🕷️ Crawler Inspector

[\[Jump to code hierarchy\]](https://errorcorrectionzoo.org/c/qecc#code_hierarchy) # Quantum error-correcting code (QECC) ## Description Encodes quantum information in a (logical) subspace of a (physical) Hilbert space such that it is possible to recover said information from errors that act as linear maps on the physical space. The state space of a QECC is contained in the space of complex \\(L^2\\)\-normalizable functions of some configuration space, which usually corresponds to the alphabet of a classical code. Since quantum information is encoded in quantum superpositions, an additional source of noise (not relevant to classical encodings) can affect the relative phase of such superpositions. Quantum error-correcting codes have to protect against such phase-flip noise while also protecting against conventional classical bit-flip noise. Codes are not expected to protect against both types of noise perfectly, and there is generally a tradeoff. The logical subspace is spanned by a basis comprised of code basis states or codewords. Codewords may not be normalizable if the physical Hilbert space in infinite, so approximate versions have to be constructed in practice. While all considered QECC states are complex functions, real or quaternionic function spaces can also be considered for QEC \[[1](https://errorcorrectionzoo.org/c/qecc#citation-1),[2](https://errorcorrectionzoo.org/c/qecc#citation-2)\]. ## Protection Denoting Hilbert spaces by the letter \\(\\mathsf{H}\\), a quantum code \\((U,\\cal{E})\\) is a partial isometry \\(U:\\mathsf{H}\_{\\text{logical}}\\to\\mathsf{H}\_{\\text{physical}}\\) with a set of correctable errors \\(\\cal{E}\\) with the following property: there exists a quantum operation \\(\\cal{D}\\) such that for all \\(E\\in\\cal{E}\\) and states \\(\|\\psi\\rangle\\in\\mathsf{H}\_{\\text{logical}}\\), \\begin{align} {\\cal D} (EU\|\\psi\\rangle\\langle\\psi\|U^{\\dagger}E^{\\dagger}) = c(E,\|\\psi\\rangle)\|\\psi\\rangle\\langle\\psi\| \\tag\*{(1)}\\end{align} for some constant \\(c\\). Equivalently, correction capability is determined by the [Knill-Laflamme conditions](https://errorcorrectionzoo.org/c/qecc_finite#defterm-Knill-Laflamme_20Xconditions), which may admit infinite terms due to non-normalizability of ideal code states in the case of codes with infinite-dimenional physical spaces. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an [approximate code](https://errorcorrectionzoo.org/c/approximate_qecc). These conditions can also be formulated in terms of a dual Heisenberg picture, where correctability is checked for some algebra of observables [\[3\]](https://errorcorrectionzoo.org/c/qecc#citation-3). Pseudo-threshold (a.k.a. break-even point): The ultimate goal of error correction is to make sure that the logical error rate is smaller than the underlying physical error rate. For a noise model parameterized by a single physical error rate \\(p\\), the pseudo-threshold or break-even point is the smallest \\(p\\) at which the logical error rate after error correction is equal to \\(p\\). ## Rate The quantum channel capacity, i.e., the regularized coherent information, is the highest rate of quantum information transmission through a quantum channel with arbitrarily small error rate \[[4](https://errorcorrectionzoo.org/c/qecc#citation-4)–[6](https://errorcorrectionzoo.org/c/qecc#citation-6)\]. In other words, the capacity formula implies that one can achieve a transmission rate \\(r\\) over a quantum channel \\(\\mathcal{E}\\) iff, for sufficiently large \\(n\\), \\(m=\\lfloor r n \\rfloor\\), and for all \\(\\epsilon\>0\\), \\begin{align} \|\|\\mathcal{D}\\mathcal{E}\\mathcal{U}-I^{\\otimes m}\|\|\_1\\leq \\epsilon \\tag\*{(2)}\\end{align} for some encoding channel \\(\\mathcal{U}\\) and some recovery channel \\(\\mathcal{D}\\). The quantum capacity \\(Q\\) of \\(\\mathcal{E}\\) is defined as the supremum over \\(n\\) of achievable transmission rates [\[7\]](https://errorcorrectionzoo.org/c/qecc#citation-7). See [\[8; Ch. 24\]](https://errorcorrectionzoo.org/c/qecc#citation-8) for definitions and a history. The fault-tolerant capacity is the capacity for the more general case where the encoding and decoding maps are also assumed to undergo noise [\[9\]](https://errorcorrectionzoo.org/c/qecc#citation-9). Doublin coefficients [\[10\]](https://errorcorrectionzoo.org/c/qecc#citation-10) for quantum channels have been studied [\[11\]](https://errorcorrectionzoo.org/c/qecc#citation-11). ## Decoding The effect of an error is a mapping of the code subspace into another, potentially overlapping, subspace. To determine, or diagnose, the effect of the error in what is known as syndrome-based decoding, one can measure one or more operators called check operators, which resolve code and error spaces without collapsing the quantum information inside the spaces. The eigenvalues of check operators are called error syndromes. One round or cycle of quantum error correction proceeds by extracting syndromes and performing correcting operations to map the error space containing the logical information back into the codespace. For some codes, correcting operations are not necessary because one can instead track which error space contains the logical information. ## Notes See Refs. \[[12](https://errorcorrectionzoo.org/c/qecc#citation-12)–[29](https://errorcorrectionzoo.org/c/qecc#citation-29)\] for overviews of quantum error correction.See Refs. \[[30](https://errorcorrectionzoo.org/c/qecc#citation-30)–[32](https://errorcorrectionzoo.org/c/qecc#citation-32)\] for books on quantum error correction.See video tutorials by [V. V. Albert](https://www.youtube.com/watch?v=_ls3KczZL2c), [S. M. Girvin](https://www.youtube.com/watch?v=uD69GCYF9Zg), [P. Shor](https://www.youtube.com/watch?v=buIbd_aXAHw), [B. Terhal](https://www.youtube.com/watch?v=Je7sVJGKMgU), and [J. Wright](https://www.youtube.com/watch?v=mcwpe8iJ5uo).Quantum error correction was initially claimed not to be theoretically possible \[[33](https://errorcorrectionzoo.org/c/qecc#citation-33),[34](https://errorcorrectionzoo.org/c/qecc#citation-34)\] and has been criticized since [\[35\]](https://errorcorrectionzoo.org/c/qecc#citation-35).Resource-theoretic interpretations of quantum error correction have been developed, including those that think of codes together with recovery operations as superchannels (a.k.a. quantum combs or bipartite operations) \[[36](https://errorcorrectionzoo.org/c/qecc#citation-36)–[40](https://errorcorrectionzoo.org/c/qecc#citation-40)\].QECC can be used as a mechanism for securing quantum computation [\[41\]](https://errorcorrectionzoo.org/c/qecc#citation-41). ## Cousins - [Approximate quantum error-correcting code (AQECC)](https://errorcorrectionzoo.org/c/approximate_qecc)— QAECCs correcting a noise channel exactly reduce to QECCs. - [Error-correcting code (ECC)](https://errorcorrectionzoo.org/c/ecc)— Quantum information cannot be copied using a linear process [\[42\]](https://errorcorrectionzoo.org/c/qecc#citation-42), so one cannot send several copies of a quantum state through a channel as can be done for classical information. The [Knill-Laflamme conditions](https://errorcorrectionzoo.org/c/qecc_finite#defterm-Knill-Laflamme_20Xconditions) can similarly be formulated for classical codes [\[43; Sec. 3\]](https://errorcorrectionzoo.org/c/qecc#citation-43), although they are not as widely as used as those for quantum codes. - [Metrological code](https://errorcorrectionzoo.org/c/metrological)— Metrological codes are logical-qubit codes that satisfy the [Knill-Laflamme conditions](https://errorcorrectionzoo.org/c/qecc_finite#defterm-Knill-Laflamme_20Xconditions) conditions only partially, and codes that satisfy them fully are QECCs. - [Entanglement-assisted (EA) QECC](https://errorcorrectionzoo.org/c/eaqecc)— EA QECCs utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to QECCs when said subsystems are interpreted as noiseless physical subsystems. - [Hybrid QECC](https://errorcorrectionzoo.org/c/hybridqecc)— A hybrid QECC storing no classical information reduces to a QECC. Conversely, any QECC can be converted into a hybrid QECC by using a portion of its logical subspace to store only classical information. - [Subsystem QECC](https://errorcorrectionzoo.org/c/oecc)— A subsystem QECC reduces to an ordinary (i.e., subspace) QECC when the gauge subsystem is trivial. Conversely, any QECC with a tensor-product logical subspace can be turned into a subsystem code by treating a logical tensor factor as a gauge subsystem. - [Holographic tensor-network code](https://errorcorrectionzoo.org/c/holographic_tensor)— Quantum encoding maps are isometries, but non-isometric encodings are relevant to describing mappings into the interior of a black hole [\[44\]](https://errorcorrectionzoo.org/c/qecc#citation-44) and de Sitter time evolution [\[45\]](https://errorcorrectionzoo.org/c/qecc#citation-45). Trace-norm preserving encodings have also been studied [\[46\]](https://errorcorrectionzoo.org/c/qecc#citation-46). - [Quantum error-transmuting code (QETC)](https://errorcorrectionzoo.org/c/qetc)— QETCs are quantum codes which satisfy a generalization of the [Knill-Laflamme conditions](https://errorcorrectionzoo.org/c/qecc_finite#defterm-Knill-Laflamme_20Xconditions). QETCs for which the admissible logical error set consists solely of the identity are QECCs. - [\\(\[\[9,1,3\]\]\\) Shor code](https://errorcorrectionzoo.org/c/shor_nine)— The Shor code is the first quantum error-correcting code. ## Member of code lists - [Approximate quantum codes](https://errorcorrectionzoo.org/list/approximate_qecc) - [Entanglement-assisted quantum codes and friends](https://errorcorrectionzoo.org/list/eaqecc) - [Quantum codes](https://errorcorrectionzoo.org/list/quantum) - [Quantum codes with a rate](https://errorcorrectionzoo.org/list/quantum_with_rate) - [Quantum codes with notable decoders](https://errorcorrectionzoo.org/list/quantum_decoders) ## Primary Hierarchy [Quantum Domain](https://errorcorrectionzoo.org/domain/quantum_domain) [Quantum code](https://errorcorrectionzoo.org/c/quantum_into_quantum) [Operator-algebra QECC (OAQECC)](https://errorcorrectionzoo.org/c/oaecc) Parents [Operator-algebra QECC (OAQECC)](https://errorcorrectionzoo.org/c/oaecc) An OAQECC which has no gauge structure (e.g., gauge qubits) and no block structure is a QECC. Quantum error-correcting code (QECC) Children [Category-based quantum code](https://errorcorrectionzoo.org/c/category_quantum)[Fock-state bosonic](https://errorcorrectionzoo.org/c/fock_state "Category-based quantum code → Group-based quantum code → Mixed oscillator code → Bosonic code → Qudit-into-oscillator code → Fock-state bosonic code") [Stabilizer](https://errorcorrectionzoo.org/c/stabilizer "Category-based quantum code → Group-based quantum code → Stabilizer code") [CSS](https://errorcorrectionzoo.org/c/css "Category-based quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code") [QLDPC](https://errorcorrectionzoo.org/c/qldpc "Category-based quantum code → Group-based quantum code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code") [Generalized homological-product](https://errorcorrectionzoo.org/c/generalized_homological_product "Category-based quantum code → Group-based quantum code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Generalized homological-product code") [Lattice stabilizer](https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer "Category-based quantum code → Group-based quantum code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Quasi-cyclic QLDPC (QC-QLDPC) code → Lattice stabilizer code") [Bosonic stabilizer](https://errorcorrectionzoo.org/c/oscillator_stabilizer "Category-based quantum code → Group-based quantum code → Stabilizer code → Bosonic stabilizer code") [Quantum lattice](https://errorcorrectionzoo.org/c/quantum_lattice "Category-based quantum code → Group-based quantum code → Mixed oscillator code → Locally compact Abelian (LCA) stabilizer code → Quantum lattice code") [Analog stabilizer](https://errorcorrectionzoo.org/c/analog_stabilizer "Category-based quantum code → Group-based quantum code → Stabilizer code → Bosonic stabilizer code → Analog stabilizer code") [Fracton stabilizer](https://errorcorrectionzoo.org/c/fracton "Category-based quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → 3D lattice stabilizer code → Fracton stabilizer code") [Qubit](https://errorcorrectionzoo.org/c/qubits_into_qubits "Category-based quantum code → Group-based quantum code → Modular-qudit code → Qubit code") [Hastings-Haah Floquet](https://errorcorrectionzoo.org/c/floquet "Category-based quantum code → Group-based quantum code → Modular-qudit code → Qubit code → Dynamical code → Hastings-Haah Floquet code") [Fermion](https://errorcorrectionzoo.org/c/fermions "Category-based quantum code → Group-based quantum code → Modular-qudit code → Qubit code → Fermion code") [Self-complementary qubit](https://errorcorrectionzoo.org/c/self_complementary "Category-based quantum code → Group-based quantum code → Modular-qudit code → Qubit code → Self-complementary qubit code") [Quantum QR code](https://errorcorrectionzoo.org/c/galois_quad_residue "Category-based quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Galois-qudit CSS code → Quantum quadratic-residue (QR) code") [BB](https://errorcorrectionzoo.org/c/qcga "Category-based quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Bivariate bicycle (BB) code") [Homological](https://errorcorrectionzoo.org/c/higher_dimensional_surface "Category-based quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Homological code") [Fermion-into-qubit](https://errorcorrectionzoo.org/c/fermions_into_qubits "Category-based quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Fermion-into-qubit code") [Quantum RM code](https://errorcorrectionzoo.org/c/quantum_reed_muller "Category-based quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Modular-qudit CSS code → Prime-qudit RM code → Quantum Reed-Muller (RM) code") [Color](https://errorcorrectionzoo.org/c/color "Category-based quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → \(k\)-orthogonal code → Quantum pin code → Color code") [Twist-defect color](https://errorcorrectionzoo.org/c/twist_defect_color "Category-based quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect color code") [CDSC](https://errorcorrectionzoo.org/c/clifford-deformed_surface "Category-based quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Clifford-deformed surface code (CDSC)") [Twist-defect surface](https://errorcorrectionzoo.org/c/twist_defect_surface "Category-based quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect surface code") [Kitaev surface](https://errorcorrectionzoo.org/c/surface "Category-based quantum code → Two-gauge theory code → Dijkgraaf-Witten gauge theory code → Twisted quantum double (TQD) code → Quantum-double code → Galois-qudit surface code → Kitaev surface code") [Homogeneous-space quantum code](https://errorcorrectionzoo.org/c/homogeneous_space_quantum)[Fock-state bosonic](https://errorcorrectionzoo.org/c/fock_state "Homogeneous-space quantum code → Group-based quantum code → Mixed oscillator code → Bosonic code → Qudit-into-oscillator code → Fock-state bosonic code") [Stabilizer](https://errorcorrectionzoo.org/c/stabilizer "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code") [CSS](https://errorcorrectionzoo.org/c/css "Homogeneous-space quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code") [QLDPC](https://errorcorrectionzoo.org/c/qldpc "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code") [Generalized homological-product](https://errorcorrectionzoo.org/c/generalized_homological_product "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Generalized homological-product code") [Lattice stabilizer](https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Quasi-cyclic QLDPC (QC-QLDPC) code → Lattice stabilizer code") [Bosonic stabilizer](https://errorcorrectionzoo.org/c/oscillator_stabilizer "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Bosonic stabilizer code") [Quantum lattice](https://errorcorrectionzoo.org/c/quantum_lattice "Homogeneous-space quantum code → Group-based quantum code → Mixed oscillator code → Locally compact Abelian (LCA) stabilizer code → Quantum lattice code") [Analog stabilizer](https://errorcorrectionzoo.org/c/analog_stabilizer "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Bosonic stabilizer code → Analog stabilizer code") [Fracton stabilizer](https://errorcorrectionzoo.org/c/fracton "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → 3D lattice stabilizer code → Fracton stabilizer code") [Qubit](https://errorcorrectionzoo.org/c/qubits_into_qubits "Homogeneous-space quantum code → Group-based quantum code → Modular-qudit code → Qubit code") [Hastings-Haah Floquet](https://errorcorrectionzoo.org/c/floquet "Homogeneous-space quantum code → Group-based quantum code → Modular-qudit code → Qubit code → Dynamical code → Hastings-Haah Floquet code") [Fermion](https://errorcorrectionzoo.org/c/fermions "Homogeneous-space quantum code → Group-based quantum code → Modular-qudit code → Qubit code → Fermion code") [Self-complementary qubit](https://errorcorrectionzoo.org/c/self_complementary "Homogeneous-space quantum code → Group-based quantum code → Modular-qudit code → Qubit code → Self-complementary qubit code") [Quantum QR code](https://errorcorrectionzoo.org/c/galois_quad_residue "Homogeneous-space quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Galois-qudit CSS code → Quantum quadratic-residue (QR) code") [BB](https://errorcorrectionzoo.org/c/qcga "Homogeneous-space quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Bivariate bicycle (BB) code") [Homological](https://errorcorrectionzoo.org/c/higher_dimensional_surface "Homogeneous-space quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Homological code") [Fermion-into-qubit](https://errorcorrectionzoo.org/c/fermions_into_qubits "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Fermion-into-qubit code") [Quantum RM code](https://errorcorrectionzoo.org/c/quantum_reed_muller "Homogeneous-space quantum code → Group-based quantum code → Group GKP code → Calderbank-Shor-Steane (CSS) stabilizer code → Modular-qudit CSS code → Prime-qudit RM code → Quantum Reed-Muller (RM) code") [Color](https://errorcorrectionzoo.org/c/color "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → \(k\)-orthogonal code → Quantum pin code → Color code") [Twist-defect color](https://errorcorrectionzoo.org/c/twist_defect_color "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect color code") [CDSC](https://errorcorrectionzoo.org/c/clifford-deformed_surface "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Clifford-deformed surface code (CDSC)") [Twist-defect surface](https://errorcorrectionzoo.org/c/twist_defect_surface "Homogeneous-space quantum code → Group-based quantum code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect surface code") [Kitaev surface](https://errorcorrectionzoo.org/c/surface "Homogeneous-space quantum code → Group-based quantum code → Dijkgraaf-Witten gauge theory code → Twisted quantum double (TQD) code → Quantum-double code → Galois-qudit surface code → Kitaev surface code") [Asymmetric quantum code](https://errorcorrectionzoo.org/c/asymmetric_qecc) [Block quantum code](https://errorcorrectionzoo.org/c/block_quantum)[Amplitude-damping (AD)](https://errorcorrectionzoo.org/c/ampdamp "Block quantum code → Amplitude-damping (AD) code") [Fock-state bosonic](https://errorcorrectionzoo.org/c/fock_state "Block quantum code → Amplitude-damping (AD) code → Fock-state bosonic code") [CE](https://errorcorrectionzoo.org/c/constant_excitation "Block quantum code → Amplitude-damping (AD) code → Constant-excitation (CE) code") [Stabilizer](https://errorcorrectionzoo.org/c/stabilizer "Block quantum code → Covariant block quantum code → Group-representation code → Knill code → Stabilizer code") [Bosonic stabilizer](https://errorcorrectionzoo.org/c/oscillator_stabilizer "Block quantum code → Bosonic code → Bosonic stabilizer code") [Analog stabilizer](https://errorcorrectionzoo.org/c/analog_stabilizer "Block quantum code → Bosonic code → Oscillator-into-oscillator code → Analog stabilizer code") [Quantum lattice](https://errorcorrectionzoo.org/c/quantum_lattice "Block quantum code → Bosonic code → Coherent-state constellation code → Quantum lattice code") [CSS](https://errorcorrectionzoo.org/c/css "Block quantum code → Covariant block quantum code → Group-representation code → Knill code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code") [Dynamically generated](https://errorcorrectionzoo.org/c/dynamic_gen "Block quantum code → Dynamically generated QECC") [QLDPC](https://errorcorrectionzoo.org/c/qldpc "Block quantum code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code") [Generalized homological-product](https://errorcorrectionzoo.org/c/generalized_homological_product "Block quantum code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Generalized homological-product code") [Quantum MDS](https://errorcorrectionzoo.org/c/quantum_mds "Block quantum code → Quantum maximum-distance-separable (MDS) code") [Single-shot](https://errorcorrectionzoo.org/c/single_shot "Block quantum code → Single-shot code") [Small-distance block quantum](https://errorcorrectionzoo.org/c/small_distance_quantum "Block quantum code → Small-distance block quantum code") [Perfect quantum](https://errorcorrectionzoo.org/c/quantum_perfect "Block quantum code → Small-distance block quantum code → Perfect quantum code") [Lattice stabilizer](https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer "Block quantum code → Quasi-cyclic quantum code → Quasi-cyclic QLDPC (QC-QLDPC) code → Lattice stabilizer code") [Cyclic quantum](https://errorcorrectionzoo.org/c/quantum_cyclic "Block quantum code → Quasi-cyclic quantum code → Cyclic quantum code") [PI](https://errorcorrectionzoo.org/c/permutation_invariant "Block quantum code → Quasi-cyclic quantum code → Cyclic quantum code → Permutation-invariant (PI) code") [Concatenated quantum](https://errorcorrectionzoo.org/c/quantum_concatenated "Block quantum code → Tensor-network code → Concatenated quantum code") [Topological](https://errorcorrectionzoo.org/c/topological "Block quantum code → Topological code") [Abelian topological](https://errorcorrectionzoo.org/c/topological_abelian "Block quantum code → Topological code → Abelian topological code") [QLTC](https://errorcorrectionzoo.org/c/qltc "Block quantum code → Quantum locally testable code (QLTC)") [Fracton stabilizer](https://errorcorrectionzoo.org/c/fracton "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → 3D lattice stabilizer code → Fracton stabilizer code") [Qubit](https://errorcorrectionzoo.org/c/qubits_into_qubits "Block quantum code → Modular-qudit code → Qubit code") [Self-complementary qubit](https://errorcorrectionzoo.org/c/self_complementary "Block quantum code → Amplitude-damping (AD) code → Self-complementary qubit code") [Hastings-Haah Floquet](https://errorcorrectionzoo.org/c/floquet "Block quantum code → Dynamically generated QECC → Modular-qudit dynamical code → Dynamical code → Hastings-Haah Floquet code") [Fermion](https://errorcorrectionzoo.org/c/fermions "Block quantum code → Modular-qudit code → Qubit code → Fermion code") [Fermion-into-qubit](https://errorcorrectionzoo.org/c/fermions_into_qubits "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Qubit stabilizer code → Fermion-into-qubit code") [Twist-defect color](https://errorcorrectionzoo.org/c/twist_defect_color "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect color code") [CDSC](https://errorcorrectionzoo.org/c/clifford-deformed_surface "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Qubit stabilizer code → Clifford-deformed surface code (CDSC)") [Twist-defect surface](https://errorcorrectionzoo.org/c/twist_defect_surface "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect surface code") [Color](https://errorcorrectionzoo.org/c/color "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Qubit stabilizer code → \(k\)-orthogonal code → Quantum pin code → Color code") [Homological](https://errorcorrectionzoo.org/c/higher_dimensional_surface "Block quantum code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Generalized homological-product code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Homological code") [Kitaev surface](https://errorcorrectionzoo.org/c/surface "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Qubit stabilizer code → Clifford-deformed surface code (CDSC) → Kitaev surface code") [BB](https://errorcorrectionzoo.org/c/qcga "Block quantum code → Quasi-cyclic quantum code → Quasi-cyclic QLDPC (QC-QLDPC) code → Lattice stabilizer code → 2D lattice stabilizer code → Bivariate bicycle (BB) code") [Quantum QR code](https://errorcorrectionzoo.org/c/galois_quad_residue "Block quantum code → Galois-qudit code → Galois-qudit USt code → Galois-qudit stabilizer code → True Galois-qudit stabilizer code → Galois-qudit CSS code → Quantum quadratic-residue (QR) code") [Quantum RM code](https://errorcorrectionzoo.org/c/quantum_reed_muller "Block quantum code → Tensor-network code → Modular-qudit stabilizer code → Modular-qudit CSS code → Prime-qudit RM code → Quantum Reed-Muller (RM) code") [Holographic code](https://errorcorrectionzoo.org/c/holographic) [Hamiltonian-based code](https://errorcorrectionzoo.org/c/hamiltonian)[Topological](https://errorcorrectionzoo.org/c/topological "Hamiltonian-based code → Topological code") [Abelian topological](https://errorcorrectionzoo.org/c/topological_abelian "Hamiltonian-based code → Topological code → Abelian topological code") [CE](https://errorcorrectionzoo.org/c/constant_excitation "Hamiltonian-based code → Constant-excitation (CE) code") [Stabilizer](https://errorcorrectionzoo.org/c/stabilizer "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code") [CSS](https://errorcorrectionzoo.org/c/css "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code") [QLDPC](https://errorcorrectionzoo.org/c/qldpc "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code") [Generalized homological-product](https://errorcorrectionzoo.org/c/generalized_homological_product "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Generalized homological-product code") [Lattice stabilizer](https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Quantum low-weight check (QLWC) code → Quantum LDPC (QLDPC) code → Quasi-cyclic QLDPC (QC-QLDPC) code → Lattice stabilizer code") [Bosonic stabilizer](https://errorcorrectionzoo.org/c/oscillator_stabilizer "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Bosonic stabilizer code") [Quantum lattice](https://errorcorrectionzoo.org/c/quantum_lattice "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Locally compact Abelian (LCA) stabilizer code → Quantum lattice code") [Analog stabilizer](https://errorcorrectionzoo.org/c/analog_stabilizer "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Bosonic stabilizer code → Analog stabilizer code") [Fracton stabilizer](https://errorcorrectionzoo.org/c/fracton "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Modular-qudit stabilizer code → 3D lattice stabilizer code → Fracton stabilizer code") [BB](https://errorcorrectionzoo.org/c/qcga "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Bivariate bicycle (BB) code") [Homological](https://errorcorrectionzoo.org/c/higher_dimensional_surface "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Generalized homological-product qubit CSS code → Homological code") [Fermion-into-qubit](https://errorcorrectionzoo.org/c/fermions_into_qubits "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Fermion-into-qubit code") [Color](https://errorcorrectionzoo.org/c/color "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → \(k\)-orthogonal code → Quantum pin code → Color code") [Twist-defect color](https://errorcorrectionzoo.org/c/twist_defect_color "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect color code") [CDSC](https://errorcorrectionzoo.org/c/clifford-deformed_surface "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Clifford-deformed surface code (CDSC)") [Twist-defect surface](https://errorcorrectionzoo.org/c/twist_defect_surface "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Modular-qudit stabilizer code → Qubit stabilizer code → Twist-defect surface code") [Kitaev surface](https://errorcorrectionzoo.org/c/surface "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code → Generalized homological-product CSS code → Modular-qudit surface code → Kitaev surface code") [Quantum QR code](https://errorcorrectionzoo.org/c/galois_quad_residue "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code → Galois-qudit CSS code → Quantum quadratic-residue (QR) code") [Quantum RM code](https://errorcorrectionzoo.org/c/quantum_reed_muller "Hamiltonian-based code → Commuting-projector Hamiltonian code → Stabilizer code → Calderbank-Shor-Steane (CSS) stabilizer code → Modular-qudit CSS code → Prime-qudit RM code → Quantum Reed-Muller (RM) code") [QLTC](https://errorcorrectionzoo.org/c/qltc "Hamiltonian-based code → Commuting-projector Hamiltonian code → Quantum locally testable code (QLTC)") [Self-correcting quantum](https://errorcorrectionzoo.org/c/self_correct "Hamiltonian-based code → Symmetry-protected self-correcting quantum code → Self-correcting quantum code") [Finite-dimensional quantum error-correcting code](https://errorcorrectionzoo.org/c/qecc_finite)[Quantum MDS](https://errorcorrectionzoo.org/c/quantum_mds "Finite-dimensional quantum error-correcting code → Quantum maximum-distance-separable (MDS) code") [Perfect quantum](https://errorcorrectionzoo.org/c/quantum_perfect "Finite-dimensional quantum error-correcting code → Perfect quantum code") [Fracton stabilizer](https://errorcorrectionzoo.org/c/fracton "Finite-dimensional quantum error-correcting code → Modular-qudit code → Modular-qudit USt code → Modular-qudit stabilizer code → 3D lattice stabilizer code → Fracton stabilizer code") [Quantum QR code](https://errorcorrectionzoo.org/c/galois_quad_residue "Finite-dimensional quantum error-correcting code → Galois-qudit code → Galois-qudit USt code → Galois-qudit stabilizer code → True Galois-qudit stabilizer code → Galois-qudit CSS code → Quantum quadratic-residue (QR) code") [Qubit](https://errorcorrectionzoo.org/c/qubits_into_qubits "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code") [Hastings-Haah Floquet](https://errorcorrectionzoo.org/c/floquet "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Dynamical code → Hastings-Haah Floquet code") [Fermion](https://errorcorrectionzoo.org/c/fermions "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Fermion code") [Self-complementary qubit](https://errorcorrectionzoo.org/c/self_complementary "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Self-complementary qubit code") [BB](https://errorcorrectionzoo.org/c/qcga "Finite-dimensional quantum error-correcting code → Modular-qudit code → Modular-qudit USt code → Modular-qudit stabilizer code → Modular-qudit CSS code → Qubit CSS code → Generalized homological-product qubit CSS code → Bivariate bicycle (BB) code") [Homological](https://errorcorrectionzoo.org/c/higher_dimensional_surface "Finite-dimensional quantum error-correcting code → Modular-qudit code → Modular-qudit USt code → Modular-qudit stabilizer code → Modular-qudit CSS code → Qubit CSS code → Generalized homological-product qubit CSS code → Homological code") [Fermion-into-qubit](https://errorcorrectionzoo.org/c/fermions_into_qubits "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Union stabilizer (USt) code → Qubit stabilizer code → Fermion-into-qubit code") [Quantum RM code](https://errorcorrectionzoo.org/c/quantum_reed_muller "Finite-dimensional quantum error-correcting code → Modular-qudit code → Modular-qudit USt code → Modular-qudit stabilizer code → Modular-qudit CSS code → Prime-qudit RM code → Quantum Reed-Muller (RM) code") [Color](https://errorcorrectionzoo.org/c/color "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Union stabilizer (USt) code → Qubit stabilizer code → \(k\)-orthogonal code → Quantum pin code → Color code") [Twist-defect color](https://errorcorrectionzoo.org/c/twist_defect_color "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Union stabilizer (USt) code → Qubit stabilizer code → Twist-defect color code") [CDSC](https://errorcorrectionzoo.org/c/clifford-deformed_surface "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Union stabilizer (USt) code → Qubit stabilizer code → Clifford-deformed surface code (CDSC)") [Twist-defect surface](https://errorcorrectionzoo.org/c/twist_defect_surface "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Union stabilizer (USt) code → Qubit stabilizer code → Twist-defect surface code") [Kitaev surface](https://errorcorrectionzoo.org/c/surface "Finite-dimensional quantum error-correcting code → Modular-qudit code → Qubit code → Union stabilizer (USt) code → Qubit stabilizer code → Clifford-deformed surface code (CDSC) → Kitaev surface code") [Monolithic quantum code](https://errorcorrectionzoo.org/c/single_subsystem) ## References \[1\] A. Yu. Vlasov, “Error Correction with Euclidean Qubits”, (1999) [arXiv:quant-ph/9911074](https://arxiv.org/abs/quant-ph/9911074) \[2\] V. Nyirahafashimana, N. M. Shah, U. A. Halim, M. Othman, and S. K. S. Husain, “Quantum Error Correction in Quaternionic Hilbert Spaces”, (2025) [arXiv:2504.19833](https://arxiv.org/abs/2504.19833) \[3\] C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction on infinite-dimensional Hilbert spaces”, Journal of Mathematical Physics 50, (2009) [arXiv:0811.0421](https://arxiv.org/abs/0811.0421) [DOI](https://doi.org/10.1063%2F1.3155783) \[4\] S. Lloyd, “Capacity of the noisy quantum channel”, Physical Review A 55, 1613 (1997) [arXiv:quant-ph/9604015](https://arxiv.org/abs/quant-ph/9604015) [DOI](https://doi.org/10.1103%2Fphysreva.55.1613) \[5\] Peter W. Shor, [The quantum channel capacity and coherent information](https://github.com/errorcorrectionzoo/eczoo_data/files/7808154/shor.pdf), 2002 (obtained from the MSRI Workshop on Quantum Computation website). \[6\] I. Devetak, “The private classical capacity and quantum capacity of a quantum channel”, (2004) [arXiv:quant-ph/0304127](https://arxiv.org/abs/quant-ph/0304127) \[7\] J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018) [DOI](https://doi.org/10.1017%2F9781316848142) \[8\] “Preface to the Second Edition”, Quantum Information Theory xi (2016) [arXiv:1106.1445](https://arxiv.org/abs/1106.1445) [DOI](https://doi.org/10.1017%2F9781316809976.001) \[9\] M. Christandl and A. Müller-Hermes, “Fault-Tolerant Coding for Quantum Communication”, IEEE Transactions on Information Theory 70, 282 (2024) [arXiv:2009.07161](https://arxiv.org/abs/2009.07161) [DOI](https://doi.org/10.1109%2Ftit.2022.3169438) \[10\] Doeblin, W. (1937). Sur les propriétés asymptotiques de mouvement régis par certains types de chaines simples. Bulletin mathématique de la Société roumaine des sciences, 39(1), 57-115. \[11\] A. Makur and J. Singh, “Doeblin Coefficients and Related Measures”, IEEE Transactions on Information Theory 70, 4667 (2024) [arXiv:2309.08475](https://arxiv.org/abs/2309.08475) [DOI](https://doi.org/10.1109%2Ftit.2024.3367856) \[12\] J. Preskill, “Fault-tolerant quantum computation”, (1997) [arXiv:quant-ph/9712048](https://arxiv.org/abs/quant-ph/9712048) \[13\] D. Gottesman, “An Introduction to Quantum Error Correction”, (2000) [arXiv:quant-ph/0004072](https://arxiv.org/abs/quant-ph/0004072) \[14\] A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (American Mathematical Society, 2002) [DOI](https://doi.org/10.1090%2Fgsm%2F047) \[15\] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) [DOI](https://doi.org/10.1017%2Fcbo9780511976667) \[16\] D. Gottesman, “Quantum Error Correction and Fault-Tolerance”, (2005) [arXiv:quant-ph/0507174](https://arxiv.org/abs/quant-ph/0507174) \[17\] J. Kempe, “Approaches to Quantum Error Correction”, (2006) [arXiv:quant-ph/0612185](https://arxiv.org/abs/quant-ph/0612185) \[18\] D. Gottesman, “An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation”, (2009) [arXiv:0904.2557](https://arxiv.org/abs/0904.2557) \[19\] S. J. Devitt, W. J. Munro, and K. Nemoto, “Quantum error correction for beginners”, Reports on Progress in Physics 76, 076001 (2013) [arXiv:0905.2794](https://arxiv.org/abs/0905.2794) [DOI](https://doi.org/10.1088%2F0034-4885%2F76%2F7%2F076001) \[20\] B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) [arXiv:1302.3428](https://arxiv.org/abs/1302.3428) [DOI](https://doi.org/10.1103%2Frevmodphys.87.307) \[21\] D. Suter and G. A. Álvarez, “Colloquium: Protecting quantum information against environmental noise”, Reviews of Modern Physics 88, (2016) [DOI](https://doi.org/10.1103%2Frevmodphys.88.041001) \[22\] M. Grassl, “Classical Information Theory and Classical Error Correction”, Quantum Information 1 (2016) [DOI](https://doi.org/10.1002%2F9783527805785.ch1) \[23\] A. Paler and S. J. Devitt, “An introduction to Fault-tolerant Quantum Computing”, (2015) [arXiv:1508.03695](https://arxiv.org/abs/1508.03695) \[24\] J. Roffe, “Quantum error correction: an introductory guide”, Contemporary Physics 60, 226 (2019) [arXiv:1907.11157](https://arxiv.org/abs/1907.11157) [DOI](https://doi.org/10.1080%2F00107514.2019.1667078) \[25\] J. Preskill. Lecture notes on Quantum Computation. (1997–2020) [URL](http://theory.caltech.edu/~preskill/ph219/) \[26\] T. A. Brun, “Quantum Error Correction”, Oxford Research Encyclopedia of Physics (2020) [arXiv:1910.03672](https://arxiv.org/abs/1910.03672) [DOI](https://doi.org/10.1093%2Facrefore%2F9780190871994.013.35) \[27\] S. Ball, A. Centelles, and F. Huber, “Quantum error-correcting codes and their geometries”, Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions 10, 337 (2023) [arXiv:2007.05992](https://arxiv.org/abs/2007.05992) [DOI](https://doi.org/10.4171%2Faihpd%2F160) \[28\] “Quantum Error Correction”, Artificial Intelligence and Quantum Computing for Advanced Wireless Networks 451 (2022) [DOI](https://doi.org/10.1002%2F9781119790327.ch10) \[29\] P. J. Nadkarni, N. Rengaswamy, and B. Vasić, “Tutorial on Quantum Error Correction for 2024 Quantum Information Knowledge (QuIK) Workshop”, (2024) [arXiv:2407.12737](https://arxiv.org/abs/2407.12737) \[30\] D. A. Lidar and T. A. Brun, editors , Quantum Error Correction (Cambridge University Press, 2013) [DOI](https://doi.org/10.1017%2Fcbo9781139034807) \[31\] F. Gaitan, Quantum Error Correction and Fault Tolerant Quantum Computing (CRC Press, 2018) [DOI](https://doi.org/10.1201%2Fb15868) \[32\] D. Gottesman. Surviving as a quantum computer in a classical world (2024) [URL](https://www.cs.umd.edu/class/spring2024/cmsc858G) \[33\] W. G. Unruh, “Maintaining coherence in quantum computers”, Physical Review A 51, 992 (1995) [arXiv:hep-th/9406058](https://arxiv.org/abs/hep-th/9406058) [DOI](https://doi.org/10.1103%2Fphysreva.51.992) \[34\] “Is quantum mechanics useful?”, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 353, 367 (1995) [DOI](https://doi.org/10.1098%2Frsta.1995.0106) \[35\] R. Alicki, “Critique of Fault-Tolerant Quantum Information Processing”, (2013) [arXiv:1310.8457](https://arxiv.org/abs/1310.8457) \[36\] O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order”, Nature Communications 3, (2012) [arXiv:1105.4464](https://arxiv.org/abs/1105.4464) [DOI](https://doi.org/10.1038%2Fncomms2076) \[37\] W. Matthews and S. Wehner, “Finite Blocklength Converse Bounds for Quantum Channels”, IEEE Transactions on Information Theory 60, 7317 (2014) [arXiv:1210.4722](https://arxiv.org/abs/1210.4722) [DOI](https://doi.org/10.1109%2Ftit.2014.2353614) \[38\] D. Leung and W. Matthews, “On the Power of PPT-Preserving and Non-Signalling Codes”, IEEE Transactions on Information Theory 61, 4486 (2015) [arXiv:1406.7142](https://arxiv.org/abs/1406.7142) [DOI](https://doi.org/10.1109%2Ftit.2015.2439953) \[39\] A. Tanggara, M. Gu, and K. Bharti, “Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction”, (2024) [arXiv:2405.17567](https://arxiv.org/abs/2405.17567) \[40\] D.-S. Wang, Y.-D. Liu, Y.-J. Wang, and S. Luo, “Quantum resource theory of coding for error correction”, Physical Review A 110, (2024) [arXiv:2409.09416](https://arxiv.org/abs/2409.09416) [DOI](https://doi.org/10.1103%2Fphysreva.110.032413) \[41\] V. Mutolo, D. Campbell, Q. Manning, H. W. Dubourg, R. Lyu, S. Sethumadhavan, D. Rubenstein, and S. Stolfo, “Quantum Computer Fingerprinting using Error Syndromes”, (2025) [arXiv:2506.16614](https://arxiv.org/abs/2506.16614) \[42\] W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature 299, 802 (1982) [DOI](https://doi.org/10.1038%2F299802a0) \[43\] B. Yoshida, “Decoding the Entanglement Structure of Monitored Quantum Circuits”, (2021) [arXiv:2109.08691](https://arxiv.org/abs/2109.08691) \[44\] C. Akers, N. Engelhardt, D. Harlow, G. Penington, and S. Vardhan, “The black hole interior from non-isometric codes and complexity”, (2022) [arXiv:2207.06536](https://arxiv.org/abs/2207.06536) \[45\] J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) [arXiv:2201.11658](https://arxiv.org/abs/2201.11658) \[46\] F. Ticozzi and L. Viola, “Quantum information encoding, protection, and correction from trace-norm isometries”, Physical Review A 81, (2010) [arXiv:0912.0963](https://arxiv.org/abs/0912.0963) [DOI](https://doi.org/10.1103%2Fphysreva.81.032313) Page edit log - [Victor V. Albert](https://errorcorrectionzoo.org/team#u-VictorVAlbert) (2022-07-30) — most recent - [Philippe Faist](https://errorcorrectionzoo.org/team#u-PhilippeFaist) (2022-07-15) - [Victor V. Albert](https://errorcorrectionzoo.org/team#u-VictorVAlbert) (2022-01-03) ### Your contribution is welcome\! [on github.com (edit & pull request)](https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/qecc.yml "Edit code information directly on github.com")— see [instructions](https://github.com/errorcorrectionzoo/eczoo_data/blob/main/README.md "edit source instructions") [edit on this site](https://errorcorrectionzoo.org/edit_code#%7B%22code_id%22%3A%22qecc%22%2C%22code_yml_filename%22%3A%22codes%2Fquantum%2Fproperties%2Fqecc.yml%22%7D "edit code information on this site without using github") Zoo Code ID: `qecc` Cite as: “Quantum error-correcting code (QECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc BibTeX: @incollection{eczoo\_qecc, title={Quantum error-correcting code (QECC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qecc} } Share via: [Twitter](https://twitter.com/intent/tweet?text=Quantum%20error-correcting%20code%20\(QECC\)%20https%3A%2F%2Ferrorcorrectionzoo.org%2Fc%2Fqecc) \| [Mastodon](https://toot.kytta.dev/?text=Quantum%20error-correcting%20code%20\(QECC\)%20https%3A%2F%2Ferrorcorrectionzoo.org%2Fc%2Fqecc&instance=https%3A%2F%2Fqubit-social.xyz) \| [Facebook](http://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Ferrorcorrectionzoo.org%2Fc%2Fqecc&t=Quantum%20error-correcting%20code%20\(QECC\)) \| [E-mail](mailto:?subject=Quantum%20error-correcting%20code%20\(QECC\)&body=Quantum%20error-correcting%20code%20\(QECC\)%20https%3A%2F%2Ferrorcorrectionzoo.org%2Fc%2Fqecc) Permanent link: https://errorcorrectionzoo.org/c/qecc ## Cite as: “Quantum error-correcting code (QECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc Github: <https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/qecc.yml>. - [OAQECC](https://errorcorrectionzoo.org/c/oaecc) - [Quantum](https://errorcorrectionzoo.org/c/quantum_into_quantum) - [Category-based quantum](https://errorcorrectionzoo.org/c/category_quantum) - [Homogeneous-space quantum](https://errorcorrectionzoo.org/c/homogeneous_space_quantum) - [Asymmetric quantum](https://errorcorrectionzoo.org/c/asymmetric_qecc) - [Block quantum](https://errorcorrectionzoo.org/c/block_quantum) - [AQECC](https://errorcorrectionzoo.org/c/approximate_qecc) - [ECC](https://errorcorrectionzoo.org/c/ecc) - [Metrological](https://errorcorrectionzoo.org/c/metrological) - [EAQECC](https://errorcorrectionzoo.org/c/eaqecc) - [Hybrid QECC](https://errorcorrectionzoo.org/c/hybridqecc) - [Subsystem QECC](https://errorcorrectionzoo.org/c/oecc) # [Classical Domain](https://errorcorrectionzoo.org/domain/classical_domain) # [Quantum Domain](https://errorcorrectionzoo.org/domain/quantum_domain) # [Classical-quantum Domain](https://errorcorrectionzoo.org/domain/classical_into_quantum_domain) # - [Home](https://errorcorrectionzoo.org/) - [Team](https://errorcorrectionzoo.org/team) - [About](https://errorcorrectionzoo.org/about) - [Code graph](https://errorcorrectionzoo.org/code_graph) - [Lists](https://errorcorrectionzoo.org/lists) - [Concepts glossary](https://errorcorrectionzoo.org/concepts) - [Add new code](https://errorcorrectionzoo.org/edit_code) - [Search](https://errorcorrectionzoo.org/search) # - [🌒]() [≡](https://errorcorrectionzoo.org/c/qecc#navigation) [Error correction zoo](https://errorcorrectionzoo.org/about) by Victor V. Albert, Philippe Faist, and [many contributors](https://errorcorrectionzoo.org/team). This work is licensed under a [CC-BY-SA License](http://creativecommons.org/licenses/by-sa/4.0/). See [how to contribute](https://github.com/errorcorrectionzoo/eczoo_data/blob/main/CONTRIBUTING.md).