01323)).
P ( ϕ ) \= 1 0 0 e i ϕ
(39)
For QFT, the rotation angle ϕ is determined by the qubit positions: ϕ \= 2 π / 2 k, where k is the distance between the control and target qubits in a controlled-phase operation.
#### Three-Qubit QFT Procedure
Let the input state be \| x 1 x 2 x 3 ⟩, where x 1 is the most significant qubit. The QFT is performed as follows:
- Apply a Hadamard gate to the first qubit x 1.
- Apply a controlled-R 2 gate between x 1 and x 2, where the R 2 phase gate is defined in Equation ([40](https://www.mdpi.com/2079-9292/15/6/1323#FD40-electronics-15-01323)).
R 2 \= 1 0 0 e i π / 2
(40)
- Apply a controlled-R 3 gate between x 1 and x 3, where the R 3 phase gate is defined in Equation ([41](https://www.mdpi.com/2079-9292/15/6/1323#FD41-electronics-15-01323)).
R 3 \= 1 0 0 e i π / 4
(41)
- Perform a Hadamard gate on x 2, followed by a controlled-R 2 gate between x 2 and x 3.
- Apply a Hadamard gate to x 3.
- Swap qubits x 1 ↔ x 3 to correct the qubit order in the computational basis.
The resulting three-qubit QFT state can be expressed as in Equation ([42](https://www.mdpi.com/2079-9292/15/6/1323#FD42-electronics-15-01323)).
QFT ( \| x 1 x 2 x 3 ⟩ ) \= 1 8 \| 0 ⟩ \+ e 2 π i 0 . x 3 \| 1 ⟩ ⊗ \| 0 ⟩ \+ e 2 π i 0 . x 2 x 3 \| 1 ⟩ ⊗ \| 0 ⟩ \+ e 2 π i 0 . x 1 x 2 x 3 \| 1 ⟩
(42)
This formulation demonstrates that each qubit sequentially accumulates phase contributions from less significant qubits, effectively encoding the input into the frequency domain. The output of this circuit matches the theoretical QFT computations for all basis states, confirming the correctness of the implementation \[52\]. Similarly, this physical implementation can be generalized to accommodate arbitrary qubit sizes.
### 3\.5. Quantum and Classical Components of the Framework
The proposed system follows a hybrid quantum–classical architecture. The genuinely quantum operations in the framework include: (i) preparation of the n\-qubit computational basis state corresponding to each data segment, (ii) application of the QFT using quantum gate circuits composed of Hadamard and controlled phase-rotation gates, (iii) propagation of the quantum state through the quantum communication channel modeled by quantum noise processes such as bit-flip, phase-flip, depolarizing, amplitude damping, and phase damping channels, (iv) application of the IQFT at the receiver, and (v) measurement of the qubits in the computational basis to obtain classical outcomes.
The classical components of the framework include preprocessing and post-processing stages. Specifically, the video data is first compressed using the classical VVC encoder, and the resulting bitstream is segmented into n\-bit blocks before quantum encoding. In scenarios where uncompressed video is used, the raw pixel data is directly converted into a binary bitstream and segmented into n\-bit blocks without applying source compression. After quantum measurement at the receiver, the recovered binary data is concatenated and decoded using the classical VVC decoder to reconstruct the video frames when compressed inputs are used. For uncompressed inputs, the recovered bitstream is directly mapped back to the corresponding pixel representation. In addition, classical detection procedures such as nearest-neighbor symbol estimation are used to determine the most likely transmitted index from the received noisy observations. Although the system is evaluated through numerical simulation, the evolution of the quantum states is modeled according to the standard formalism of quantum mechanics, including unitary QFT/IQFT operations and quantum channel noise represented by completely positive trace-preserving maps
### 3\.6. Performance Evaluation Methodology
The performance of the proposed system is evaluated across multiple scenarios to assess compression efficiency, error resilience, and robustness under different channel conditions. The evaluation framework is described as follows:
- Uncompressed video inputs: Multi-qubit configurations with n \= 1 to 8 are tested using raw video sequences. These experiments examine how the qubit dimension affects the compression ratio and robustness to quantum channel noise. Results are reported for both cases without and with classical channel coding.
- Compressed video inputs: To study transmission efficiency improvements, multi-qubit systems (n \= 1 to 8) are applied to VVC-compressed video sequences. Two GOP structures, 8 and 32, are considered. Performance is evaluated both with and without classical channel coding.
- Performance of QFT encoding without compression: The QFT-based full-vector transmission, which sends the complete quantum state without compression \[19\], is compared against alternative methods under the same bandwidth constraints. These alternatives include a time-domain Hadamard-based multi-qubit system with qubit sizes n \= 1 to 8, and a classical system using binary phase-shift keying (BPSK). In the Hadamard-based approach, the multi-qubit encoding matrix is generated via the tensor product of the Hadamard matrix defined in Equation ([36](https://www.mdpi.com/2079-9292/15/6/1323#FD36-electronics-15-01323)), and decoding is performed using the inverse Hadamard transform to recover the transmitted quantum state \[32\].
In the proposed framework, the single-qubit configuration is considered the reference transmission system. A single qubit can represent two computational basis states, \| 0 ⟩ and \| 1 ⟩, which correspond to a two-point constellation. This is directly comparable to the two-symbol constellation used in a classical BPSK modulation scheme. Therefore, BPSK provides a natural classical counterpart for evaluating the transmission behavior of the QFT-based frequency-domain representation under equivalent binary symbol conditions.
To maintain equivalent bandwidth usage across all qubit configurations, the input bitstream is compressed using different quantization parameter (QP) settings in the VVC encoder. This produces progressively shorter bitstreams as the qubit grouping size increases. As a result, the total amount of transmitted information remains comparable across different configurations, allowing the impact of the QFT-based transmission process to be evaluated without introducing bandwidth bias.
### 3\.7. Simulation Configuration
The proposed system is evaluated using a numerical simulation framework that implements the complete transmission and reconstruction pipeline described in the previous subsections. The results are obtained using a Monte Carlo simulation framework to evaluate system performance under stochastic noise conditions. For each video sequence and each SNR value, the complete transmission and reconstruction process is repeated over 1000 independent trials. In each trial, a new realization of the composite quantum noise channel is generated by drawing random weights for the individual noise processes from a uniform distribution and normalizing them to satisfy the total noise probability constraint. The transmitted states are propagated through the resulting channel, and the reconstructed video quality is evaluated using bit error rate (BER), peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM) \[53\], and video multi-method assessment fusion (VMAF) \[54\] metrics. The final curves shown in the figures correspond to the average performance obtained across these 1000 trials, which explains the smooth appearance of the plotted results despite the underlying stochastic noise model. The key parameters and implementation settings used in the experiments are summarized in [Table 1](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-t001).
**Table 1.** Experimental Setup Parameters.
## 4\. Results and Discussion
In this section, we present and analyze the performance of the proposed QFT-based video compression and transmission system under varying quantum channel conditions. The evaluation examines how the system handles videos with different spatial and temporal complexities, providing insight into its robustness and effectiveness. Quantitative results are reported using BER, PSNR, SSIM, and VMAF. Together, these metrics evaluate reconstruction fidelity, perceptual quality, and temporal consistency, offering a comprehensive view of system performance. The discussion highlights trends observed in average results across multiple test videos, emphasizing the effects of motion complexity, channel noise, and compression on video quality, while summarizing overall system behavior through aggregated performance metrics.
Each comparison scenario introduced in [Section 3.6](https://www.mdpi.com/2079-9292/15/6/1323#sec3dot6-electronics-15-01323) is explained in detail in the following subsections.
### 4\.1. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for Uncompressed Video Inputs Using Multi-Qubit Configurations Without Channel Coding
The results of the proposed QFT-based video compression and transmission system, evaluated without channel coding, is summarized in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005). The figure illustrates the system’s behavior across four key metrics, BER, PSNR, SSIM, and VMAF, as a function of the channel SNR for different qubit encoding sizes (F1 to F8, corresponding to n = 1 to n = 8 qubits). The BER results in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)a reveal a critical trade-off governed by the qubit encoding size, n. In this system, a block of video data is mapped via the QFT into a 2 n\-dimensional frequency vector. The unitary nature of the QFT enables powerful, exponential compression; an n\-qubit encoding achieves a theoretical compression ratio of 2 n : 1, as only the dominant frequency coefficient needs to be transmitted to reconstruct the original block. However, this compression gain comes at the cost of increased susceptibility to noise. The angular separation between adjacent frequency components decreases exponentially with n, as given by Equation ([43](https://www.mdpi.com/2079-9292/15/6/1323#FD43-electronics-15-01323)). A smaller Δ θ causes the constellation of frequency states to become denser, making it more difficult for the receiver to discriminate between them in the presence of phase noise induced by the channel.
Δ θ \= 2 π 2 n
(43)
**Figure 5.** Performance proposed QFT-based video compression and transmission system for the uncompressed video inputs without channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
Consequently, while the F8 configuration (n \= 8) achieves a very high 256:1 compression ratio, it exhibits the highest BER across all SNR levels, as shown in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)a. In contrast, the F2 configuration (n \= 2) benefits from a more robust angular separation of π / 2 radians and a moderate 4:1 compression ratio, resulting in superior error resilience. The F1 configuration, although offering the highest error tolerance, provides only minimal compression (2:1). Overall, these results indicate that intermediate encoding sizes (e.g., F2–F4) offer the most practical operating point, achieving a favorable trade-off between substantial compression and acceptable error rates under typical noisy channel conditions.
The video reconstruction quality metrics, PSNR, SSIM, and VMAF, shown in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)b, [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)c, and [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005)d, respectively, follow consistent trends. As expected, all metrics generally improve with increasing SNR, as a cleaner channel allows for more accurate reconstruction of the transmitted video data. The performance hierarchy across different encoding sizes mirrors the findings from the BER analysis. Systems with lower qubit counts (F1, F2) consistently achieve higher PSNR, SSIM, and VMAF scores at a given SNR due to their inherent noise resilience. The superior performance of F2, in particular, confirms its optimal balance between compression efficiency and reconstruction quality.
An interesting observation is the behavior of the F8 configuration. While its BER is significantly higher, its PSNR does not degrade as catastrophically as one might expect. This can be attributed to the nature of the QFT/IQFT process and the structure of video data. Channel noise primarily perturbs the finer, less significant details in the frequency domain. During the inverse QFT, these errors are distributed across the pixel block. Furthermore, natural video has significant spatial and temporal correlation, allowing adjacent pixels to mask these distributed errors. As a result, the overall structural integrity and perceptual quality, as captured by PSNR, SSIM and VMAF, are preserved to a greater degree than the raw BER might suggest. However, the F8 system requires a very high SNRs to achieve the highest quality levels that lower-qubit systems can achieve at moderate SNRs.
Therefore, the results in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005) highlight a fundamental trade-off in the proposed system: higher qubit encoding sizes achieve greater compression but are more susceptible to channel noise. Under realistic noisy channel conditions, smaller or intermediate qubit configurations, such as F2 to F4, provide a more favorable balance between compression and error resilience, ensuring reliable video reconstruction.
### 4\.2. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for Uncompressed Video Inputs Using Multi-Qubit Configurations with Channel Coding
The introduction of channel coding dramatically alters the performance landscape of the proposed system, as illustrated in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006). When compared to the uncoded results in [Figure 5](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f005), the application of forward error correction provides a significant coding gain, effectively shifting all performance curves to the left and enabling reliable operation at much lower SNR levels. The BER characteristics, depicted in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)a, demonstrate the substantial channel coding gain achieved through forward error correction. Although the trade-off between qubit encoding size and error susceptibility persists, the inclusion of channel coding substantially reduces the absolute BER across all SNR levels relative to the uncoded scenario. The channel coding effectively compensates for the inherent vulnerability of larger encoding sizes, enabling their practical deployment in noisy environments where they would otherwise be unusable. The error correction manifests most prominently in the intermediate SNR regime (10–25 dB), where the coding gain provides the greatest marginal benefit. In this region, the BER curves exhibit the characteristic steep descent associated with effective coding schemes, transitioning rapidly from high to low error probability as SNR increases.
**Figure 6.** Performance proposed QFT video compression and transmission system for the uncompressed video inputs with channel coding (**a**) bit error rate (BER), (**b**) peak signal-to-noise ratio (PSNR), (**c**) structural similarity index measure (SSIM), and (**d**) video multi-method assessment fusion (VMAF).
The PSNR results in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)b reveal a significant qualitative shift in system behavior compared to the uncoded scenario. Channel coding eliminates the anomalous performance previously observed with F8, establishing a consistent monotonic relationship between encoding dimension and reconstruction fidelity. The F8, which demonstrated unexpected resilience in the uncoded system due to its direct pixel mapping characteristics, now exhibits the most pronounced sensitivity to channel impairments. This normalization of behavior arises because the error correction process fundamentally alters how quantization errors propagate through the system. The coding redundancy, while essential for error protection, disrupts the spatial correlation properties that previously provided natural error masking for certain encoding configurations. Consequently, the theoretical vulnerability of high-dimensional encodings to angular separation constraints becomes fully manifested in the channel coded system.
The SSIM metrics, presented in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)c, corroborate the PSNR findings while providing additional insights into structural preservation. Channel coding maintains superior structural similarity across all encoding sizes, particularly in the critical mid-range SNR conditions where perceptual quality is most vulnerable. However, the progressive degradation with increasing encoding size remains evident, confirming that while channel coding improves absolute performance, it does not alter the fundamental compression-robustness trade-off intrinsic to the QFT encoding approach.
The VMAF results in [Figure 6](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f006)d provide the most comprehensive assessment of perceptual video quality. This metric, which incorporates characteristics of the human visual system and accounts for temporal relationships across consecutive frames, reveals that channel coding substantially improves the viewing experience across all encoding sizes. Nonetheless, the hierarchical pattern remains: smaller encoding sizes (F1–F3) maintain excellent perceptual quality (VMAF \> 80) at moderate SNR levels, while larger encodings require progressively higher SNR to achieve comparable viewing quality. Notably, VMAF highlights the superiority of the two-qubit encoding (F2), which achieves an optimal balance between compression efficiency and perceptual quality preservation.
The comparative analysis across all four metrics provides crucial insights for system optimization. The single-qubit configuration demonstrates the highest robustness, maintaining reliable performance under noisy conditions while achieving a 2:1 compression ratio. However, the two-qubit system emerges as the optimal compromise, delivering a favorable 4:1 compression ratio while requiring only modest SNR to maintain acceptable quality across diverse channel conditions, as consistently reflected in its superior BER, PSNR, SSIM, and VMAF performance. In contrast, higher-qubit configurations (e.g., F4–F8) offer greater compression but demand significantly higher SNR to achieve comparable quality, highlighting the trade-off between compression efficiency and error resilience.
### 4\.3. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for VVC Encoded Video Inputs Using Multi-Qubit Configurations Without Channel Coding
The performance of the system with pre-compressed VVC input using GOP 8 exhibits a notably different behavior compared to uncompressed video, as illustrated in [Figure 7](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f007). All three quality metrics, PSNR, SSIM, and VMAF, follow a consistent pattern across different qubit encoding sizes. The smaller qubit encodings (F1–F3) maintain superior performance throughout the SNR range, while larger encodings (F6–F8) show significant degradation, particularly at low to moderate SNR levels. The PSNR results in [Figure 7](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f007)a indicate that VVC-compressed inputs are more susceptible to quality degradation under channel noise than uncompressed content. The SSIM and VMAF metrics, shown in [Figure 7](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f007)b,c, reveal similar overall trends, albeit with different sensitivity characteristics. SSIM values for smaller encodings remain above 0.8 for SNR \> 10 dB, indicating good structural preservation. However, larger encodings struggle to maintain acceptable structural similarity, with the F8 configuration performing the worst under channel noise. VMAF scores show the most pronounced separation between encoding sizes. The F1 configuration maintains excellent perceptual quality (VMAF \> 80) across most of the SNR range, whereas F8 only exceeds a VMAF of 40 at around 40 dB channel SNR, indicating a poor viewing experience across nearly all channel conditions.
**Figure 7.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
The results clearly indicate that VVC-encoded content is less tolerant to the additional compression introduced by larger qubit encodings compared to uncompressed inputs. Moreover, the combination of VVC compression artifacts and the noise sensitivity of high-dimensional QFT encodings produces a compounding effect that significantly degrades reconstruction quality. For practical systems using pre-compressed video content, smaller qubit encodings (F1–F2) are essential to maintain acceptable quality. The theoretical compression benefits of larger encodings are negated by the severe quality degradation.
Based on the results shown in [Figure 8](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f008), which depicts the system performance with a GOP size of 32 and without channel coding, the effect of larger GOP sizes on video quality is clearly observed. The results for PSNR, SSIM, and VMAF consistently show that all configurations (F1 to F8) exhibit reduced quality compared to scenarios with GOP size 8. As the GOP size increases to 32, the number of inter-coded frames rises, leading to greater reliance on predictive coding. Errors introduced in early frames within the GOP propagate through subsequent inter-frames, amplifying quality loss. This propagation effect is evident in [Figure 8](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f008), where even robust encodings such as F1 and F2 exhibit lower PSNR, SSIM, and VMAF scores compared to the corresponding results for GOP size 8. To mitigate this, adaptive error correction strategies tailored to the GOP structure, such as strengthening protection for key frames or using error-resilient encoding techniques, could be employed to reduce error propagation and enhance overall performance.
**Figure 8.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs without channel coding (GOP32) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
### 4\.4. Performance Analysis of the Proposed QFT-Based Video Compression and Transmission System for VVC Encoded Video Inputs Using Multi-Qubit Configurations with Channel Coding
The results for VVC-encoded video (GOP size 8, shown here as an example) with channel coding, presented in [Figure 9](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f009), demonstrate a significant improvement compared to uncoded VVC transmission. Channel coding effectively mitigates the error propagation issues inherent in predictive coding with GOP size 8. All three quality metrics, PSNR, SSIM, and VMAF, show substantial enhancement compared to the uncoded case. The system now maintains acceptable quality levels even at moderate SNR conditions, where previously severe degradation occurred. The hierarchical relationship between different qubit encodings persists, but with improved absolute performance. Smaller encodings (F1–F3) achieve excellent reconstruction quality, with F1 and F2 maintaining PSNR above 25 dB, SSIM above 0.8, and VMAF above 80 across most of the SNR range. Even larger encodings (F4–F6) now provide usable quality at sufficient SNR levels, though F7 and F8 still show limitations due to their sensitivity to phase noise. The combination of VVC encoding, QFT-based compression, and channel coding represents a viable operational point for practical systems. The two-qubit encoding (F2) emerges as particularly advantageous, offering a favorable 4:1 compression ratio while maintaining robust performance across all quality metrics. This configuration balances compression efficiency with error resilience, making it suitable for bandwidth-constrained applications requiring reliable video transmission.
**Figure 9.** Performance of the proposed QFT-based video compression and transmission system for VVC-encoded video inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
### 4\.5. Performance of Frequency-Domain Encoding Without Compression for VVC-Encoded Inputs with Channel Coding
Based on the results shown in [Figure 10](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f010) for VVC-encoded videos, a clear performance hierarchy emerges between the quantum-inspired encoding systems operating without additional compression and the classical baseline approach. Both the QFT-based frequency-domain system (F1–F8) and the time-domain Hadamard system (T1–T8) consistently outperform the bandwidth equivalent classical system (C) across all quality metrics, PSNR, SSIM, and VMAF. Notably, unlike the scenarios involving QFT-based compression where larger qubit sizes degrade performance, in this case increasing the qubit encoding size (n) actually improves reconstruction quality for both quantum-inspired systems. This improvement occurs because, in this scenario, no QFT-based compression is applied and the full quantum state vector is transmitted. As a result, the video does not contain the artifacts and inter-frame dependencies that typically amplify transmission errors in compressed content. As a result, the superior noise resilience and representational capacity of higher-dimensional quantum encodings become evident. Larger qubit configurations are able to capture and represent more complex quantum states, allowing them to encode greater amounts of information per transmitted symbol. This leads to improved reconstruction fidelity and robustness against channel noise, which is why larger-qubit configurations such as F7/F8 and T7/T8 consistently outperform smaller qubit systems.
**Figure 10.** Performance of frequency-domain encoding without compression for VVC-encoded inputs with channel coding (GOP8) (**a**) peak Signal-to-Noise Ratio (PSNR), (**b**) structural similarity index measure (SSIM), and (**c**) video multi-method assessment Fusion (VMAF).
In particular, the QFT-based encoding system benefits from high-dimensional frequency-domain representations in Hilbert space, utilizing both amplitude and phase encoding to capture intricate correlations within the video data. In contrast, the Hadamard-based encoding operates in the time domain, relying on an orthogonal basis structure with amplitude encoding alone. The dual encoding capability of the QFT system enables it to more accurately preserve the full quantum state and resist errors, resulting in superior performance compared to Hadamard-based systems, especially in high-dimensional qubit configurations. Consequently, the combination of larger qubit sizes and QFT-based frequency-domain encoding provides the highest reconstruction quality, demonstrating the fundamental advantages of quantum-inspired encoding for high-fidelity transmission of uncompressed video. This encoding scheme differs from the proposed method in that it does not achieve compression; instead, it transmits the full quantum state vector without reducing the data volume.
### 4\.6. Performance Assessment of the Proposed QFT-Based Compression and Transmission System Across Different Resolutions and Qubit Configurations
As shown in [Table 2](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-t002), the channel SNR gains compared to the eight-qubit encoding under the channel-coded system remain consistent across all tested video resolutions and frame rates. This indicates that the resolution and frame rate of the video do not significantly impact the performance of the QFT-based compression and transmission system. Instead, the maximum SNR gains are primarily determined by the qubit encoding size, demonstrating that the system’s error resilience and compression efficiency depend on the quantum encoding configuration rather than the specific characteristics of the video content.
**Table 2.** Maximum SNR Gains (in dB) Achieved by the QFT-Based Compression and Transmission System for Each Qubit Configuration Compared to the Eight-Qubit System Across Different Video Resolutions and Frame Rates.
An illustrative example of reconstructed video frames using the proposed QFT-based system under varying channel conditions is shown in [Figure 11](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f011). The frames span SNR levels from 38 dB down to − 10 dB, with each subfigure labeled alphabetically for easy reference. The system employs eight-qubit encoding (F8) with a 256:1 compression ratio applied to uncompressed video inputs, demonstrating the high compression efficiency of the proposed QFT-based video compression and transmission framework. At higher SNR levels, the reconstructed frames closely preserve the original visual details, indicating effective retention of critical image information. As the SNR decreases, visual degradations such as blurring and minor artifacts become more apparent, reflecting the impact of channel noise on the transmitted quantum-encoded data. Despite these distortions, the system demonstrates graceful degradation, highlighting the robustness of the QFT-based video compression and transmission system with eight-qubit encoding against channel impairments. This illustrative example complements quantitative evaluations using PSNR, SSIM, and VMAF, providing a clear visual demonstration of the proposed method’s performance across a wide range of channel conditions.
**Figure 11.** Reconstructed video frames at SNR levels from 38 dB to − 10 dB for the QFT-based video compression and transmission system with eight-qubit encoding (F8) and 256:1 compression of uncompressed video inputs.
### 4\.7. Performance Evaluation Compared to Classical Communication Systems
To contextualize the performance of the proposed framework in a video transmission setting, we compare it with representative classical baselines, as illustrated in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012) for the two-qubit quantum system. QPSK serves as a modulation-level reference with comparable bandwidth, while both HEVC and VVC represent widely used state-of-the-art classical video compression standards. All systems are evaluated under equivalent bitrate constraints to ensure fair bandwidth utilization.
**Figure 12.** Comparison of the proposed system with classical communication systems: (**a**) Peak Signal-to-Noise Ratio (PSNR), (**b**) Structural Similarity Index Measure (SSIM).
The QFT-based system enables frequency-domain video transmission by encoding frame-level information into quantum states, allowing efficient representation of spatial content. Even without channel coding, the proposed approach achieves performance comparable to uncompressed QPSK transmission while benefiting from intrinsic compression (4:1) due to the quantum state representation. When polar coding with a rate of 1/4 is applied, the QFT-based system significantly improves robustness against channel noise, achieving substantial PSNR gains (up to 10 dB) over uncompressed QPSK transmission, as shown in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012)a, under the same bitrate.
Unlike classical video compression methods such as HEVC and VVC, which rely on predictive coding and inter-frame dependencies, the proposed QFT-based framework demonstrates a more gradual degradation in the presence of channel noise. This behavior is evident in the PSNR trends in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012)a, where the QFT-coded system maintains high reconstruction quality at lower SNR values compared to both uncoded QPSK and classical codec-based transmission schemes. Similarly, the SSIM results in [Figure 12](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-f012)b confirm that the proposed method preserves structural information more effectively under channel impairments.
While HEVC and VVC provide strong rate–distortion performance under ideal channel conditions, they rely on inherently lossy compression mechanisms and predictive coding structures that make them highly sensitive to transmission errors. Even small bitstream corruptions can propagate across multiple frames due to inter-frame dependencies, leading to abrupt visual degradation. In contrast, the proposed QFT-based framework operates on independently encoded frequency-domain representations, which reduces error propagation and enables more stable reconstruction quality in noisy channels. Furthermore, since the QFT and its inverse are unitary transformations, the proposed system can theoretically achieve lossless reconstruction under ideal channel conditions.
Although classical codecs such as HEVC and VVC benefit from mature implementations and highly optimized compression efficiency in error-free environments, the proposed quantum-inspired framework offers a promising alternative for video transmission in bandwidth-limited and error-prone channels, providing improved robustness and graceful degradation characteristics.
### 4\.8. Advantages and Practical Implications of the Proposed QFT-Based Symbol-Level Compression and Transmission Framework
It is important to emphasize that the proposed method introduces quantum domain symbol-level compression. In this framework, compression occurs during quantum encoding by selectively transmitting a reduced set of QFT-domain coefficients. This process decreases the transmission payload but does not reduce the file size of the original video for storage. Unlike classical codecs, which replace the stored representation with a compressed version, the proposed QFT-based approach requires access to the full quantum state vector for decoding. As such, the method provides compression strictly within the communication pipeline, not for persistent storage.
The mathematical foundation of the QFT ensures theoretically perfect reconstruction (lossless) under ideal noise-free conditions. Since the QFT and its inverse (IQFT) are unitary operations, a computational basis state can be perfectly recovered when no channel noise, decoherence, or measurement errors are present. Therefore, the lossless property represents the theoretical upper bound of the system rather than a guarantee of lossless performance in practical quantum communication environments. Under practical noisy channel conditions, however, the system exhibits graceful degradation: increasing the qubit dimension reduces the angular separation between encoded frequency components, making higher-dimensional states more susceptible to noise. Importantly, this controlled degradation profile offers system designers well-defined trade-off parameters between compression efficiency and noise resilience, providing a level of tunability not typically available in conventional communication systems.
Based on the comprehensive evaluation of the proposed QFT-based compression and transmission system, several key advantages emerge that highlight its potential for practical video transmission applications. The system exhibits strong adaptability, allowing the optimal qubit encoding size to be selected according to specific operational requirements and channel conditions. Smaller encoding configurations (n \= 1–2 qubits) excel in low-SNR scenarios, providing robust and reliable performance that is well suited for mission-critical applications where transmission reliability outweighs compression efficiency. Medium-sized encodings (n \= 3–5 qubits) offer an effective balance between compression gains and robustness, making them appropriate for typical multimedia transmission environments. Larger encoding sizes (n \= 6–8 qubits) achieve substantial compression ratios and are most beneficial in high-SNR channels or in scenarios where bandwidth conservation is a primary objective.
### 4\.9. Channel Use, Resource Cost, and Noise Sensitivity
In the proposed framework, the encoded video bitstream is segmented into blocks containing n classical bits, as described in the encoding stage. Each block is mapped to the computational basis of an n\-qubit register and processed using the QFT to obtain a frequency-domain representation of the encoded data.
A single representative coefficient from the QFT representation is selected and encoded into a transmitted quantum state. Therefore, a channel use in the proposed system is defined as the transmission of one encoded quantum state corresponding to a QFT-transformed data block. Under this definition, each segmented block requires one channel use. Increasing the qubit grouping size n increases the number of classical bits represented by each block and therefore reduces the number of channel uses required to transmit the entire bitstream.
From an information-theoretic perspective, the channel capacity can be related to the classical Shannon capacity formula, as shown in Equation ([44](https://www.mdpi.com/2079-9292/15/6/1323#FD44-electronics-15-01323)).
C raw \= B log 2 ( 1 \+ SNR )
(44)
In Equation ([44](https://www.mdpi.com/2079-9292/15/6/1323#FD44-electronics-15-01323)), C raw denotes the raw channel capacity in bits per second, B represents the channel bandwidth in Hertz, and SNR is the signal-to-noise ratio.
Because the proposed QFT-based representation transmits only one coefficient out of the 2 n possible frequency components produced by the QFT, the effective transmitted information rate can be expressed as in Equation ([45](https://www.mdpi.com/2079-9292/15/6/1323#FD45-electronics-15-01323)).
C eff ( n ) \= C raw 2 n
(45)
In Equation ([45](https://www.mdpi.com/2079-9292/15/6/1323#FD45-electronics-15-01323)), C eff ( n ) represents the effective channel capacity associated with the transmitted coefficient when an n\-qubit encoding is used, and 2 n denotes the number of spectral components generated by the QFT.
The transmitted resource cost can therefore be interpreted as the number of quantum states required to represent the encoded bitstream. If L denotes the total number of bits in the compressed bitstream, the number of transmitted blocks is given as in Equation ([46](https://www.mdpi.com/2079-9292/15/6/1323#FD46-electronics-15-01323)).
N b \= L n
(46)
In Equation ([46](https://www.mdpi.com/2079-9292/15/6/1323#FD46-electronics-15-01323)), N b represents the number of encoded blocks generated from the bitstream and therefore corresponds directly to the total number of channel uses required for transmission.
To evaluate the robustness of the proposed framework under different channel conditions, the transmission process is analyzed across multiple SNR levels. Channel disturbances are modeled as noise perturbations applied to the transmitted quantum state before the IQFT is performed at the receiver. By analyzing system performance across different SNR values, the sensitivity of the proposed framework to channel noise can be systematically characterized.
### 4\.10. Computational Complexity and System Scalability
The computational complexity of the proposed QFT-based compression system is one of its key advantages. For an n\-qubit encoding, the total gate count scales as O ( n 2 ), primarily due to the use of standard Hadamard gates and controlled-phase rotations, which are commonly supported in most quantum hardware. Circuit depth, which determines the temporal length of the computation, can be reduced to O ( n ) when gates acting on independent qubits are executed in parallel; however, hardware connectivity constraints, such as linear nearest-neighbor layouts, introduce additional SWAP operations that can increase depth toward O ( n 2 ). The use of widely available Hadamard and phase gates, along with approximate QFT methods that truncate small-angle rotations, ensures that both gate count and depth remain manageable while maintaining high fidelity.
Moreover, the system is scalable to higher qubit counts depending on the application requirements. Increasing the qubit dimension allows for higher compression ratios, as more information can be encoded per quantum state. However, larger qubit sizes can reduce error resilience, since each qubit becomes more susceptible to noise. Therefore, qubit count selection involves a trade-off between compression efficiency and robustness, allowing the system to be tailored to specific application scenarios.
### 4\.11. Hardware Feasibility and Resource Requirements
The proposed compression and transmission framework is compatible with practical quantum communication architectures because it relies on standard quantum operations that are widely studied and experimentally realizable. The encoding stage requires only computational basis state preparation, which can be implemented by initializing qubits in the ground state \| 0 ⟩ and applying Pauli-X gates when the corresponding classical bit equals one. The transformation stage employs the QFT, which can be implemented using Hadamard gates and controlled phase-rotation gates with polynomial circuit complexity. For an n\-qubit QFT circuit, the number of Hadamard gates is H \= n, while the number of controlled phase gates is n ( n − 1 ) 2, resulting in an approximate total gate count of G ( n ) \= n \+ n ( n − 1 ) 2. After transmission through the quantum channel, the receiver applies the IQFT, followed by computational-basis measurement to recover the classical bitstream.
From a hardware perspective, the proposed system requires relatively small quantum registers and a moderate number of gate operations. In the experiments presented in this work, the encoding size ranges from n \= 1 to n \= 8 qubits, which is well within the capabilities of current noisy intermediate-scale quantum (NISQ) devices and experimental quantum communication platforms. The required gate operations scale quadratically with the number of qubits due to the structure of the QFT circuit, while the readout stage requires only standard computational-basis measurements.
[Table 3](https://www.mdpi.com/2079-9292/15/6/1323#electronics-15-01323-t003) summarizes the approximate hardware resources required for different qubit encoding sizes considered in this study.
**Table 3.** Estimated QFT Hardware Resources for Different Qubit Sizes.
### 4\.12. Simulation Methodology and Assumptions
The proposed quantum-inspired compression framework is evaluated through classical simulations on conventional computing platforms, specifically an Intel Core i5-1345U processor (Intel Corporation, Santa Clara, CA, USA) with 16 GB of RAM. Classical simulation is widely used in quantum communication research to analyze algorithmic behavior and system-level performance prior to implementation on physical quantum hardware. Current quantum processors remain limited in terms of qubit count, coherence time, and circuit depth, which makes large-scale multimedia transmission experiments impractical. Therefore, numerical simulation provides a controlled and reproducible environment for modeling quantum state preparation, QFT/IQFT transformations, and channel noise processes according to the standard formalism of quantum mechanics.
These simulations allow systematic evaluation of the proposed framework under different channel conditions and encoding configurations while avoiding hardware-specific constraints that may obscure algorithmic performance. The experimental evaluation considers video sequences with different motion characteristics (low, medium, and high-motion content) across multiple resolutions and frame rates, using both uncompressed and VVC-encoded inputs. The results should therefore be interpreted as a proof-of-concept validation of the proposed QFT-based video transmission framework, demonstrating its theoretical feasibility and system-level behavior rather than claiming immediate practical deployment on existing quantum hardware.
By abstracting some hardware-related limitations, this approach focuses on evaluating the conceptual feasibility of the proposed quantum communication framework before practical deployment. The simulation results provide theoretical support for the system design and help assess its potential effectiveness. As quantum technologies continue to advance, the findings from these studies can support future prototype development, experimental validation, and hardware-level testing, facilitating the gradual transition from simulation-based analysis to real-world implementation.
## 5\. Conclusions
This study presents and evaluates a novel QFT-based framework for efficient video compression and transmission over noisy communication channels. Through comprehensive analysis across different encoding parameters and channel conditions, several key insights emerge regarding the trade-offs inherent in quantum-inspired video transmission systems. The results indicate that the qubit encoding size plays a crucial role in balancing compression efficiency and robustness to channel noise. Larger qubit encodings enable significantly higher compression ratios, reaching up to 256:1, by representing more classical information within a single quantum state; however, these higher-dimensional encodings become increasingly sensitive to channel impairments due to the reduced angular separation between frequency-domain components. Conversely, smaller qubit group sizes provide stronger resilience to noise at the expense of lower compression efficiency. Under ideal noise-free channel conditions, the proposed framework achieves theoretically lossless reconstruction for all qubit sizes, since the QFT and its inverse are unitary operations that preserve the encoded information. In practical noisy environments, the optimal encoding size depends on the intended application—larger qubit configurations are advantageous when maximizing compression efficiency is the primary objective, whereas smaller encodings are preferable when transmission reliability is critical. Among the evaluated configurations, the two-qubit system provides a particularly effective compromise between compression and robustness, achieving a moderate compression ratio of 4:1 while maintaining stable performance across diverse channel conditions. These findings highlight the potential of QFT-based encoding as a flexible and robust approach for future multimedia transmission systems operating in bandwidth-constrained and error-prone environments.
Looking ahead, this work lays the foundation for several important research directions. Future studies should address practical challenges such as hardware imperfections, computational efficiency, and the development of adaptive qubit allocation strategies based on channel conditions or bandwidth constraints. Designing specialized error correction schemes tailored to quantum frequency-domain representations offers a promising approach to enhancing system reliability while maintaining high compression ratios. Additionally, the development of new compression algorithms capable of optimizing both transmission and storage efficiency could further improve overall system performance. Therefore, this research contributes to the growing field of quantum-inspired signal processing, demonstrating practical approaches for efficient communication systems that leverage quantum principles while remaining compatible with emerging quantum hardware.
## Author Contributions
Conceptualization, U.J.; methodology, U.J.; software, U.J.; validation, U.J. and A.F.; formal analysis, A.F.; investigation, A.F.; resources, U.J.; data curation, U.J.; writing—original draft preparation, U.J.; writing—review and editing, A.F.; visualization, U.J.; supervision, A.F.; project administration, A.F. All authors have read and agreed to the published version of the manuscript.
## Funding
This research received no external funding.
## Data Availability Statement
The original data presented in the study are openly available at [https://www.pexels.com](https://www.pexels.com/) (accessed on 11 December 2025) under the Creative Commons Zero (CC0) license, which allows free use, distribution, and modification without attribution.
## Conflicts of Interest
The authors declare no conflicts of interest.
## Abbreviations
The following abbreviations are used in this manuscript:
| | |
|---|---|
| AVC | Advanced Video Coding |
| BER | Bit Error Rate |
| GOP | Group of Pictures |
| HEVC | High Efficiency Video Coding |
| IQFT | Inverse Quantum Fourier Transform |
| LDPC | Low-Density Parity Check |
| MIMO | Multi-Input Multi-Output |
| OFDM | Orthogonal Frequency-Division Multiplexing |
| PSNR | Peak Signal-to-Noise Ratio |
| QEC | Quantum Error Correction |
| QFT | Quantum Fourier Transform |
| QKD | Quantum Key Distribution |
| SI | Structural Information |
| SNR | Signal-to-Noise Ratio |
| SSIM | Structural Similarity Index Measure |
| TI | Temporal Information |
| VMAF | Video Multi-Method Assessment Fusion |
| VVC | Versatile Video Coding |
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