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1. Shard Calculation

Query:

Response:

Calculated Shard: 15 (from laksa152)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa015.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://www.quantinuum.com/glossary-item/quantum-error-correcting-code\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.quantinuum.com/glossary-item/quantum-error-correcting-code\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.quantinuum.com\/glossary-item\/quantum-error-correcting-code","crawl_time":1775033785,"first_indexed_time":1736918967,"http_code":200,"src_unparsed":"com,quantinuum!www,\/glossary-item\/quantum-error-correcting-code s443","src_root_hash":"264960727261934015","history_drop_reason":null,"meta_title":"Quantinuum | Quantum Error Correcting Code","meta_descriptions":[],"attrs_boilerpipe_text":"We take the original definition\n: “A quantum error-correcting code is defined to be a unitary mapping (encoding) of\nk\nqubits\n(2-state quantum systems) into a subspace of the quantum state space of\nn\nqubits such that if any\nt\nof the qubits undergo arbitrary decoherence, not necessarily independently, the resulting\nn\nqubits can be used to faithfully reconstruct the original quantum state of the\nk\nencoded qubits.”\nAn [[\nn,k,d\n]] code is a quantum error correction code which encodes\nk\nqubits in an\nn\n-qubit state, in such a way that any operation which maps some encoded state to another encoded state must act on at least dd qubits. (So, for example, any encoded state which has been subjected to an error consisting of at most ⌊\n(d−1)\/2\n⌋ Pauli operations can in principle be recovered perfectly).\nThis notation generalizes the notation [\nn,k,d\n] for classical error correction codes, in which\nk\n-bit \"plaintext\" strings are encoded in\nn\n-bit \"codeword\" strings, in such a way that at least\nd\nbits must be flipped to transform between any two codewords representing different plain texts. (In this context and in the quantum case,\nd\nis referred to as the code distance.) The double-brackets are used simply to denote that the code being referred to is a quantum error correction code rather than a classical code.\nAnother way to understand quantum error correcting codes is offered by\nJohn Preskill\n: “A quantum error-correcting code can be viewed as a mapping of\nk\nqubits (a Hilbert space of dimension\n2k\n) into n qubits (a Hilbert space of dimension 2n), where\nn\n>\nk\n. The\nk\nqubits are the “\nlogical qubits\n” or “encoded qubits” that we wish to protect from error. The additional\nn − k\nqubits allow us to store the\nk\nlogical qubits in a redundant fashion, so that the encoded information is not easily damaged.”","attrs_markdown":"Products & Solutions\n\n#### Quantum Computers\n[Quantinuum Systems Discover the world’s highest performing quantum computers](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems)\n\n[System Model H1 Our first-generation quantum computer with a linear architecture](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/system-model-h1)\n\n[System Model H2 Our second-generation quantum computer with a racetrack architecture](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/system-model-h2)\n\n[Helios Our Hardware-as-a-Service provides access on-premise, in the cloud, or both](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/helios)\n\n[Hardware Roadmap Explore our accelerated roadmap aimed at advancing commercial quantum computing](https:\/\/www.quantinuum.com\/blog\/technical-perspective-by-the-end-of-the-decade-we-will-deliver-universal-fault-tolerant-quantum-computing)\n\n#### SOlutions\n[Quantum Origin Our advanced quantum random number generator (QRNG), designed to strengthen encryption and protect data](https:\/\/www.quantinuum.com\/products-solutions\/quantum-origin)\n\n[InQuanto Accelerate quantum computational chemistry for complex molecular & materials simulations](https:\/\/www.quantinuum.com\/products-solutions\/inquanto)\n\n[Nexus All-in-one quantum computing platform for accessing and managing the highest performance full-stack](https:\/\/www.quantinuum.com\/products-solutions\/nexus)\n\n[Cloud Access Explore using our world-leading quantum hardware for commercial applications](https:\/\/www.quantinuum.com\/products-solutions\/hardware-h-series#access)\n\n#### Developer Tools\n[TKET Develop and execute gate-level quantum computation](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools#tket)\n\n[Guppy Program the next generation of quantum computers with our purpose-built language hosted in Python](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools#guppy)\n\n[Lambeq Experiment with the structural relationships between quantum theory and natural languages](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools#lambeq)\n\nResearch\n\n[Research Areas Discover the vast landscape of our scientific expertise](https:\/\/www.quantinuum.com\/research\/research-areas)\n\n[Technical Blog Get educated on the world of quantum computing](https:\/\/www.quantinuum.com\/research\/research-areas#technical-blog)\n\n[Publications Explore research and journal publications from our scientific teams](https:\/\/www.quantinuum.com\/research\/research-areas#publications)\n\nCompany\n\n[About Quantinuum Learn more about the world’s largest integrated quantum computing company](https:\/\/www.quantinuum.com\/company\/about)\n\n[Careers Join us in the center of gravity for quantum computing](https:\/\/www.quantinuum.com\/company\/careers)\n\n[Contact Us Get in touch with our team](https:\/\/www.quantinuum.com\/contact)\n\nNews\n\n[Press Releases See how Quantinuum is pushing the boundaries in the world of quantum](https:\/\/www.quantinuum.com\/news\/news#press-release)\n\n[Blog Taking steps forward across the quantum computing landscape](https:\/\/www.quantinuum.com\/news\/blog)\n\n[Glossary Better understand quantum computing with these terms and definitions](https:\/\/www.quantinuum.com\/glossary)\n\nCommunity\n\n[Q-Net Join our user community aimed at shared learning, support and collaboration for our full-stack technologies](https:\/\/www.quantinuum.com\/q-net)\n\n[Events Find out where we’ll be next](https:\/\/www.quantinuum.com\/news\/events)\n\n[Startup Partner Program Pioneering startups connect with us to accelerate ecosystem innovation and access our world-leading technologies](https:\/\/www.quantinuum.com\/startup-partner-program)\n\nLet's Talk\n\n[Button Text](https:\/\/www.quantinuum.com\/contact)\n\nlets talk\n\n[Get Access](https:\/\/www.quantinuum.com\/contact)\n\nProducts & Solutions\n\n#### Quantum Computers\n[Quantinuum Systems Discover the world’s highest performing quantum computers](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems)\n\n[System Model H1 Our first-generation quantum computer with a linear architecture](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/system-model-h1)\n\n[System Model H2 Our latest-generation quantum computer with a racetrack architecture](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/system-model-h2)\n\n[Helios Our Hardware-as-a-Service provides access on-premise, in the cloud, or both](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/helios)\n\n[Hardware Roadmap Explore our accelerated roadmap aimed at advancing commercial quantum computing](https:\/\/www.quantinuum.com\/blog\/quantinuum-accelerates-the-path-to-universal-fault-tolerant-quantum-computing-supports-microsofts-ai-and-quantum-powered-compute-platform-and-the-path-to-a-quantum-supercomputer)\n\n#### SOlutions\n[Quantum Origin Our advanced quantum random number generator (QRNG), designed to strengthen encryption and protect data](https:\/\/www.quantinuum.com\/products-solutions\/quantum-origin)\n\n[InQuanto Accelerate quantum computational chemistry and for complex molecular & materials simulations](https:\/\/www.quantinuum.com\/products-solutions\/inquanto)\n\n[Nexus (Beta) All-in-one quantum computing platform for accessing and managing the highest performance full-stack](https:\/\/www.quantinuum.com\/products-solutions\/nexus)\n\n[Cloud Access Explore using our world-leading quantum hardware for commercial applications](https:\/\/www.quantinuum.com\/products-solutions\/hardware-h-series#access)\n\n#### Developer Tools\n[TKET Develop and execute gate-level quantum computation](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools#tket)\n\n[Lambeq Experiment with the structural relationships between quantum theory and natural languages](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools#lambeq)\n\n[Qermit Run error-mitigation protocols on quantum processors](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools#qermit)\n\nResearch\n\n[Research Areas Discover the vast landscape of our scientific expertise](https:\/\/www.quantinuum.com\/research\/research-areas)\n\n[Technical Blog Get educated on the world of quantum computing](https:\/\/www.quantinuum.com\/research\/research-areas#technical-blog)\n\n[Publications Explore research and journal publications from our scientific teams](https:\/\/www.quantinuum.com\/research\/research-areas#publications)\n\nCompany\n\n[About Quantinuum Learn more about the world’s largest integrated quantum computing company](https:\/\/www.quantinuum.com\/company\/about)\n\n[Careers Join us in the center of gravity for quantum computing](https:\/\/www.quantinuum.com\/company\/careers)\n\n[Contact Us Get in touch with our team](https:\/\/www.quantinuum.com\/contact)\n\nNews\n\n[Press Releases See how Quantinuum is pushing the boundaries in the world of quantum](https:\/\/www.quantinuum.com\/news\/news#press-release)\n\n[Blog Taking steps forward across the quantum computing landscape](https:\/\/www.quantinuum.com\/news\/blog)\n\n[Events Find out where we’ll be next](https:\/\/www.quantinuum.com\/news\/events)\n\n[Glossary Better understand quantum computing with these terms and definitions](https:\/\/www.quantinuum.com\/glossary)\n\nCommunity\n\n[Q-Net Join our user community aimed at shared learning, support and collaboration for our full-stack technologies](https:\/\/www.quantinuum.com\/q-net)\n\n[Startup Partner Program Pioneering startups connect with us to accelerate ecosystem innovation and access our world-leading technologies](https:\/\/www.quantinuum.com\/startup-partner-program)\n\nGet Access\n\n[Get Access](https:\/\/www.quantinuum.com\/glossary-item\/quantum-error-correcting-code)\n\nLet's Talk\n\n[Get Access](https:\/\/www.quantinuum.com\/contact)\n\n[Back to glossary](https:\/\/www.quantinuum.com\/glossary)\n\n# **Quantum Error Correcting Code**\n[We take the original definition](https:\/\/arxiv.org\/pdf\/quant-ph\/9512032): “A quantum error-correcting code is defined to be a unitary mapping (encoding) of *k* [qubits](https:\/\/www.quantinuum.com\/glossary-item\/qubit) (2-state quantum systems) into a subspace of the quantum state space of *n* qubits such that if any *t* of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting *n* qubits can be used to faithfully reconstruct the original quantum state of the *k* encoded qubits.”\n\nAn \\[\\[*n,k,d*\\]\\] code is a quantum error correction code which encodes *k* qubits in an *n*\\-qubit state, in such a way that any operation which maps some encoded state to another encoded state must act on at least dd qubits. (So, for example, any encoded state which has been subjected to an error consisting of at most ⌊*(d−1)\/2*⌋ Pauli operations can in principle be recovered perfectly).\n\nThis notation generalizes the notation \\[*n,k,d*\\] for classical error correction codes, in which *k*\\-bit \"plaintext\" strings are encoded in *n*\\-bit \"codeword\" strings, in such a way that at least *d* bits must be flipped to transform between any two codewords representing different plain texts. (In this context and in the quantum case, *d* is referred to as the code distance.) The double-brackets are used simply to denote that the code being referred to is a quantum error correction code rather than a classical code.\n\nAnother way to understand quantum error correcting codes is offered by [John Preskill](http:\/\/theory.caltech.edu\/~preskill\/ph229\/notes\/chap7.pdf): “A quantum error-correcting code can be viewed as a mapping of *k* qubits (a Hilbert space of dimension *2k*) into n qubits (a Hilbert space of dimension 2n), where *n* \\> *k*. The *k* qubits are the “[logical qubits](https:\/\/www.quantinuum.com\/glossary-item\/logical-qubit)” or “encoded qubits” that we wish to protect from error. The additional *n − k* qubits allow us to store the *k* logical qubits in a redundant fashion, so that the encoded information is not easily damaged.”\n\nBlog\n\nMarch 25, 2026\n\nCelebrating Our First Annual Q-Net Connect\\!\n\n[Read More](https:\/\/www.quantinuum.com\/blog\/celebrating-our-first-annual-q-net-connect)\n\nBlog\n\nMarch 16, 2026\n\nWe’re Using AI to Discover New Quantum Algorithms\n\n[Read More](https:\/\/www.quantinuum.com\/blog\/were-using-ai-to-discover-new-quantum-algorithms)\n\nBlog\n\nMarch 16, 2026\n\nReal Time Error Correction at Increased Scale\n\n[Read More](https:\/\/www.quantinuum.com\/blog\/real-time-error-correction-at-increased-scale)\n\n###### Company\n[About Us](https:\/\/www.quantinuum.com\/company\/about)\n\n[Careers](https:\/\/www.quantinuum.com\/company\/careers)\n\n[Leadership](https:\/\/www.quantinuum.com\/company\/about#leadership)\n\n[Newsroom](https:\/\/www.quantinuum.com\/news\/news)\n\n[Publications](https:\/\/www.quantinuum.com\/research\/research-areas#publications)\n\n[Contact Us](https:\/\/www.quantinuum.com\/contact)\n\n[Quantinuum Japan](https:\/\/quantinuum.co.jp\/)\n\n###### Products & Solutions\n[Quantinuum Systems](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems)\n\n[Helios](https:\/\/www.quantinuum.com\/products-solutions\/quantinuum-systems\/helios)\n\n[Quantum Origin](https:\/\/www.quantinuum.com\/products-solutions\/quantum-origin)\n\n[InQuanto](https:\/\/www.quantinuum.com\/products-solutions\/inquanto)\n\n[Nexus](https:\/\/www.quantinuum.com\/products-solutions\/nexus)\n\n[Developer Tools](https:\/\/www.quantinuum.com\/products-solutions\/developer-tools)\n\n###### Get Started\n[Hardware Access](https:\/\/www.quantinuum.com\/products-solutions\/hardware-h-series#access)\n\n[Cloud Access](https:\/\/www.quantinuum.com\/products-solutions\/hardware-h-series#access)\n\n[Documentation](https:\/\/docs.quantinuum.com\/)\n\n###### Stay Connected\n[Q-Net](https:\/\/www.quantinuum.com\/q-net)\n\n[Startup Partner Program](https:\/\/www.quantinuum.com\/startup-partner-program)\n\n###### Legal\n[Privacy Statement](https:\/\/www.quantinuum.com\/privacy)\n\n[Cookie Notice](https:\/\/www.quantinuum.com\/cookie-notice)\n\n[Terms & Conditions](https:\/\/www.quantinuum.com\/terms-conditions)\n\n[Trademark Usage Guidelines](https:\/\/cdn.prod.website-files.com\/669960f53cd73aedb80c8eea\/6733a7ff524a2043654ad9a1_Quantinuum%20Trademark%20Guidelines%20v1.1%20Q4%20Dec%202024.pdf)\n\n[License Agreements](https:\/\/www.quantinuum.com\/license-agreements)\n\nCookie Preferences\n\n[Integrity & Compliance](https:\/\/www.quantinuum.com\/integrity-and-compliance)\n\n© 2026 Quantinuum. All rights reserved.","attrs_readable_markdown":"[We take the original definition](https:\/\/arxiv.org\/pdf\/quant-ph\/9512032): “A quantum error-correcting code is defined to be a unitary mapping (encoding) of *k* [qubits](https:\/\/www.quantinuum.com\/glossary-item\/qubit) (2-state quantum systems) into a subspace of the quantum state space of *n* qubits such that if any *t* of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting *n* qubits can be used to faithfully reconstruct the original quantum state of the *k* encoded qubits.”\n\nAn \\[\\[*n,k,d*\\]\\] code is a quantum error correction code which encodes *k* qubits in an *n*\\-qubit state, in such a way that any operation which maps some encoded state to another encoded state must act on at least dd qubits. (So, for example, any encoded state which has been subjected to an error consisting of at most ⌊*(d−1)\/2*⌋ Pauli operations can in principle be recovered perfectly).\n\nThis notation generalizes the notation \\[*n,k,d*\\] for classical error correction codes, in which *k*\\-bit \"plaintext\" strings are encoded in *n*\\-bit \"codeword\" strings, in such a way that at least *d* bits must be flipped to transform between any two codewords representing different plain texts. (In this context and in the quantum case, *d* is referred to as the code distance.) The double-brackets are used simply to denote that the code being referred to is a quantum error correction code rather than a classical code.\n\nAnother way to understand quantum error correcting codes is offered by [John Preskill](http:\/\/theory.caltech.edu\/~preskill\/ph229\/notes\/chap7.pdf): “A quantum error-correcting code can be viewed as a mapping of *k* qubits (a Hilbert space of dimension *2k*) into n qubits (a Hilbert space of dimension 2n), where *n* \\> *k*. The *k* qubits are the “[logical qubits](https:\/\/www.quantinuum.com\/glossary-item\/logical-qubit)” or “encoded qubits” that we wish to protect from error. The additional *n − k* qubits allow us to store the *k* logical qubits in a redundant fashion, so that the encoded information is not easily damaged.”","meta_canonical":null}

3. Robots.txt Check

Query:

Response:

4. Spam/Ban Check

Query:

Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

5 days ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0.2 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://www.quantinuum.com/glossary-item/quantum-error-correcting-code
Last Crawled	2026-04-01 08:56:25 (5 days ago)
First Indexed	2025-01-15 05:29:27 (1 year ago)
HTTP Status Code	200
Meta Title	Quantinuum \| Quantum Error Correcting Code
Meta Description	null
Meta Canonical	null
Boilerpipe Text	We take the original definition : “A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits.” An [[ n,k,d ]] code is a quantum error correction code which encodes k qubits in an n -qubit state, in such a way that any operation which maps some encoded state to another encoded state must act on at least dd qubits. (So, for example, any encoded state which has been subjected to an error consisting of at most ⌊ (d−1)/2 ⌋ Pauli operations can in principle be recovered perfectly). This notation generalizes the notation [ n,k,d ] for classical error correction codes, in which k -bit "plaintext" strings are encoded in n -bit "codeword" strings, in such a way that at least d bits must be flipped to transform between any two codewords representing different plain texts. (In this context and in the quantum case, d is referred to as the code distance.) The double-brackets are used simply to denote that the code being referred to is a quantum error correction code rather than a classical code. Another way to understand quantum error correcting codes is offered by John Preskill : “A quantum error-correcting code can be viewed as a mapping of k qubits (a Hilbert space of dimension 2k ) into n qubits (a Hilbert space of dimension 2n), where n > k . The k qubits are the “ logical qubits ” or “encoded qubits” that we wish to protect from error. The additional n − k qubits allow us to store the k logical qubits in a redundant fashion, so that the encoded information is not easily damaged.”
Markdown	Products & Solutions #### Quantum Computers [Quantinuum Systems Discover the world’s highest performing quantum computers](https://www.quantinuum.com/products-solutions/quantinuum-systems) [System Model H1 Our first-generation quantum computer with a linear architecture](https://www.quantinuum.com/products-solutions/quantinuum-systems/system-model-h1) [System Model H2 Our second-generation quantum computer with a racetrack architecture](https://www.quantinuum.com/products-solutions/quantinuum-systems/system-model-h2) [Helios Our Hardware-as-a-Service provides access on-premise, in the cloud, or both](https://www.quantinuum.com/products-solutions/quantinuum-systems/helios) [Hardware Roadmap Explore our accelerated roadmap aimed at advancing commercial quantum computing](https://www.quantinuum.com/blog/technical-perspective-by-the-end-of-the-decade-we-will-deliver-universal-fault-tolerant-quantum-computing) #### SOlutions [Quantum Origin Our advanced quantum random number generator (QRNG), designed to strengthen encryption and protect data](https://www.quantinuum.com/products-solutions/quantum-origin) [InQuanto Accelerate quantum computational chemistry for complex molecular & materials simulations](https://www.quantinuum.com/products-solutions/inquanto) [Nexus All-in-one quantum computing platform for accessing and managing the highest performance full-stack](https://www.quantinuum.com/products-solutions/nexus) [Cloud Access Explore using our world-leading quantum hardware for commercial applications](https://www.quantinuum.com/products-solutions/hardware-h-series#access) #### Developer Tools [TKET Develop and execute gate-level quantum computation](https://www.quantinuum.com/products-solutions/developer-tools#tket) [Guppy Program the next generation of quantum computers with our purpose-built language hosted in Python](https://www.quantinuum.com/products-solutions/developer-tools#guppy) [Lambeq Experiment with the structural relationships between quantum theory and natural languages](https://www.quantinuum.com/products-solutions/developer-tools#lambeq) Research [Research Areas Discover the vast landscape of our scientific expertise](https://www.quantinuum.com/research/research-areas) [Technical Blog Get educated on the world of quantum computing](https://www.quantinuum.com/research/research-areas#technical-blog) [Publications Explore research and journal publications from our scientific teams](https://www.quantinuum.com/research/research-areas#publications) Company [About Quantinuum Learn more about the world’s largest integrated quantum computing company](https://www.quantinuum.com/company/about) [Careers Join us in the center of gravity for quantum computing](https://www.quantinuum.com/company/careers) [Contact Us Get in touch with our team](https://www.quantinuum.com/contact) News [Press Releases See how Quantinuum is pushing the boundaries in the world of quantum](https://www.quantinuum.com/news/news#press-release) [Blog Taking steps forward across the quantum computing landscape](https://www.quantinuum.com/news/blog) [Glossary Better understand quantum computing with these terms and definitions](https://www.quantinuum.com/glossary) Community [Q-Net Join our user community aimed at shared learning, support and collaboration for our full-stack technologies](https://www.quantinuum.com/q-net) [Events Find out where we’ll be next](https://www.quantinuum.com/news/events) [Startup Partner Program Pioneering startups connect with us to accelerate ecosystem innovation and access our world-leading technologies](https://www.quantinuum.com/startup-partner-program) Let's Talk [Button Text](https://www.quantinuum.com/contact) lets talk [Get Access](https://www.quantinuum.com/contact) Products & Solutions #### Quantum Computers [Quantinuum Systems Discover the world’s highest performing quantum computers](https://www.quantinuum.com/products-solutions/quantinuum-systems) [System Model H1 Our first-generation quantum computer with a linear architecture](https://www.quantinuum.com/products-solutions/quantinuum-systems/system-model-h1) [System Model H2 Our latest-generation quantum computer with a racetrack architecture](https://www.quantinuum.com/products-solutions/quantinuum-systems/system-model-h2) [Helios Our Hardware-as-a-Service provides access on-premise, in the cloud, or both](https://www.quantinuum.com/products-solutions/quantinuum-systems/helios) [Hardware Roadmap Explore our accelerated roadmap aimed at advancing commercial quantum computing](https://www.quantinuum.com/blog/quantinuum-accelerates-the-path-to-universal-fault-tolerant-quantum-computing-supports-microsofts-ai-and-quantum-powered-compute-platform-and-the-path-to-a-quantum-supercomputer) #### SOlutions [Quantum Origin Our advanced quantum random number generator (QRNG), designed to strengthen encryption and protect data](https://www.quantinuum.com/products-solutions/quantum-origin) [InQuanto Accelerate quantum computational 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expertise](https://www.quantinuum.com/research/research-areas) [Technical Blog Get educated on the world of quantum computing](https://www.quantinuum.com/research/research-areas#technical-blog) [Publications Explore research and journal publications from our scientific teams](https://www.quantinuum.com/research/research-areas#publications) Company [About Quantinuum Learn more about the world’s largest integrated quantum computing company](https://www.quantinuum.com/company/about) [Careers Join us in the center of gravity for quantum computing](https://www.quantinuum.com/company/careers) [Contact Us Get in touch with our team](https://www.quantinuum.com/contact) News [Press Releases See how Quantinuum is pushing the boundaries in the world of quantum](https://www.quantinuum.com/news/news#press-release) [Blog Taking steps forward across the quantum computing landscape](https://www.quantinuum.com/news/blog) [Events Find out where we’ll be next](https://www.quantinuum.com/news/events) [Glossary Better understand quantum computing with these terms and definitions](https://www.quantinuum.com/glossary) Community [Q-Net Join our user community aimed at shared learning, support and collaboration for our full-stack technologies](https://www.quantinuum.com/q-net) [Startup Partner Program Pioneering startups connect with us to accelerate ecosystem innovation and access our world-leading technologies](https://www.quantinuum.com/startup-partner-program) Get Access [Get Access](https://www.quantinuum.com/glossary-item/quantum-error-correcting-code) Let's Talk [Get Access](https://www.quantinuum.com/contact) [Back to glossary](https://www.quantinuum.com/glossary) # Quantum Error Correcting Code [We take the original definition](https://arxiv.org/pdf/quant-ph/9512032): “A quantum error-correcting code is defined to be a unitary mapping (encoding) of k [qubits](https://www.quantinuum.com/glossary-item/qubit) (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits.” An \[\[n,k,d\]\] code is a quantum error correction code which encodes k qubits in an n\-qubit state, in such a way that any operation which maps some encoded state to another encoded state must act on at least dd qubits. (So, for example, any encoded state which has been subjected to an error consisting of at most ⌊(d−1)/2⌋ Pauli operations can in principle be recovered perfectly). This notation generalizes the notation \[n,k,d\] for classical error correction codes, in which k\-bit "plaintext" strings are encoded in n\-bit "codeword" strings, in such a way that at least d bits must be flipped to transform between any two codewords representing different plain texts. (In this context and in the quantum case, d is referred to as the code distance.) The double-brackets are used simply to denote that the code being referred to is a quantum error correction code rather than a classical code. Another way to understand quantum error correcting codes is offered by [John Preskill](http://theory.caltech.edu/~preskill/ph229/notes/chap7.pdf): “A quantum error-correcting code can be viewed as a mapping of k qubits (a Hilbert space of dimension 2k) into n qubits (a Hilbert space of dimension 2n), where n \> k. The k qubits are the “[logical qubits](https://www.quantinuum.com/glossary-item/logical-qubit)” or “encoded qubits” that we wish to protect from error. The additional n − k qubits allow us to store the k logical qubits in a redundant fashion, so that the encoded information is not easily damaged.” Blog March 25, 2026 Celebrating Our First Annual Q-Net Connect\! 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Readable Markdown	[We take the original definition](https://arxiv.org/pdf/quant-ph/9512032): “A quantum error-correcting code is defined to be a unitary mapping (encoding) of k [qubits](https://www.quantinuum.com/glossary-item/qubit) (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits.” An \[\[n,k,d\]\] code is a quantum error correction code which encodes k qubits in an n\-qubit state, in such a way that any operation which maps some encoded state to another encoded state must act on at least dd qubits. (So, for example, any encoded state which has been subjected to an error consisting of at most ⌊(d−1)/2⌋ Pauli operations can in principle be recovered perfectly). This notation generalizes the notation \[n,k,d\] for classical error correction codes, in which k\-bit "plaintext" strings are encoded in n\-bit "codeword" strings, in such a way that at least d bits must be flipped to transform between any two codewords representing different plain texts. (In this context and in the quantum case, d is referred to as the code distance.) The double-brackets are used simply to denote that the code being referred to is a quantum error correction code rather than a classical code. Another way to understand quantum error correcting codes is offered by [John Preskill](http://theory.caltech.edu/~preskill/ph229/notes/chap7.pdf): “A quantum error-correcting code can be viewed as a mapping of k qubits (a Hilbert space of dimension 2k) into n qubits (a Hilbert space of dimension 2n), where n \> k. The k qubits are the “[logical qubits](https://www.quantinuum.com/glossary-item/logical-qubit)” or “encoded qubits” that we wish to protect from error. The additional n − k qubits allow us to store the k logical qubits in a redundant fashion, so that the encoded information is not easily damaged.”
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