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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 168 (from laksa083)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa168.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction\')), getAhrefsUnparsedNoserviceFromURL(\'https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction","crawl_time":1775411305,"first_indexed_time":1718492041,"http_code":200,"src_unparsed":"com,microsoft!learn,\/en-us\/azure\/quantum\/concepts-error-correction s443","src_root_hash":"14615152987638977768","history_drop_reason":null,"meta_title":"Quantum Error Correction Codes - Azure Quantum | Microsoft Learn","meta_descriptions":["Learn how quantum error correction works, the types of quantum errors and codes, and how to correct errors using the three-qubit code as an example."],"attrs_boilerpipe_text":"This article explains the basics of quantum error correction, the types of quantum errors, and some common quantum error correction codes. It also provides an example of how to correct errors using the three-qubit code.\nQuantum error correction (QEC) is a technique that allows us to protect quantum information from errors. Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large-scale quantum computers, which are sensitive to noise.\nThe basic principle behind quantum error correction is that the number of bits used to encode a given amount of information is increased. This redundancy allows the code to detect and correct errors.\nThe error rates for quantum computers are typically higher than classical computer's errors due to the challenges associated with building and operating quantum systems. Noise, decoherence, and imperfections in quantum gates can cause errors in quantum computations. Current quantum computers have error rates in the range of 1% to 0.1%. In other words, this means that on average one out of every 100 to 1000 quantum gate operations results in an error.\nThere are two fundamental types of quantum errors: bit flips and phase flips.\nBit flip errors occur when a qubit changes from\n|\n0\n⟩\nto\n|\n1\n⟩\nor vice versa. Bit flip errors are also known as\nσ\nx\n-errors, because they map the qubit states\nσ\nx\n|\n0\n⟩\n=\n|\n1\n⟩\nand\nσ\nx\n|\n1\n⟩\n=\n|\n0\n⟩\n. This error is analogous to a classical bit flip error.\nPhase flip errors occur when a qubit changes its phase. They are also known as\nσ\nz\n-errors, because they map the qubit states\nσ\nz\n|\n0\n⟩\n=\n|\n0\n⟩\nand\nσ\nz\n|\n1\n⟩\n=\n−\n|\n1\n⟩\n. This type of error has no classical analog.\nIn quantum computing, quantum errors can manifest as bit flips, phase flips, or a combination of both.\nQuantum error correction codes work by encoding the quantum information into a larger set of qubits, called the\nphysical qubits\n. The joint state of the physical qubits represents a\nlogical qubit\n.\nThe physical qubits are subject to errors due to decoherence and imperfections in quantum gates. The code is designed so that errors can be detected and corrected by measuring some of the qubits in the code.\nFor example, imagine you want to send the single-qubit message\n|\n0\n⟩\n. You could use three physical qubits to encode the message, sending\n|\n000\n⟩\n, which is known as a\ncodeword\n. This error-correcting code is a\nrepetition code\n, because the message is repeated three times.\nNow, imagine that a single bit-flip error occurs during transmission so that what the recipient receives is the state\n|\n010\n⟩\n. In this scenario, the recipient may be able to infer that the intended message is\n|\n000\n⟩\n. However, if the message is subject to two bit-flip errors, the recipient may infer an incorrect message. Finally, if all three bits are flipped so that the original message\n|\n000\n⟩\nbecomes\n|\n111\n⟩\n, the recipient has no way of knowing an error occurred.\nThe code distance of a QEC code is the minimum number of errors that change one codeword into another, that is, the number of errors that can't be detected. The code distance\nd\ncan be defined as\nd\n=\n2\nt\n+\n1\nwhere\nt\nis the number of errors the code can correct. For example, the three-bit code can detect and correct one bit-flip error, so\nt\n=\n1\n, and thus the code distance is\nd\n=\n3\n.\nNote that repetition codes, such as the three-bit code used in this example, can only correct bit-flip errors, and not phase flip errors. To correct both types of errors, more sophisticated quantum error correction codes are needed.\nThere are many different types of QEC codes, each with its own properties and advantages. Some common QEC codes are:\nRepetition code:\nThe simplest quantum error correction code, where a single qubit is encoded into multiple qubits by repeating it multiple times. The repetition code can correct bit flip errors, but not phase flip errors.\nShor code:\nThe first quantum error correction code, developed by Peter Shor. It encodes one logical qubit into nine physical qubits. Shor code can correct one-bit flip error or one phase flip error, but it can't correct both types of errors at the same time.\nSteane code:\nThis is a seven-qubit code that can correct both bit flip and phase flip errors. It has the advantage of being fault-tolerant, meaning that the error correction process itself doesn't introduce extra errors.\nSurface code:\nThis is a topological error correction code that uses a two-dimensional lattice of qubits to encode logical qubits. It has a high error correction threshold and is considered one of the most promising techniques for large-scale, fault-tolerant quantum computing. The surface code is used by the\nMicrosoft Quantum resource estimator\n.\nHastings-Haah code:\nThis quantum error correction code offers better space-time costs than surface codes on Majorana qubits in many regimes. For gate-based instruction sets, the overhead is larger, which makes this approach less efficient than the surface code.\nThe three-qubit error correction code is a simple repetition code that can detect and correct one bit flip error. It encodes a single logical qubit into three physical qubits by repeating the qubit three times.\nImagine you want to send an arbitrary single qubit\n|\nϕ\n⟩\n=\nα\n|\n0\n⟩\n+\nβ\n|\n1\n⟩\n. To avoid errors, you encode the basis states\n|\n0\n⟩\nand\n|\n1\n⟩\ninto a joint state of three qubits. The two logical basis states are\n|\n0\nL\n⟩\n=\n|\n000\n⟩\nand\n|\n1\nL\n⟩\n=\n|\n111\n⟩\n.\nTherefore, the single qubit\n|\nϕ\n⟩\n=\nα\n|\n0\n⟩\n+\nβ\n|\n1\n⟩\nis encoded as:\n|\nϕ\nL\n⟩\n=\nα\n|\n000\n⟩\n+\nβ\n|\n111\n⟩\n=\nα\n|\n0\nL\n⟩\n+\nβ\n|\n1\nL\n⟩\nLet's break down the steps of the three-qubit code.\nFirst, you encode your single qubit\n|\nϕ\n⟩\n=\nα\n|\n0\n⟩\n+\nβ\n|\n1\n⟩\ninto a joint state of three qubits.\nNext, you prepare two further qubits in the state\n|\n0\n⟩\n. So, the global state of all three qubits is\n(\nα\n|\n0\n⟩\n+\nβ\n|\n1\n⟩\n)\n|\n0\n⟩\n|\n0\n⟩\n=\nα\n|\n000\n⟩\n+\nβ\n|\n100\n⟩\n.\nLastly, you encode the single qubit into a joint state of three qubits, by applying two CNOT operations. The first CNOT uses the first qubit as control and acts on the second qubit, producing\nα\n|\n000\n⟩\n+\nβ\n|\n110\n⟩\n. The second CNOT uses the first qubit as control and acts on the third qubit. The state of the three qubits is now\nα\n|\n000\n⟩\n+\nβ\n|\n111\n⟩\n.\nYou send all three qubits. Assuming only one-bit flip errors can occur, the received qubits are in one of the following states:\nState\nError\nα\n|\n000\n⟩\n+\nβ\n|\n111\n⟩\nNo error\nα\n|\n100\n⟩\n+\nβ\n|\n011\n⟩\nQubit 1\nα\n|\n010\n⟩\n+\nβ\n|\n101\n⟩\nQubit 2\nα\n|\n001\n⟩\n+\nβ\n|\n110\n⟩\nQubit 3\nFirst, you introduce two more qubits, prepared in the state\n|\n00\n⟩\n. This auxiliary pair of qubits are used to extract information of the error without directly measuring or obtaining information about the logical state.\nNext, you carry out four CNOT operations: the first two operations use the first and second received qubits as control and act on the first auxiliary qubit, and last two operations use the first and third received qubits as control and act on the second auxiliary bit. The total state of all five qubits is now:\nState\nError\n(\nα\n|\n000\n⟩\n+\nβ\n|\n111\n⟩\n)\n|\n00\n⟩\nNo error\n(\nα\n|\n100\n⟩\n+\nβ\n|\n011\n⟩\n)\n|\n11\n⟩\nQubit 1\n(\nα\n|\n010\n⟩\n+\nβ\n|\n101\n⟩\n)\n|\n10\n⟩\nQubit 2\n(\nα\n|\n001\n⟩\n+\nβ\n|\n110\n⟩\n)\n|\n01\n⟩\nQubit 3\nTo retrieve the error information, you measure the two auxiliary qubits in the computational basis states\n|\n0\n⟩\nand\n|\n1\n⟩\n. By doing this, you recover the joint state, which is called the\nerror syndrome\nbecause it helps diagnose the errors in the received qubits.\nNow you know which of the four possible states the three received qubits are in. You can correct the error by applying the correction operation. In this case you're dealing with bit flip errors, so the correction is a\nσ\nx\noperation applied to one (or none) of the qubits.\nFor example, if the error syndrome is\n|\n00\n⟩\n, then the received qubits are in the state\nα\n|\n000\n⟩\n+\nβ\n|\n111\n⟩\n, which is the state you originally sent. If the error syndrome is\n|\n11\n⟩\n, then the received qubits are in the state\nα\n|\n100\n⟩\n+\nb\n|\n011\n⟩\n. There's a bit flip error on the first qubit, which you can correct by applying a\nσ\nx\noperation to the first qubit.\nError syndrome\nCollapse state\nCorrection\n|\n00\n⟩\nα\n|\n000\n⟩\n+\nβ\n|\n111\n⟩\nDo nothing\n|\n01\n⟩\nα\n|\n100\n⟩\n+\nβ\n|\n011\n⟩\nApply\nσ\nx\nto qubit 3\n|\n10\n⟩\nα\n|\n010\n⟩\n+\nβ\n|\n101\n⟩\nApply\nσ\nx\nto qubit 2\n|\n11\n⟩\nα\n|\n001\n⟩\n+\nβ\n|\n110\n⟩\nApply\nσ\nx\nto qubit 1\nFinally, to extract the single qubit you wanted to transmit originally, you apply two CNOT operations: one uses the first qubit as control and acts on the second qubit, and other uses the first qubit as control and acts on the third one.\nThe state of the first qubit is now\nα\n|\n0\n⟩\n+\nβ\n|\n1\n⟩\n, which is the original qubit you wanted to transmit.\nImportant\nThe QEC code doesn't gain any information regarding the coefficients\nα\nand\nβ\n, hence superpositions of the computational state remain intact during correction.","attrs_markdown":"[Skip to main content](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#main)   [Skip to Ask Learn chat experience](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction)\n\n## Microsoft Build 2026\nJune 2-3, 2026\n\nGo deep on real code and real systems in San Francisco and online\n\n[Learn more](https:\/\/aka.ms\/MSBuild_FY26_BN_MSLearn_Hero)\n\nDismiss alert\n\nThis browser is no longer supported.\n\nUpgrade to Microsoft Edge to take advantage of the latest features, security updates, and technical support.\n\n[Download Microsoft Edge](https:\/\/go.microsoft.com\/fwlink\/p\/?LinkID=2092881%20)   [More info about Internet Explorer and Microsoft Edge](https:\/\/learn.microsoft.com\/en-us\/lifecycle\/faq\/internet-explorer-microsoft-edge)\n\n[Learn](https:\/\/learn.microsoft.com\/en-us\/)\n\n[Sign in](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction)\n\n![]()\n\n![]()\n\n- [Profile](https:\/\/learn.microsoft.com\/en-us\/users\/me\/activity\/)\n- [Settings](https:\/\/learn.microsoft.com\/en-us\/users\/me\/settings\/)\n\n[Sign out](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction)\n\n[Learn](https:\/\/learn.microsoft.com\/en-us\/)\n\n- Documentation\n  \n  - [All product documentation](https:\/\/learn.microsoft.com\/en-us\/docs\/)\n  - [Azure documentation](https:\/\/learn.microsoft.com\/en-us\/azure\/?product=popular)\n  - [Dynamics 365 documentation](https:\/\/learn.microsoft.com\/en-us\/dynamics365\/)\n  - [Microsoft Copilot documentation](https:\/\/learn.microsoft.com\/en-us\/copilot\/)\n  - [Microsoft 365 documentation](https:\/\/learn.microsoft.com\/en-us\/microsoft-365\/)\n  - [Power Platform documentation](https:\/\/learn.microsoft.com\/en-us\/power-platform\/)\n  - [Code samples](https:\/\/learn.microsoft.com\/en-us\/samples\/)\n  - [Troubleshooting documentation](https:\/\/learn.microsoft.com\/en-us\/troubleshoot\/)\n  \n  Free to join. 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You can try [signing in](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction) or [changing directories]().\n\nAccess to this page requires authorization. You can try [changing directories]().\n# Introduction to quantum error correction\nFeedback\n\nSummarize this article for me\n\n## In this article\n1. [What is quantum error correction?](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#what-is-quantum-error-correction)\n2. [Types of quantum errors](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#types-of-quantum-errors)\n3. [How does quantum error correction work?](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#how-does-quantum-error-correction-work)\n4. [Types of QEC codes](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#types-of-qec-codes)\n5. [Example: The three-qubit code](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#example-the-three-qubit-code)\n\nThis article explains the basics of quantum error correction, the types of quantum errors, and some common quantum error correction codes. It also provides an example of how to correct errors using the three-qubit code.\n\n## What is quantum error correction?\nQuantum error correction (QEC) is a technique that allows us to protect quantum information from errors. Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large-scale quantum computers, which are sensitive to noise.\n\nThe basic principle behind quantum error correction is that the number of bits used to encode a given amount of information is increased. This redundancy allows the code to detect and correct errors.\n\nThe error rates for quantum computers are typically higher than classical computer's errors due to the challenges associated with building and operating quantum systems. Noise, decoherence, and imperfections in quantum gates can cause errors in quantum computations. Current quantum computers have error rates in the range of 1% to 0.1%. In other words, this means that on average one out of every 100 to 1000 quantum gate operations results in an error.\n\n## Types of quantum errors\nThere are two fundamental types of quantum errors: bit flips and phase flips.\n\nBit flip errors occur when a qubit changes from \\| 0 ⟩ to \\| 1 ⟩ or vice versa. Bit flip errors are also known as σ x\\-errors, because they map the qubit states σ x \\| 0 ⟩ \\= \\| 1 ⟩ and σ x \\| 1 ⟩ \\= \\| 0 ⟩. This error is analogous to a classical bit flip error.\n\nPhase flip errors occur when a qubit changes its phase. They are also known as σ z\\-errors, because they map the qubit states σ z \\| 0 ⟩ \\= \\| 0 ⟩ and σ z \\| 1 ⟩ \\= − \\| 1 ⟩. This type of error has no classical analog.\n\nIn quantum computing, quantum errors can manifest as bit flips, phase flips, or a combination of both.\n\n## How does quantum error correction work?\nQuantum error correction codes work by encoding the quantum information into a larger set of qubits, called the *physical qubits*. The joint state of the physical qubits represents a *logical qubit*.\n\nThe physical qubits are subject to errors due to decoherence and imperfections in quantum gates. The code is designed so that errors can be detected and corrected by measuring some of the qubits in the code.\n\nFor example, imagine you want to send the single-qubit message \\| 0 ⟩. You could use three physical qubits to encode the message, sending \\| 000 ⟩, which is known as a *codeword*. This error-correcting code is a *repetition code*, because the message is repeated three times.\n\nNow, imagine that a single bit-flip error occurs during transmission so that what the recipient receives is the state \\| 010 ⟩. In this scenario, the recipient may be able to infer that the intended message is \\| 000 ⟩. However, if the message is subject to two bit-flip errors, the recipient may infer an incorrect message. Finally, if all three bits are flipped so that the original message \\| 000 ⟩ becomes \\| 111 ⟩, the recipient has no way of knowing an error occurred.\n\nThe code distance of a QEC code is the minimum number of errors that change one codeword into another, that is, the number of errors that can't be detected. The code distance d can be defined as\n\nd \\= 2 t \\+ 1\n\nwhere t is the number of errors the code can correct. For example, the three-bit code can detect and correct one bit-flip error, so t \\= 1, and thus the code distance is d \\= 3.\n\nNote that repetition codes, such as the three-bit code used in this example, can only correct bit-flip errors, and not phase flip errors. To correct both types of errors, more sophisticated quantum error correction codes are needed.\n\n## Types of QEC codes\nThere are many different types of QEC codes, each with its own properties and advantages. Some common QEC codes are:\n\n- **Repetition code:** The simplest quantum error correction code, where a single qubit is encoded into multiple qubits by repeating it multiple times. The repetition code can correct bit flip errors, but not phase flip errors.\n- **Shor code:** The first quantum error correction code, developed by Peter Shor. It encodes one logical qubit into nine physical qubits. Shor code can correct one-bit flip error or one phase flip error, but it can't correct both types of errors at the same time.\n- **Steane code:** This is a seven-qubit code that can correct both bit flip and phase flip errors. It has the advantage of being fault-tolerant, meaning that the error correction process itself doesn't introduce extra errors.\n- **Surface code:** This is a topological error correction code that uses a two-dimensional lattice of qubits to encode logical qubits. It has a high error correction threshold and is considered one of the most promising techniques for large-scale, fault-tolerant quantum computing. The surface code is used by the [Microsoft Quantum resource estimator](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/overview-resources-estimator#quantum-error-correction-schemes).\n- **Hastings-Haah code:** This quantum error correction code offers better space-time costs than surface codes on Majorana qubits in many regimes. For gate-based instruction sets, the overhead is larger, which makes this approach less efficient than the surface code.\n## Example: The three-qubit code\nThe three-qubit error correction code is a simple repetition code that can detect and correct one bit flip error. It encodes a single logical qubit into three physical qubits by repeating the qubit three times.\n\nTip\n\nCheck out the [Q\\# code sample for the three-qubit code](https:\/\/github.com\/microsoft\/qdk\/blob\/main\/samples\/algorithms\/ThreeQubitRepetitionCode.qs).\n\nImagine you want to send an arbitrary single qubit \\| ϕ ⟩ \\= α \\| 0 ⟩ \\+ β \\| 1 ⟩. To avoid errors, you encode the basis states \\| 0 ⟩ and \\| 1 ⟩ into a joint state of three qubits. The two logical basis states are \\| 0 L ⟩ \\= \\| 000 ⟩ and \\| 1 L ⟩ \\= \\| 111 ⟩.\n\nTherefore, the single qubit \\| ϕ ⟩ \\= α \\| 0 ⟩ \\+ β \\| 1 ⟩ is encoded as:\n\n\\| ϕ L ⟩ \\= α \\| 000 ⟩ \\+ β \\| 111 ⟩ \\= α \\| 0 L ⟩ \\+ β \\| 1 L ⟩\n\nLet's break down the steps of the three-qubit code.\n\n### Preparing the qubits\nFirst, you encode your single qubit \\| ϕ ⟩ \\= α \\| 0 ⟩ \\+ β \\| 1 ⟩ into a joint state of three qubits.\n\nNext, you prepare two further qubits in the state \\| 0 ⟩. So, the global state of all three qubits is ( α \\| 0 ⟩ \\+ β \\| 1 ⟩ ) \\| 0 ⟩ \\| 0 ⟩ \\= α \\| 000 ⟩ \\+ β \\| 100 ⟩.\n\nLastly, you encode the single qubit into a joint state of three qubits, by applying two CNOT operations. The first CNOT uses the first qubit as control and acts on the second qubit, producing α \\| 000 ⟩ \\+ β \\| 110 ⟩. The second CNOT uses the first qubit as control and acts on the third qubit. The state of the three qubits is now α \\| 000 ⟩ \\+ β \\| 111 ⟩.\n\n### Sending the qubits\nYou send all three qubits. Assuming only one-bit flip errors can occur, the received qubits are in one of the following states:\n\nExpand table\n\n| State | Error |\n|---|---|\n| α \\| 000 ⟩ \\+ β \\| 111 ⟩ | No error |\n| α \\| 100 ⟩ \\+ β \\| 011 ⟩ | Qubit 1 |\n| α \\| 010 ⟩ \\+ β \\| 101 ⟩ | Qubit 2 |\n| α \\| 001 ⟩ \\+ β \\| 110 ⟩ | Qubit 3 |\n### Adding auxiliary qubits\nFirst, you introduce two more qubits, prepared in the state \\| 00 ⟩. This auxiliary pair of qubits are used to extract information of the error without directly measuring or obtaining information about the logical state.\n\nNext, you carry out four CNOT operations: the first two operations use the first and second received qubits as control and act on the first auxiliary qubit, and last two operations use the first and third received qubits as control and act on the second auxiliary bit. The total state of all five qubits is now:\n\nExpand table\n\n| State | Error |\n|---|---|\n| ( α \\| 000 ⟩ \\+ β \\| 111 ⟩ ) \\| 00 ⟩ | No error |\n| ( α \\| 100 ⟩ \\+ β \\| 011 ⟩ ) \\| 11 ⟩ | Qubit 1 |\n| ( α \\| 010 ⟩ \\+ β \\| 101 ⟩ ) \\| 10 ⟩ | Qubit 2 |\n| ( α \\| 001 ⟩ \\+ β \\| 110 ⟩ ) \\| 01 ⟩ | Qubit 3 |\n### Retrieving the error syndrome\nTo retrieve the error information, you measure the two auxiliary qubits in the computational basis states \\| 0 ⟩ and \\| 1 ⟩. By doing this, you recover the joint state, which is called the *error syndrome* because it helps diagnose the errors in the received qubits.\n\nNow you know which of the four possible states the three received qubits are in. You can correct the error by applying the correction operation. In this case you're dealing with bit flip errors, so the correction is a σ x operation applied to one (or none) of the qubits.\n\nFor example, if the error syndrome is \\| 00 ⟩, then the received qubits are in the state α \\| 000 ⟩ \\+ β \\| 111 ⟩, which is the state you originally sent. If the error syndrome is \\| 11 ⟩, then the received qubits are in the state α \\| 100 ⟩ \\+ b \\| 011 ⟩. There's a bit flip error on the first qubit, which you can correct by applying a σ x operation to the first qubit.\n\nExpand table\n\n| Error syndrome | Collapse state | Correction |\n|---|---|---|\n| \\| 00 ⟩ | α \\| 000 ⟩ \\+ β \\| 111 ⟩ | Do nothing |\n| \\| 01 ⟩ | α \\| 100 ⟩ \\+ β \\| 011 ⟩ | Apply σ x to qubit 3 |\n| \\| 10 ⟩ | α \\| 010 ⟩ \\+ β \\| 101 ⟩ | Apply σ x to qubit 2 |\n| \\| 11 ⟩ | α \\| 001 ⟩ \\+ β \\| 110 ⟩ | Apply σ x to qubit 1 |\n### Extracting the original qubit\nFinally, to extract the single qubit you wanted to transmit originally, you apply two CNOT operations: one uses the first qubit as control and acts on the second qubit, and other uses the first qubit as control and acts on the third one.\n\nThe state of the first qubit is now α \\| 0 ⟩ \\+ β \\| 1 ⟩, which is the original qubit you wanted to transmit.\n\nImportant\n\nThe QEC code doesn't gain any information regarding the coefficients α and β, hence superpositions of the computational state remain intact during correction.\n\n***\n## Feedback\nWas this page helpful?\n\nYes\n\nNo\n\nNo\n\nNeed help with this topic?\n\nWant to try using Ask Learn to clarify or guide you through this topic?\n\nAsk Learn\n\nAsk Learn\n\nSuggest a fix?\n\n***\n## Additional resources\nTraining\n\nModule\n\n[Teleport entangled qubits with Q\\# - Training](https:\/\/learn.microsoft.com\/en-us\/training\/modules\/explore-entanglement\/?source=recommendations)\n\nTeleport entangled qubits with Q\\#\n\n***\n- Last updated on\n  \n  09\/16\/2024\n\n## In this article\n1. [What is quantum error correction?](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#what-is-quantum-error-correction)\n2. [Types of quantum errors](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#types-of-quantum-errors)\n3. [How does quantum error correction work?](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#how-does-quantum-error-correction-work)\n4. [Types of QEC codes](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#types-of-qec-codes)\n5. [Example: The three-qubit code](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction#example-the-three-qubit-code)\n\nWas this page helpful?\n\nYes\n\nNo\n\nNo\n\nNeed help with this topic?\n\nWant to try using Ask Learn to clarify or guide you through this topic?\n\nAsk Learn\n\nAsk Learn\n\nSuggest a fix?\n\n## Ask Learn\nPreview\n\nAsk Learn is an AI assistant that can answer questions, clarify concepts, and define terms using trusted Microsoft documentation.\n\nPlease sign in to use Ask Learn.\n\n[Sign in](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/concepts-error-correction)\n\n[English (United States)](https:\/\/learn.microsoft.com\/en-us\/locale?target=https%3A%2F%2Flearn.microsoft.com%2Fen-us%2Fazure%2Fquantum%2Fconcepts-error-correction)\n\n[Your Privacy Choices](https:\/\/aka.ms\/yourcaliforniaprivacychoices)\n\nTheme\n\n- Light\n- Dark\n- High contrast\n\n- [AI Disclaimer](https:\/\/learn.microsoft.com\/en-us\/principles-for-ai-generated-content)\n- [Previous Versions](https:\/\/learn.microsoft.com\/en-us\/previous-versions\/)\n- [Blog](https:\/\/techcommunity.microsoft.com\/t5\/microsoft-learn-blog\/bg-p\/MicrosoftLearnBlog)\n- [Contribute](https:\/\/learn.microsoft.com\/en-us\/contribute)\n- [Privacy](https:\/\/go.microsoft.com\/fwlink\/?LinkId=521839)\n- [Consumer Health Privacy](https:\/\/go.microsoft.com\/fwlink\/?linkid=2259814)\n- [Terms of Use](https:\/\/learn.microsoft.com\/en-us\/legal\/termsofuse)\n- [Trademarks](https:\/\/www.microsoft.com\/legal\/intellectualproperty\/Trademarks\/)\n- © Microsoft 2026","attrs_readable_markdown":"This article explains the basics of quantum error correction, the types of quantum errors, and some common quantum error correction codes. It also provides an example of how to correct errors using the three-qubit code.\n\nQuantum error correction (QEC) is a technique that allows us to protect quantum information from errors. Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large-scale quantum computers, which are sensitive to noise.\n\nThe basic principle behind quantum error correction is that the number of bits used to encode a given amount of information is increased. This redundancy allows the code to detect and correct errors.\n\nThe error rates for quantum computers are typically higher than classical computer's errors due to the challenges associated with building and operating quantum systems. Noise, decoherence, and imperfections in quantum gates can cause errors in quantum computations. Current quantum computers have error rates in the range of 1% to 0.1%. In other words, this means that on average one out of every 100 to 1000 quantum gate operations results in an error.\n\nThere are two fundamental types of quantum errors: bit flips and phase flips.\n\nBit flip errors occur when a qubit changes from \\| 0 ⟩ to \\| 1 ⟩ or vice versa. Bit flip errors are also known as σ x\\-errors, because they map the qubit states σ x \\| 0 ⟩ \\= \\| 1 ⟩ and σ x \\| 1 ⟩ \\= \\| 0 ⟩. This error is analogous to a classical bit flip error.\n\nPhase flip errors occur when a qubit changes its phase. They are also known as σ z\\-errors, because they map the qubit states σ z \\| 0 ⟩ \\= \\| 0 ⟩ and σ z \\| 1 ⟩ \\= − \\| 1 ⟩. This type of error has no classical analog.\n\nIn quantum computing, quantum errors can manifest as bit flips, phase flips, or a combination of both.\n\nQuantum error correction codes work by encoding the quantum information into a larger set of qubits, called the *physical qubits*. The joint state of the physical qubits represents a *logical qubit*.\n\nThe physical qubits are subject to errors due to decoherence and imperfections in quantum gates. The code is designed so that errors can be detected and corrected by measuring some of the qubits in the code.\n\nFor example, imagine you want to send the single-qubit message \\| 0 ⟩. You could use three physical qubits to encode the message, sending \\| 000 ⟩, which is known as a *codeword*. This error-correcting code is a *repetition code*, because the message is repeated three times.\n\nNow, imagine that a single bit-flip error occurs during transmission so that what the recipient receives is the state \\| 010 ⟩. In this scenario, the recipient may be able to infer that the intended message is \\| 000 ⟩. However, if the message is subject to two bit-flip errors, the recipient may infer an incorrect message. Finally, if all three bits are flipped so that the original message \\| 000 ⟩ becomes \\| 111 ⟩, the recipient has no way of knowing an error occurred.\n\nThe code distance of a QEC code is the minimum number of errors that change one codeword into another, that is, the number of errors that can't be detected. The code distance d can be defined as\n\nd \\= 2 t \\+ 1\n\nwhere t is the number of errors the code can correct. For example, the three-bit code can detect and correct one bit-flip error, so t \\= 1, and thus the code distance is d \\= 3.\n\nNote that repetition codes, such as the three-bit code used in this example, can only correct bit-flip errors, and not phase flip errors. To correct both types of errors, more sophisticated quantum error correction codes are needed.\n\nThere are many different types of QEC codes, each with its own properties and advantages. Some common QEC codes are:\n\n- **Repetition code:** The simplest quantum error correction code, where a single qubit is encoded into multiple qubits by repeating it multiple times. The repetition code can correct bit flip errors, but not phase flip errors.\n- **Shor code:** The first quantum error correction code, developed by Peter Shor. It encodes one logical qubit into nine physical qubits. Shor code can correct one-bit flip error or one phase flip error, but it can't correct both types of errors at the same time.\n- **Steane code:** This is a seven-qubit code that can correct both bit flip and phase flip errors. It has the advantage of being fault-tolerant, meaning that the error correction process itself doesn't introduce extra errors.\n- **Surface code:** This is a topological error correction code that uses a two-dimensional lattice of qubits to encode logical qubits. It has a high error correction threshold and is considered one of the most promising techniques for large-scale, fault-tolerant quantum computing. The surface code is used by the [Microsoft Quantum resource estimator](https:\/\/learn.microsoft.com\/en-us\/azure\/quantum\/overview-resources-estimator#quantum-error-correction-schemes).\n- **Hastings-Haah code:** This quantum error correction code offers better space-time costs than surface codes on Majorana qubits in many regimes. For gate-based instruction sets, the overhead is larger, which makes this approach less efficient than the surface code.\n\nThe three-qubit error correction code is a simple repetition code that can detect and correct one bit flip error. It encodes a single logical qubit into three physical qubits by repeating the qubit three times.\n\nImagine you want to send an arbitrary single qubit \\| ϕ ⟩ \\= α \\| 0 ⟩ \\+ β \\| 1 ⟩. To avoid errors, you encode the basis states \\| 0 ⟩ and \\| 1 ⟩ into a joint state of three qubits. The two logical basis states are \\| 0 L ⟩ \\= \\| 000 ⟩ and \\| 1 L ⟩ \\= \\| 111 ⟩.\n\nTherefore, the single qubit \\| ϕ ⟩ \\= α \\| 0 ⟩ \\+ β \\| 1 ⟩ is encoded as:\n\n\\| ϕ L ⟩ \\= α \\| 000 ⟩ \\+ β \\| 111 ⟩ \\= α \\| 0 L ⟩ \\+ β \\| 1 L ⟩\n\nLet's break down the steps of the three-qubit code.\n\nFirst, you encode your single qubit \\| ϕ ⟩ \\= α \\| 0 ⟩ \\+ β \\| 1 ⟩ into a joint state of three qubits.\n\nNext, you prepare two further qubits in the state \\| 0 ⟩. So, the global state of all three qubits is ( α \\| 0 ⟩ \\+ β \\| 1 ⟩ ) \\| 0 ⟩ \\| 0 ⟩ \\= α \\| 000 ⟩ \\+ β \\| 100 ⟩.\n\nLastly, you encode the single qubit into a joint state of three qubits, by applying two CNOT operations. The first CNOT uses the first qubit as control and acts on the second qubit, producing α \\| 000 ⟩ \\+ β \\| 110 ⟩. The second CNOT uses the first qubit as control and acts on the third qubit. The state of the three qubits is now α \\| 000 ⟩ \\+ β \\| 111 ⟩.\n\nYou send all three qubits. Assuming only one-bit flip errors can occur, the received qubits are in one of the following states:\n\n| State | Error |\n|---|---|\n| α \\| 000 ⟩ \\+ β \\| 111 ⟩ | No error |\n| α \\| 100 ⟩ \\+ β \\| 011 ⟩ | Qubit 1 |\n| α \\| 010 ⟩ \\+ β \\| 101 ⟩ | Qubit 2 |\n| α \\| 001 ⟩ \\+ β \\| 110 ⟩ | Qubit 3 |\n\nFirst, you introduce two more qubits, prepared in the state \\| 00 ⟩. This auxiliary pair of qubits are used to extract information of the error without directly measuring or obtaining information about the logical state.\n\nNext, you carry out four CNOT operations: the first two operations use the first and second received qubits as control and act on the first auxiliary qubit, and last two operations use the first and third received qubits as control and act on the second auxiliary bit. The total state of all five qubits is now:\n\n| State | Error |\n|---|---|\n| ( α \\| 000 ⟩ \\+ β \\| 111 ⟩ ) \\| 00 ⟩ | No error |\n| ( α \\| 100 ⟩ \\+ β \\| 011 ⟩ ) \\| 11 ⟩ | Qubit 1 |\n| ( α \\| 010 ⟩ \\+ β \\| 101 ⟩ ) \\| 10 ⟩ | Qubit 2 |\n| ( α \\| 001 ⟩ \\+ β \\| 110 ⟩ ) \\| 01 ⟩ | Qubit 3 |\n\nTo retrieve the error information, you measure the two auxiliary qubits in the computational basis states \\| 0 ⟩ and \\| 1 ⟩. By doing this, you recover the joint state, which is called the *error syndrome* because it helps diagnose the errors in the received qubits.\n\nNow you know which of the four possible states the three received qubits are in. You can correct the error by applying the correction operation. In this case you're dealing with bit flip errors, so the correction is a σ x operation applied to one (or none) of the qubits.\n\nFor example, if the error syndrome is \\| 00 ⟩, then the received qubits are in the state α \\| 000 ⟩ \\+ β \\| 111 ⟩, which is the state you originally sent. If the error syndrome is \\| 11 ⟩, then the received qubits are in the state α \\| 100 ⟩ \\+ b \\| 011 ⟩. There's a bit flip error on the first qubit, which you can correct by applying a σ x operation to the first qubit.\n\n| Error syndrome | Collapse state | Correction |\n|---|---|---|\n| \\| 00 ⟩ | α \\| 000 ⟩ \\+ β \\| 111 ⟩ | Do nothing |\n| \\| 01 ⟩ | α \\| 100 ⟩ \\+ β \\| 011 ⟩ | Apply σ x to qubit 3 |\n| \\| 10 ⟩ | α \\| 010 ⟩ \\+ β \\| 101 ⟩ | Apply σ x to qubit 2 |\n| \\| 11 ⟩ | α \\| 001 ⟩ \\+ β \\| 110 ⟩ | Apply σ x to qubit 1 |\n\nFinally, to extract the single qubit you wanted to transmit originally, you apply two CNOT operations: one uses the first qubit as control and acts on the second qubit, and other uses the first qubit as control and acts on the third one.\n\nThe state of the first qubit is now α \\| 0 ⟩ \\+ β \\| 1 ⟩, which is the original qubit you wanted to transmit.\n\nImportant\n\nThe QEC code doesn't gain any information regarding the coefficients α and β, hence superpositions of the computational state remain intact during correction.","meta_canonical":null}

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Meta Title	Quantum Error Correction Codes - Azure Quantum \| Microsoft Learn
Meta Description	Learn how quantum error correction works, the types of quantum errors and codes, and how to correct errors using the three-qubit code as an example.
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Boilerpipe Text	This article explains the basics of quantum error correction, the types of quantum errors, and some common quantum error correction codes. It also provides an example of how to correct errors using the three-qubit code. Quantum error correction (QEC) is a technique that allows us to protect quantum information from errors. Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large-scale quantum computers, which are sensitive to noise. The basic principle behind quantum error correction is that the number of bits used to encode a given amount of information is increased. This redundancy allows the code to detect and correct errors. The error rates for quantum computers are typically higher than classical computer's errors due to the challenges associated with building and operating quantum systems. Noise, decoherence, and imperfections in quantum gates can cause errors in quantum computations. Current quantum computers have error rates in the range of 1% to 0.1%. In other words, this means that on average one out of every 100 to 1000 quantum gate operations results in an error. There are two fundamental types of quantum errors: bit flips and phase flips. Bit flip errors occur when a qubit changes from \| 0 ⟩ to \| 1 ⟩ or vice versa. Bit flip errors are also known as σ x -errors, because they map the qubit states σ x \| 0 ⟩ = \| 1 ⟩ and σ x \| 1 ⟩ = \| 0 ⟩ . This error is analogous to a classical bit flip error. Phase flip errors occur when a qubit changes its phase. They are also known as σ z -errors, because they map the qubit states σ z \| 0 ⟩ = \| 0 ⟩ and σ z \| 1 ⟩ = − \| 1 ⟩ . This type of error has no classical analog. In quantum computing, quantum errors can manifest as bit flips, phase flips, or a combination of both. Quantum error correction codes work by encoding the quantum information into a larger set of qubits, called the physical qubits . The joint state of the physical qubits represents a logical qubit . The physical qubits are subject to errors due to decoherence and imperfections in quantum gates. The code is designed so that errors can be detected and corrected by measuring some of the qubits in the code. For example, imagine you want to send the single-qubit message \| 0 ⟩ . You could use three physical qubits to encode the message, sending \| 000 ⟩ , which is known as a codeword . This error-correcting code is a repetition code , because the message is repeated three times. Now, imagine that a single bit-flip error occurs during transmission so that what the recipient receives is the state \| 010 ⟩ . In this scenario, the recipient may be able to infer that the intended message is \| 000 ⟩ . However, if the message is subject to two bit-flip errors, the recipient may infer an incorrect message. Finally, if all three bits are flipped so that the original message \| 000 ⟩ becomes \| 111 ⟩ , the recipient has no way of knowing an error occurred. The code distance of a QEC code is the minimum number of errors that change one codeword into another, that is, the number of errors that can't be detected. The code distance d can be defined as d = 2 t + 1 where t is the number of errors the code can correct. For example, the three-bit code can detect and correct one bit-flip error, so t = 1 , and thus the code distance is d = 3 . Note that repetition codes, such as the three-bit code used in this example, can only correct bit-flip errors, and not phase flip errors. To correct both types of errors, more sophisticated quantum error correction codes are needed. There are many different types of QEC codes, each with its own properties and advantages. Some common QEC codes are: Repetition code: The simplest quantum error correction code, where a single qubit is encoded into multiple qubits by repeating it multiple times. The repetition code can correct bit flip errors, but not phase flip errors. Shor code: The first quantum error correction code, developed by Peter Shor. It encodes one logical qubit into nine physical qubits. Shor code can correct one-bit flip error or one phase flip error, but it can't correct both types of errors at the same time. Steane code: This is a seven-qubit code that can correct both bit flip and phase flip errors. It has the advantage of being fault-tolerant, meaning that the error correction process itself doesn't introduce extra errors. Surface code: This is a topological error correction code that uses a two-dimensional lattice of qubits to encode logical qubits. It has a high error correction threshold and is considered one of the most promising techniques for large-scale, fault-tolerant quantum computing. The surface code is used by the Microsoft Quantum resource estimator . Hastings-Haah code: This quantum error correction code offers better space-time costs than surface codes on Majorana qubits in many regimes. For gate-based instruction sets, the overhead is larger, which makes this approach less efficient than the surface code. The three-qubit error correction code is a simple repetition code that can detect and correct one bit flip error. It encodes a single logical qubit into three physical qubits by repeating the qubit three times. Imagine you want to send an arbitrary single qubit \| ϕ ⟩ = α \| 0 ⟩ + β \| 1 ⟩ . To avoid errors, you encode the basis states \| 0 ⟩ and \| 1 ⟩ into a joint state of three qubits. The two logical basis states are \| 0 L ⟩ = \| 000 ⟩ and \| 1 L ⟩ = \| 111 ⟩ . Therefore, the single qubit \| ϕ ⟩ = α \| 0 ⟩ + β \| 1 ⟩ is encoded as: \| ϕ L ⟩ = α \| 000 ⟩ + β \| 111 ⟩ = α \| 0 L ⟩ + β \| 1 L ⟩ Let's break down the steps of the three-qubit code. First, you encode your single qubit \| ϕ ⟩ = α \| 0 ⟩ + β \| 1 ⟩ into a joint state of three qubits. Next, you prepare two further qubits in the state \| 0 ⟩ . So, the global state of all three qubits is ( α \| 0 ⟩ + β \| 1 ⟩ ) \| 0 ⟩ \| 0 ⟩ = α \| 000 ⟩ + β \| 100 ⟩ . Lastly, you encode the single qubit into a joint state of three qubits, by applying two CNOT operations. The first CNOT uses the first qubit as control and acts on the second qubit, producing α \| 000 ⟩ + β \| 110 ⟩ . The second CNOT uses the first qubit as control and acts on the third qubit. The state of the three qubits is now α \| 000 ⟩ + β \| 111 ⟩ . You send all three qubits. Assuming only one-bit flip errors can occur, the received qubits are in one of the following states: State Error α \| 000 ⟩ + β \| 111 ⟩ No error α \| 100 ⟩ + β \| 011 ⟩ Qubit 1 α \| 010 ⟩ + β \| 101 ⟩ Qubit 2 α \| 001 ⟩ + β \| 110 ⟩ Qubit 3 First, you introduce two more qubits, prepared in the state \| 00 ⟩ . This auxiliary pair of qubits are used to extract information of the error without directly measuring or obtaining information about the logical state. Next, you carry out four CNOT operations: the first two operations use the first and second received qubits as control and act on the first auxiliary qubit, and last two operations use the first and third received qubits as control and act on the second auxiliary bit. The total state of all five qubits is now: State Error ( α \| 000 ⟩ + β \| 111 ⟩ ) \| 00 ⟩ No error ( α \| 100 ⟩ + β \| 011 ⟩ ) \| 11 ⟩ Qubit 1 ( α \| 010 ⟩ + β \| 101 ⟩ ) \| 10 ⟩ Qubit 2 ( α \| 001 ⟩ + β \| 110 ⟩ ) \| 01 ⟩ Qubit 3 To retrieve the error information, you measure the two auxiliary qubits in the computational basis states \| 0 ⟩ and \| 1 ⟩ . By doing this, you recover the joint state, which is called the error syndrome because it helps diagnose the errors in the received qubits. Now you know which of the four possible states the three received qubits are in. You can correct the error by applying the correction operation. In this case you're dealing with bit flip errors, so the correction is a σ x operation applied to one (or none) of the qubits. For example, if the error syndrome is \| 00 ⟩ , then the received qubits are in the state α \| 000 ⟩ + β \| 111 ⟩ , which is the state you originally sent. If the error syndrome is \| 11 ⟩ , then the received qubits are in the state α \| 100 ⟩ + b \| 011 ⟩ . There's a bit flip error on the first qubit, which you can correct by applying a σ x operation to the first qubit. Error syndrome Collapse state Correction \| 00 ⟩ α \| 000 ⟩ + β \| 111 ⟩ Do nothing \| 01 ⟩ α \| 100 ⟩ + β \| 011 ⟩ Apply σ x to qubit 3 \| 10 ⟩ α \| 010 ⟩ + β \| 101 ⟩ Apply σ x to qubit 2 \| 11 ⟩ α \| 001 ⟩ + β \| 110 ⟩ Apply σ x to qubit 1 Finally, to extract the single qubit you wanted to transmit originally, you apply two CNOT operations: one uses the first qubit as control and acts on the second qubit, and other uses the first qubit as control and acts on the third one. The state of the first qubit is now α \| 0 ⟩ + β \| 1 ⟩ , which is the original qubit you wanted to transmit. Important The QEC code doesn't gain any information regarding the coefficients α and β , hence superpositions of the computational state remain intact during correction.
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You can try [signing in](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction) or [changing directories](). Access to this page requires authorization. You can try [changing directories](). # Introduction to quantum error correction Feedback Summarize this article for me ## In this article 1. [What is quantum error correction?](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#what-is-quantum-error-correction) 2. [Types of quantum errors](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#types-of-quantum-errors) 3. [How does quantum error correction work?](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#how-does-quantum-error-correction-work) 4. [Types of QEC codes](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#types-of-qec-codes) 5. [Example: The three-qubit code](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#example-the-three-qubit-code) This article explains the basics of quantum error correction, the types of quantum errors, and some common quantum error correction codes. It also provides an example of how to correct errors using the three-qubit code. ## What is quantum error correction? Quantum error correction (QEC) is a technique that allows us to protect quantum information from errors. Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large-scale quantum computers, which are sensitive to noise. The basic principle behind quantum error correction is that the number of bits used to encode a given amount of information is increased. This redundancy allows the code to detect and correct errors. The error rates for quantum computers are typically higher than classical computer's errors due to the challenges associated with building and operating quantum systems. Noise, decoherence, and imperfections in quantum gates can cause errors in quantum computations. Current quantum computers have error rates in the range of 1% to 0.1%. In other words, this means that on average one out of every 100 to 1000 quantum gate operations results in an error. ## Types of quantum errors There are two fundamental types of quantum errors: bit flips and phase flips. Bit flip errors occur when a qubit changes from \\| 0 ⟩ to \\| 1 ⟩ or vice versa. Bit flip errors are also known as σ x\-errors, because they map the qubit states σ x \\| 0 ⟩ \= \\| 1 ⟩ and σ x \\| 1 ⟩ \= \\| 0 ⟩. This error is analogous to a classical bit flip error. Phase flip errors occur when a qubit changes its phase. They are also known as σ z\-errors, because they map the qubit states σ z \\| 0 ⟩ \= \\| 0 ⟩ and σ z \\| 1 ⟩ \= − \\| 1 ⟩. This type of error has no classical analog. In quantum computing, quantum errors can manifest as bit flips, phase flips, or a combination of both. ## How does quantum error correction work? Quantum error correction codes work by encoding the quantum information into a larger set of qubits, called the physical qubits. The joint state of the physical qubits represents a logical qubit. The physical qubits are subject to errors due to decoherence and imperfections in quantum gates. The code is designed so that errors can be detected and corrected by measuring some of the qubits in the code. For example, imagine you want to send the single-qubit message \\| 0 ⟩. You could use three physical qubits to encode the message, sending \\| 000 ⟩, which is known as a codeword. This error-correcting code is a repetition code, because the message is repeated three times. Now, imagine that a single bit-flip error occurs during transmission so that what the recipient receives is the state \\| 010 ⟩. In this scenario, the recipient may be able to infer that the intended message is \\| 000 ⟩. However, if the message is subject to two bit-flip errors, the recipient may infer an incorrect message. Finally, if all three bits are flipped so that the original message \\| 000 ⟩ becomes \\| 111 ⟩, the recipient has no way of knowing an error occurred. The code distance of a QEC code is the minimum number of errors that change one codeword into another, that is, the number of errors that can't be detected. The code distance d can be defined as d \= 2 t \+ 1 where t is the number of errors the code can correct. For example, the three-bit code can detect and correct one bit-flip error, so t \= 1, and thus the code distance is d \= 3. Note that repetition codes, such as the three-bit code used in this example, can only correct bit-flip errors, and not phase flip errors. To correct both types of errors, more sophisticated quantum error correction codes are needed. ## Types of QEC codes There are many different types of QEC codes, each with its own properties and advantages. Some common QEC codes are: - Repetition code: The simplest quantum error correction code, where a single qubit is encoded into multiple qubits by repeating it multiple times. The repetition code can correct bit flip errors, but not phase flip errors. - Shor code: The first quantum error correction code, developed by Peter Shor. It encodes one logical qubit into nine physical qubits. Shor code can correct one-bit flip error or one phase flip error, but it can't correct both types of errors at the same time. - Steane code: This is a seven-qubit code that can correct both bit flip and phase flip errors. It has the advantage of being fault-tolerant, meaning that the error correction process itself doesn't introduce extra errors. - Surface code: This is a topological error correction code that uses a two-dimensional lattice of qubits to encode logical qubits. It has a high error correction threshold and is considered one of the most promising techniques for large-scale, fault-tolerant quantum computing. The surface code is used by the [Microsoft Quantum resource estimator](https://learn.microsoft.com/en-us/azure/quantum/overview-resources-estimator#quantum-error-correction-schemes). - Hastings-Haah code: This quantum error correction code offers better space-time costs than surface codes on Majorana qubits in many regimes. For gate-based instruction sets, the overhead is larger, which makes this approach less efficient than the surface code. ## Example: The three-qubit code The three-qubit error correction code is a simple repetition code that can detect and correct one bit flip error. It encodes a single logical qubit into three physical qubits by repeating the qubit three times. Tip Check out the [Q\# code sample for the three-qubit code](https://github.com/microsoft/qdk/blob/main/samples/algorithms/ThreeQubitRepetitionCode.qs). Imagine you want to send an arbitrary single qubit \\| ϕ ⟩ \= α \\| 0 ⟩ \+ β \\| 1 ⟩. To avoid errors, you encode the basis states \\| 0 ⟩ and \\| 1 ⟩ into a joint state of three qubits. The two logical basis states are \\| 0 L ⟩ \= \\| 000 ⟩ and \\| 1 L ⟩ \= \\| 111 ⟩. Therefore, the single qubit \\| ϕ ⟩ \= α \\| 0 ⟩ \+ β \\| 1 ⟩ is encoded as: \\| ϕ L ⟩ \= α \\| 000 ⟩ \+ β \\| 111 ⟩ \= α \\| 0 L ⟩ \+ β \\| 1 L ⟩ Let's break down the steps of the three-qubit code. ### Preparing the qubits First, you encode your single qubit \\| ϕ ⟩ \= α \\| 0 ⟩ \+ β \\| 1 ⟩ into a joint state of three qubits. Next, you prepare two further qubits in the state \\| 0 ⟩. So, the global state of all three qubits is ( α \\| 0 ⟩ \+ β \\| 1 ⟩ ) \\| 0 ⟩ \\| 0 ⟩ \= α \\| 000 ⟩ \+ β \\| 100 ⟩. Lastly, you encode the single qubit into a joint state of three qubits, by applying two CNOT operations. The first CNOT uses the first qubit as control and acts on the second qubit, producing α \\| 000 ⟩ \+ β \\| 110 ⟩. The second CNOT uses the first qubit as control and acts on the third qubit. The state of the three qubits is now α \\| 000 ⟩ \+ β \\| 111 ⟩. ### Sending the qubits You send all three qubits. Assuming only one-bit flip errors can occur, the received qubits are in one of the following states: Expand table \| State \| Error \| \|---\|---\| \| α \\| 000 ⟩ \+ β \\| 111 ⟩ \| No error \| \| α \\| 100 ⟩ \+ β \\| 011 ⟩ \| Qubit 1 \| \| α \\| 010 ⟩ \+ β \\| 101 ⟩ \| Qubit 2 \| \| α \\| 001 ⟩ \+ β \\| 110 ⟩ \| Qubit 3 \| ### Adding auxiliary qubits First, you introduce two more qubits, prepared in the state \\| 00 ⟩. This auxiliary pair of qubits are used to extract information of the error without directly measuring or obtaining information about the logical state. Next, you carry out four CNOT operations: the first two operations use the first and second received qubits as control and act on the first auxiliary qubit, and last two operations use the first and third received qubits as control and act on the second auxiliary bit. The total state of all five qubits is now: Expand table \| State \| Error \| \|---\|---\| \| ( α \\| 000 ⟩ \+ β \\| 111 ⟩ ) \\| 00 ⟩ \| No error \| \| ( α \\| 100 ⟩ \+ β \\| 011 ⟩ ) \\| 11 ⟩ \| Qubit 1 \| \| ( α \\| 010 ⟩ \+ β \\| 101 ⟩ ) \\| 10 ⟩ \| Qubit 2 \| \| ( α \\| 001 ⟩ \+ β \\| 110 ⟩ ) \\| 01 ⟩ \| Qubit 3 \| ### Retrieving the error syndrome To retrieve the error information, you measure the two auxiliary qubits in the computational basis states \\| 0 ⟩ and \\| 1 ⟩. By doing this, you recover the joint state, which is called the error syndrome because it helps diagnose the errors in the received qubits. Now you know which of the four possible states the three received qubits are in. You can correct the error by applying the correction operation. In this case you're dealing with bit flip errors, so the correction is a σ x operation applied to one (or none) of the qubits. For example, if the error syndrome is \\| 00 ⟩, then the received qubits are in the state α \\| 000 ⟩ \+ β \\| 111 ⟩, which is the state you originally sent. If the error syndrome is \\| 11 ⟩, then the received qubits are in the state α \\| 100 ⟩ \+ b \\| 011 ⟩. There's a bit flip error on the first qubit, which you can correct by applying a σ x operation to the first qubit. Expand table \| Error syndrome \| Collapse state \| Correction \| \|---\|---\|---\| \| \\| 00 ⟩ \| α \\| 000 ⟩ \+ β \\| 111 ⟩ \| Do nothing \| \| \\| 01 ⟩ \| α \\| 100 ⟩ \+ β \\| 011 ⟩ \| Apply σ x to qubit 3 \| \| \\| 10 ⟩ \| α \\| 010 ⟩ \+ β \\| 101 ⟩ \| Apply σ x to qubit 2 \| \| \\| 11 ⟩ \| α \\| 001 ⟩ \+ β \\| 110 ⟩ \| Apply σ x to qubit 1 \| ### Extracting the original qubit Finally, to extract the single qubit you wanted to transmit originally, you apply two CNOT operations: one uses the first qubit as control and acts on the second qubit, and other uses the first qubit as control and acts on the third one. The state of the first qubit is now α \\| 0 ⟩ \+ β \\| 1 ⟩, which is the original qubit you wanted to transmit. Important The QEC code doesn't gain any information regarding the coefficients α and β, hence superpositions of the computational state remain intact during correction. * ## Feedback Was this page helpful? Yes No No Need help with this topic? Want to try using Ask Learn to clarify or guide you through this topic? Ask Learn Ask Learn Suggest a fix? * ## Additional resources Training Module [Teleport entangled qubits with Q\# - Training](https://learn.microsoft.com/en-us/training/modules/explore-entanglement/?source=recommendations) Teleport entangled qubits with Q\# *** - Last updated on 09/16/2024 ## In this article 1. [What is quantum error correction?](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#what-is-quantum-error-correction) 2. [Types of quantum errors](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#types-of-quantum-errors) 3. [How does quantum error correction work?](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#how-does-quantum-error-correction-work) 4. [Types of QEC codes](https://learn.microsoft.com/en-us/azure/quantum/concepts-error-correction#types-of-qec-codes) 5. 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Readable Markdown	This article explains the basics of quantum error correction, the types of quantum errors, and some common quantum error correction codes. It also provides an example of how to correct errors using the three-qubit code. Quantum error correction (QEC) is a technique that allows us to protect quantum information from errors. Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large-scale quantum computers, which are sensitive to noise. The basic principle behind quantum error correction is that the number of bits used to encode a given amount of information is increased. This redundancy allows the code to detect and correct errors. The error rates for quantum computers are typically higher than classical computer's errors due to the challenges associated with building and operating quantum systems. Noise, decoherence, and imperfections in quantum gates can cause errors in quantum computations. Current quantum computers have error rates in the range of 1% to 0.1%. In other words, this means that on average one out of every 100 to 1000 quantum gate operations results in an error. There are two fundamental types of quantum errors: bit flips and phase flips. Bit flip errors occur when a qubit changes from \\| 0 ⟩ to \\| 1 ⟩ or vice versa. Bit flip errors are also known as σ x\-errors, because they map the qubit states σ x \\| 0 ⟩ \= \\| 1 ⟩ and σ x \\| 1 ⟩ \= \\| 0 ⟩. This error is analogous to a classical bit flip error. Phase flip errors occur when a qubit changes its phase. They are also known as σ z\-errors, because they map the qubit states σ z \\| 0 ⟩ \= \\| 0 ⟩ and σ z \\| 1 ⟩ \= − \\| 1 ⟩. This type of error has no classical analog. In quantum computing, quantum errors can manifest as bit flips, phase flips, or a combination of both. Quantum error correction codes work by encoding the quantum information into a larger set of qubits, called the physical qubits. The joint state of the physical qubits represents a logical qubit. The physical qubits are subject to errors due to decoherence and imperfections in quantum gates. The code is designed so that errors can be detected and corrected by measuring some of the qubits in the code. For example, imagine you want to send the single-qubit message \\| 0 ⟩. You could use three physical qubits to encode the message, sending \\| 000 ⟩, which is known as a codeword. This error-correcting code is a repetition code, because the message is repeated three times. Now, imagine that a single bit-flip error occurs during transmission so that what the recipient receives is the state \\| 010 ⟩. In this scenario, the recipient may be able to infer that the intended message is \\| 000 ⟩. However, if the message is subject to two bit-flip errors, the recipient may infer an incorrect message. Finally, if all three bits are flipped so that the original message \\| 000 ⟩ becomes \\| 111 ⟩, the recipient has no way of knowing an error occurred. The code distance of a QEC code is the minimum number of errors that change one codeword into another, that is, the number of errors that can't be detected. The code distance d can be defined as d \= 2 t \+ 1 where t is the number of errors the code can correct. For example, the three-bit code can detect and correct one bit-flip error, so t \= 1, and thus the code distance is d \= 3. Note that repetition codes, such as the three-bit code used in this example, can only correct bit-flip errors, and not phase flip errors. To correct both types of errors, more sophisticated quantum error correction codes are needed. There are many different types of QEC codes, each with its own properties and advantages. Some common QEC codes are: - Repetition code: The simplest quantum error correction code, where a single qubit is encoded into multiple qubits by repeating it multiple times. The repetition code can correct bit flip errors, but not phase flip errors. - Shor code: The first quantum error correction code, developed by Peter Shor. It encodes one logical qubit into nine physical qubits. Shor code can correct one-bit flip error or one phase flip error, but it can't correct both types of errors at the same time. - Steane code: This is a seven-qubit code that can correct both bit flip and phase flip errors. It has the advantage of being fault-tolerant, meaning that the error correction process itself doesn't introduce extra errors. - Surface code: This is a topological error correction code that uses a two-dimensional lattice of qubits to encode logical qubits. It has a high error correction threshold and is considered one of the most promising techniques for large-scale, fault-tolerant quantum computing. The surface code is used by the [Microsoft Quantum resource estimator](https://learn.microsoft.com/en-us/azure/quantum/overview-resources-estimator#quantum-error-correction-schemes). - Hastings-Haah code: This quantum error correction code offers better space-time costs than surface codes on Majorana qubits in many regimes. For gate-based instruction sets, the overhead is larger, which makes this approach less efficient than the surface code. The three-qubit error correction code is a simple repetition code that can detect and correct one bit flip error. It encodes a single logical qubit into three physical qubits by repeating the qubit three times. Imagine you want to send an arbitrary single qubit \\| ϕ ⟩ \= α \\| 0 ⟩ \+ β \\| 1 ⟩. To avoid errors, you encode the basis states \\| 0 ⟩ and \\| 1 ⟩ into a joint state of three qubits. The two logical basis states are \\| 0 L ⟩ \= \\| 000 ⟩ and \\| 1 L ⟩ \= \\| 111 ⟩. Therefore, the single qubit \\| ϕ ⟩ \= α \\| 0 ⟩ \+ β \\| 1 ⟩ is encoded as: \\| ϕ L ⟩ \= α \\| 000 ⟩ \+ β \\| 111 ⟩ \= α \\| 0 L ⟩ \+ β \\| 1 L ⟩ Let's break down the steps of the three-qubit code. First, you encode your single qubit \\| ϕ ⟩ \= α \\| 0 ⟩ \+ β \\| 1 ⟩ into a joint state of three qubits. Next, you prepare two further qubits in the state \\| 0 ⟩. So, the global state of all three qubits is ( α \\| 0 ⟩ \+ β \\| 1 ⟩ ) \\| 0 ⟩ \\| 0 ⟩ \= α \\| 000 ⟩ \+ β \\| 100 ⟩. Lastly, you encode the single qubit into a joint state of three qubits, by applying two CNOT operations. The first CNOT uses the first qubit as control and acts on the second qubit, producing α \\| 000 ⟩ \+ β \\| 110 ⟩. The second CNOT uses the first qubit as control and acts on the third qubit. The state of the three qubits is now α \\| 000 ⟩ \+ β \\| 111 ⟩. You send all three qubits. Assuming only one-bit flip errors can occur, the received qubits are in one of the following states: \| State \| Error \| \|---\|---\| \| α \\| 000 ⟩ \+ β \\| 111 ⟩ \| No error \| \| α \\| 100 ⟩ \+ β \\| 011 ⟩ \| Qubit 1 \| \| α \\| 010 ⟩ \+ β \\| 101 ⟩ \| Qubit 2 \| \| α \\| 001 ⟩ \+ β \\| 110 ⟩ \| Qubit 3 \| First, you introduce two more qubits, prepared in the state \\| 00 ⟩. This auxiliary pair of qubits are used to extract information of the error without directly measuring or obtaining information about the logical state. Next, you carry out four CNOT operations: the first two operations use the first and second received qubits as control and act on the first auxiliary qubit, and last two operations use the first and third received qubits as control and act on the second auxiliary bit. The total state of all five qubits is now: \| State \| Error \| \|---\|---\| \| ( α \\| 000 ⟩ \+ β \\| 111 ⟩ ) \\| 00 ⟩ \| No error \| \| ( α \\| 100 ⟩ \+ β \\| 011 ⟩ ) \\| 11 ⟩ \| Qubit 1 \| \| ( α \\| 010 ⟩ \+ β \\| 101 ⟩ ) \\| 10 ⟩ \| Qubit 2 \| \| ( α \\| 001 ⟩ \+ β \\| 110 ⟩ ) \\| 01 ⟩ \| Qubit 3 \| To retrieve the error information, you measure the two auxiliary qubits in the computational basis states \\| 0 ⟩ and \\| 1 ⟩. By doing this, you recover the joint state, which is called the error syndrome because it helps diagnose the errors in the received qubits. Now you know which of the four possible states the three received qubits are in. You can correct the error by applying the correction operation. In this case you're dealing with bit flip errors, so the correction is a σ x operation applied to one (or none) of the qubits. For example, if the error syndrome is \\| 00 ⟩, then the received qubits are in the state α \\| 000 ⟩ \+ β \\| 111 ⟩, which is the state you originally sent. If the error syndrome is \\| 11 ⟩, then the received qubits are in the state α \\| 100 ⟩ \+ b \\| 011 ⟩. There's a bit flip error on the first qubit, which you can correct by applying a σ x operation to the first qubit. \| Error syndrome \| Collapse state \| Correction \| \|---\|---\|---\| \| \\| 00 ⟩ \| α \\| 000 ⟩ \+ β \\| 111 ⟩ \| Do nothing \| \| \\| 01 ⟩ \| α \\| 100 ⟩ \+ β \\| 011 ⟩ \| Apply σ x to qubit 3 \| \| \\| 10 ⟩ \| α \\| 010 ⟩ \+ β \\| 101 ⟩ \| Apply σ x to qubit 2 \| \| \\| 11 ⟩ \| α \\| 001 ⟩ \+ β \\| 110 ⟩ \| Apply σ x to qubit 1 \| Finally, to extract the single qubit you wanted to transmit originally, you apply two CNOT operations: one uses the first qubit as control and acts on the second qubit, and other uses the first qubit as control and acts on the third one. The state of the first qubit is now α \\| 0 ⟩ \+ β \\| 1 ⟩, which is the original qubit you wanted to transmit. Important The QEC code doesn't gain any information regarding the coefficients α and β, hence superpositions of the computational state remain intact during correction.
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