{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "title": "Calderbank, Shor and Steane: quantum codes from nested classical codes", "summary": "Compatible nested binary linear codes produce a quantum code whose bit- and phase-error syndromes can be corrected without measuring the encoded information. The construction corrects arbitrary errors on a bounded number of physical qubits and gives asymptotically positive-rate families for a small fixed correctable error fraction.", "boundary": "This is a block-code construction and existence result, including the seven-qubit example, not a complete noisy-gate fault-tolerance threshold or an efficient explicit construction at every asymptotic bound. Correlated errors confined to the correctable support are allowed; transmission guarantees for errors on many qubits require extra noise assumptions.", "area": "Quantum information and coding", "topic": "Physics", "year": 1995, "lane": "HUMAN", "result": "THEOREM", "package_id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "humans": [ "A. R. Calderbank", "Peter W. Shor", "Andrew Steane" ], "authors_complete": true, "human_role_note": "The original named human researchers developed the exact mathematical construction or framework. Composite source families retain separate original paper bylines and predecessor roles; they are not a fictional joint publication. Individual effort shares, person identity joins and nationalities are not inferred. The source reading is not independent proof certification.", "agents": [], "related": [], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "cur20261020-arxiv-quant-ph-9512032v2", "title": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2", "date": "1996-04-16" }, { "id": "cur20261020-arxiv-quant-ph-9601029v3", "title": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3", "date": "1996-05-13" } ], "geography": { "country_codes": [ "US", "GB" ], "location_basis": "Calderbank and Shor AT&T Bell Labs, Murray Hill, US; independent Steane Clarendon Laboratory, Oxford, UK. Original source bylines, not nationality, sovereignty attribution or scientific ownership.", "source_url": "https://arxiv.org/abs/quant-ph/9512032v2", "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "basis": "RESEARCH_LOCATION_OR_PAPER_AFFILIATION", "reviewed_on": "2026-10-23", "nationality_inferred": false, "sources": [ { "title": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2" }, { "title": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3" } ], "coverage_status": "COMPLETE", "country_labels": { "US": "United States", "GB": "United Kingdom" } }, "points": "300000", "published_grade": { "id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "entry_id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "band": "S", "modifier": "-", "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "what_changed": "Compatible nested binary linear codes produce a quantum code whose bit- and phase-error syndromes can be corrected without measuring the encoded information. The construction corrects arbitrary errors on a bounded number of physical qubits and gives asymptotically positive-rate families for a small fixed correctable error fraction.", "scope": "Compatible nested binary linear codes produce a quantum code whose bit- and phase-error syndromes can be corrected without measuring the encoded information. The construction corrects arbitrary errors on a bounded number of physical qubits and gives asymptotically positive-rate families for a small fixed correctable error fraction. This is a block-code construction and existence result, including the seven-qubit example, not a complete noisy-gate fault-tolerance threshold or an efficient explicit construction at every asymptotic bound. Correlated errors confined to the correctable support are allowed; transmission guarantees for errors on many qubits require extra noise assumptions.", "status": "PROVISIONAL_EDITORIAL", "scientific_correctness_certified": false }, "assessment_disposition": { "status": "ASSESSED", "label": "Provisionally assessed", "reason": "A current grade and separate reference score identify this exact contribution.", "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "edition": "ratings-e039106f30dec177" } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-23T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [ { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "A. R. Calderbank", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Peter W. Shor", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Andrew Steane", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "This is a block-code construction and existence result, including the seven-qubit example, not a complete noisy-gate fault-tolerance threshold or an efficient explicit construction at every asymptotic bound. Correlated errors confined to the correctable support are allowed; transmission guarantees for errors on many qubits require extra noise assumptions.", "domain": "Quantum information and coding" }, "sources": [ { "citation": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2" }, { "citation": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3" } ], "statement": "# Calderbank, Shor and Steane: quantum codes from nested classical codes\n\n## How do you repair something you cannot copy?\n\nAn ordinary computer can protect a bit by keeping several copies and taking a majority vote. An unknown quantum state cannot simply be copied that way. Measuring the information directly may also change it. The challenge is therefore sharper than buying extra memory: spread one logical quantum state across several physical qubits, then learn enough about an error to repair it without learning the protected information itself. A qubit is a quantum system with two basic states.\n\n## Check relationships instead of reading the answer\n\nThe CSS construction uses compatible classical codes to organize two kinds of checks. One detects bit changes; the other detects phase changes, which alter the relative sign or orientation of a quantum superposition. Compatibility matters because arbitrary quantum measurements can disturb one another. The chosen checks can coexist. Their outcomes, called a syndrome, identify an allowed error pattern while leaving the encoded logical information hidden from the checking apparatus.\n\n## Seven physical qubits protect one logical qubit\n\nThe familiar seven-qubit example stores one logical qubit across seven physical qubits and corrects an arbitrary error on one of them, under the ideal correction model. Its coding rate is one divided by seven, not seven times as much useful information. That arithmetic is a classroom resource count, not a hardware benchmark. The cost buys protection. The larger construction also establishes families with positive rates for sufficiently small correctable error fractions.\n\n## A corrected qubit is not a complete reliable computer\n\nAn error acting on one qubit need not be a simple classical coin flip; the correction argument covers quantum errors supported on the allowed locations. But a real repair circuit contains gates that can fail and spread errors. A theorem about an ideal block code does not by itself settle that engineering problem. Calderbank and Shor developed one original route, while Steane developed an independent route. Their combined historical label does not turn two papers into one joint byline.\n\n## A useful first source check\n\nAsk your agent to make a two-column table: assumptions about errors, and guarantees about recovery. Include the number of protected logical qubits, the physical block size and whether correction operations are assumed ideal. Stop before extending the claim to a hardware threshold. Return the exact source versions and page references. Even an unfinished table can help a later reader distinguish a code theorem from an experiment or a complete fault-tolerant architecture.\n\n## Scientific scope\n\nThis is a block-code construction and existence result, including the seven-qubit example, not a complete noisy-gate fault-tolerance threshold or an efficient explicit construction at every asymptotic bound. Correlated errors confined to the correctable support are allowed; transmission guarantees for errors on many qubits require extra noise assumptions.\n\n## Sources for this explanation\n\n- [Good quantum error-correcting codes exist](https://arxiv.org/abs/quant-ph/9512032v2): Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained.\n- [Multiple Particle Interference and Quantum Error Correction](https://arxiv.org/abs/quant-ph/9601029v3): Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained.\n", "summary": "Compatible nested binary linear codes produce a quantum code whose bit- and phase-error syndromes can be corrected without measuring the encoded information. The construction corrects arbitrary errors on a bounded number of physical qubits and gives asymptotically positive-rate families for a small fixed correctable error fraction.", "title": "Calderbank, Shor and Steane: quantum codes from nested classical codes" }, "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "source_ref": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:7aad2d5ca0fd7e6c1f38ce7c5a4c9510298e5171711ede76b1d98cef8d19294d" }, "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "entry_id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "revision": 1, "title": "Calderbank, Shor and Steane: quantum codes from nested classical codes", "context": "Quantum information and coding", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": "-", "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Compatible nested binary linear codes produce a quantum code whose bit- and phase-error syndromes can be corrected without measuring the encoded information. The construction corrects arbitrary errors on a bounded number of physical qubits and gives asymptotically positive-rate families for a small fixed correctable error fraction.", "scope": "Compatible nested binary linear codes produce a quantum code whose bit- and phase-error syndromes can be corrected without measuring the encoded information. The construction corrects arbitrary errors on a bounded number of physical qubits and gives asymptotically positive-rate families for a small fixed correctable error fraction. This is a block-code construction and existence result, including the seven-qubit example, not a complete noisy-gate fault-tolerance threshold or an efficient explicit construction at every asymptotic bound. Correlated errors confined to the correctable support are allowed; transmission guarantees for errors on many qubits require extra noise assumptions.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This is a block-code construction and existence result, including the seven-qubit example, not a complete noisy-gate fault-tolerance threshold or an efficient explicit construction at every asymptotic bound. Correlated errors confined to the correctable support are allowed; transmission guarantees for errors on many qubits require extra noise assumptions.", "references": [ { "title": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." }, { "title": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "The original named human researchers developed the exact mathematical construction or framework. Composite source families retain separate original paper bylines and predecessor roles; they are not a fictional joint publication. Individual effort shares, person identity joins and nationalities are not inferred. The source reading is not independent proof certification." } ], "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "300000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "calderbank-shor-steane-quantum-error-correcting-codes-1995", "target": { "record_id": "urn:uuid:6ad17e07-fe69-4d97-a06c-7194ac9c7680", "version": 1, "content_hash": "sha256:e55bed76f34f9d64091e5f4a9dc5c9b2807e5b84ff0436cd3206fa17f44cddfd" }, "sections": [ { "title": "How do you repair something you cannot copy?", "text": "An ordinary computer can protect a bit by keeping several copies and taking a majority vote. An unknown quantum state cannot simply be copied that way. Measuring the information directly may also change it. The challenge is therefore sharper than buying extra memory: spread one logical quantum state across several physical qubits, then learn enough about an error to repair it without learning the protected information itself. A qubit is a quantum system with two basic states." }, { "title": "Check relationships instead of reading the answer", "text": "The CSS construction uses compatible classical codes to organize two kinds of checks. One detects bit changes; the other detects phase changes, which alter the relative sign or orientation of a quantum superposition. Compatibility matters because arbitrary quantum measurements can disturb one another. The chosen checks can coexist. Their outcomes, called a syndrome, identify an allowed error pattern while leaving the encoded logical information hidden from the checking apparatus." }, { "title": "Seven physical qubits protect one logical qubit", "text": "The familiar seven-qubit example stores one logical qubit across seven physical qubits and corrects an arbitrary error on one of them, under the ideal correction model. Its coding rate is one divided by seven, not seven times as much useful information. That arithmetic is a classroom resource count, not a hardware benchmark. The cost buys protection. The larger construction also establishes families with positive rates for sufficiently small correctable error fractions." }, { "title": "A corrected qubit is not a complete reliable computer", "text": "An error acting on one qubit need not be a simple classical coin flip; the correction argument covers quantum errors supported on the allowed locations. But a real repair circuit contains gates that can fail and spread errors. A theorem about an ideal block code does not by itself settle that engineering problem. Calderbank and Shor developed one original route, while Steane developed an independent route. Their combined historical label does not turn two papers into one joint byline." }, { "title": "A useful first source check", "text": "Ask your agent to make a two-column table: assumptions about errors, and guarantees about recovery. Include the number of protected logical qubits, the physical block size and whether correction operations are assumed ideal. Stop before extending the claim to a hardware threshold. Return the exact source versions and page references. Even an unfinished table can help a later reader distinguish a code theorem from an experiment or a complete fault-tolerant architecture." } ], "references": [ { "title": "Good quantum error-correcting codes exist", "url": "https://arxiv.org/abs/quant-ph/9512032v2", "locator": "Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained." }, { "title": "Multiple Particle Interference and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9601029v3", "locator": "Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained." } ] }, "reading_status": "EXPLANATORY_GUIDE" }