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Particular nondegenerate single-error Hamming bounds are not universal statements about every degenerate code, and no complete noisy-hardware threshold is claimed.", "area": "Quantum error correction and finite symplectic geometry", "topic": "Physics", "year": 1996, "lane": "HUMAN", "result": "THEOREM", "package_id": "gottesman-calderbank-rains-shor-sloane-stabilizer-codes-1996", "target": { "record_id": "urn:uuid:b476df22-bd92-4493-a9ce-48afc8286ff2", "version": 1, "content_hash": "sha256:fb7e652350be8b7133d087f026b9072f02c5973ee944a248d4925f0dab096a29" }, "humans": [ "Daniel Gottesman", "A. R. Calderbank", "E. M. Rains", "P. W. Shor", "N. J. A. Sloane" ], "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-gottesman-quant-ph-9604038v2", "title": "A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound", "url": "https://arxiv.org/abs/quant-ph/9604038v2", "date": "1996-07-24" }, { "id": "cur20261020-crss-quant-ph-9605005v3", "title": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3", "date": "1996-07-02" }, { "id": "cur20261020-gottesman-thesis-9705052v1", "title": "Stabilizer Codes and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9705052v1", "date": "1997-05-28" } ], "geography": { "country_codes": [ "US" ], "location_basis": "Gottesman Caltech Pasadena US; separate Calderbank, Shor, Sloane AT&T Murray Hill and Rains Institute for Defense Analyses Princeton US. 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Their commutation and coset structure supplies error syndromes and an algebraic correction criterion, unifying earlier CSS constructions and enabling additional quantum codes. Gottesman and the separate Calderbank-Rains-Shor-Sloane paper develop complementary formulations.", "scope": "Commuting Pauli operators with a consistent joint eigenspace define quantum coding spaces. Their commutation and coset structure supplies error syndromes and an algebraic correction criterion, unifying earlier CSS constructions and enabling additional quantum codes. Gottesman and the separate Calderbank-Rains-Shor-Sloane paper develop complementary formulations. The stabilizer excludes minus the identity; arbitrary commuting operators need not define a nonzero code. Not every quantum code is a stabilizer code. Degenerate errors can act identically on encoded information and need not be distinguished. 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Shor", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "N. J. A. Sloane", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound", "url": "https://arxiv.org/abs/quant-ph/9604038v2" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-2", "kind": "source_reference", "source": { "citation": "Stabilizer Codes and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9705052v1" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The stabilizer excludes minus the identity; arbitrary commuting operators need not define a nonzero code. Not every quantum code is a stabilizer code. Degenerate errors can act identically on encoded information and need not be distinguished. Particular nondegenerate single-error Hamming bounds are not universal statements about every degenerate code, and no complete noisy-hardware threshold is claimed.", "domain": "Quantum error correction and finite symplectic geometry" }, "sources": [ { "citation": "A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound", "url": "https://arxiv.org/abs/quant-ph/9604038v2" }, { "citation": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3" }, { "citation": "Stabilizer Codes and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9705052v1" } ], "statement": "# Gottesman and Calderbank et al.: the stabilizer framework for quantum codes\n\n## Describe a huge code with a short set of checks\n\nA quantum state involving many qubits can require an enormous list of amplitudes. Yet some useful protected subspaces have a much shorter description: specify a set of compatible checks that every valid code state satisfies. This is the stabilizer viewpoint developed in complementary papers by Gottesman and by Calderbank, Rains, Shor and Sloane. It replaces repeated manipulation of full state vectors with algebraic information about operators and their relationships.\n\n## A stabilizer leaves the encoded information unchanged\n\nAn operator stabilizes a code state when applying it leaves that state unchanged. Suitable Pauli operators either commute or anticommute. Commuting checks can have a joint eigenspace; an error that anticommutes with a check changes its recorded sign. The pattern of signs is an error syndrome. The checks must also be consistent: including minus the identity would demand that a nonzero state equal its own negative, leaving no valid coding space.\n\n## A small example explains both the power and the limit\n\nFor three qubits, the checks ZZI and IZZ select the span of 000 and 111, using familiar binary labels for basis states. Two independent checks reduce the eight-dimensional state space to a two-dimensional code space, enough for one logical qubit. This repetition-style classroom example can reveal single bit flips. It does not correct every arbitrary single-qubit error: phase errors illustrate why a full quantum code needs a stronger condition.\n\n## Different physical errors can have the same effect\n\nIn a degenerate code, distinct physical errors may act identically on the protected information. A decoder need not distinguish them merely because their operator descriptions differ. The stabilizer and coset formulation makes this distinction systematic. It also includes earlier CSS codes while supporting further constructions. That generalization is the contribution being described; the earlier examples are not newly discovered a second time, and not every possible quantum code belongs to this family.\n\n## Make the algebra inspectable\n\nAsk your agent to list the two classroom checks, their commutation relation and the eight basis labels. It can determine which labels have both required signs and then test one bit flip and one phase change. Preserve the exact convention for Pauli phases. Mark this as a finite teaching check rather than a proof for all stabilizer codes. A later participant should be able to reproduce the table without guessing the operator convention.\n\n## Scientific scope\n\nThe stabilizer excludes minus the identity; arbitrary commuting operators need not define a nonzero code. Not every quantum code is a stabilizer code. Degenerate errors can act identically on encoded information and need not be distinguished. Particular nondegenerate single-error Hamming bounds are not universal statements about every degenerate code, and no complete noisy-hardware threshold is claimed.\n\n## Sources for this explanation\n\n- [A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound](https://arxiv.org/abs/quant-ph/9604038v2): Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained.\n- [Quantum Error Correction and Orthogonal Geometry](https://arxiv.org/abs/quant-ph/9605005v3): Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained.\n- [Stabilizer Codes and Quantum Error Correction](https://arxiv.org/abs/quant-ph/9705052v1): Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained.\n", "summary": "Commuting Pauli operators with a consistent joint eigenspace define quantum coding spaces. Their commutation and coset structure supplies error syndromes and an algebraic correction criterion, unifying earlier CSS constructions and enabling additional quantum codes. Gottesman and the separate Calderbank-Rains-Shor-Sloane paper develop complementary formulations.", "title": "Gottesman and Calderbank et al.: the stabilizer framework for quantum codes" }, "record_id": "urn:uuid:b476df22-bd92-4493-a9ce-48afc8286ff2", "version": 1, "source_ref": { "record_id": "urn:uuid:b476df22-bd92-4493-a9ce-48afc8286ff2", "version": 1, "content_hash": "sha256:fb7e652350be8b7133d087f026b9072f02c5973ee944a248d4925f0dab096a29" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:e222d1defc906ede9057707aa2815581940d16d78730d34ca7ca599042f3ba65" }, "target": { "record_id": "urn:uuid:b476df22-bd92-4493-a9ce-48afc8286ff2", "version": 1, "content_hash": "sha256:fb7e652350be8b7133d087f026b9072f02c5973ee944a248d4925f0dab096a29" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "gottesman-calderbank-rains-shor-sloane-stabilizer-codes-1996", "entry_id": "gottesman-calderbank-rains-shor-sloane-stabilizer-codes-1996", "revision": 1, "title": "Gottesman and Calderbank et al.: the stabilizer framework for quantum codes", "context": "Quantum error correction and finite symplectic geometry", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Commuting Pauli operators with a consistent joint eigenspace define quantum coding spaces. Their commutation and coset structure supplies error syndromes and an algebraic correction criterion, unifying earlier CSS constructions and enabling additional quantum codes. Gottesman and the separate Calderbank-Rains-Shor-Sloane paper develop complementary formulations.", "scope": "Commuting Pauli operators with a consistent joint eigenspace define quantum coding spaces. Their commutation and coset structure supplies error syndromes and an algebraic correction criterion, unifying earlier CSS constructions and enabling additional quantum codes. Gottesman and the separate Calderbank-Rains-Shor-Sloane paper develop complementary formulations. The stabilizer excludes minus the identity; arbitrary commuting operators need not define a nonzero code. Not every quantum code is a stabilizer code. Degenerate errors can act identically on encoded information and need not be distinguished. Particular nondegenerate single-error Hamming bounds are not universal statements about every degenerate code, and no complete noisy-hardware threshold is claimed.", "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. The stabilizer excludes minus the identity; arbitrary commuting operators need not define a nonzero code. Not every quantum code is a stabilizer code. Degenerate errors can act identically on encoded information and need not be distinguished. Particular nondegenerate single-error Hamming bounds are not universal statements about every degenerate code, and no complete noisy-hardware threshold is claimed.", "references": [ { "title": "A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound", "url": "https://arxiv.org/abs/quant-ph/9604038v2", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." }, { "title": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." }, { "title": "Stabilizer Codes and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9705052v1", "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:b476df22-bd92-4493-a9ce-48afc8286ff2", "version": 1, "content_hash": "sha256:fb7e652350be8b7133d087f026b9072f02c5973ee944a248d4925f0dab096a29" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "350000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "gottesman-calderbank-rains-shor-sloane-stabilizer-codes-1996", "target": { "record_id": "urn:uuid:b476df22-bd92-4493-a9ce-48afc8286ff2", "version": 1, "content_hash": "sha256:fb7e652350be8b7133d087f026b9072f02c5973ee944a248d4925f0dab096a29" }, "sections": [ { "title": "Describe a huge code with a short set of checks", "text": "A quantum state involving many qubits can require an enormous list of amplitudes. Yet some useful protected subspaces have a much shorter description: specify a set of compatible checks that every valid code state satisfies. This is the stabilizer viewpoint developed in complementary papers by Gottesman and by Calderbank, Rains, Shor and Sloane. It replaces repeated manipulation of full state vectors with algebraic information about operators and their relationships." }, { "title": "A stabilizer leaves the encoded information unchanged", "text": "An operator stabilizes a code state when applying it leaves that state unchanged. Suitable Pauli operators either commute or anticommute. Commuting checks can have a joint eigenspace; an error that anticommutes with a check changes its recorded sign. The pattern of signs is an error syndrome. The checks must also be consistent: including minus the identity would demand that a nonzero state equal its own negative, leaving no valid coding space." }, { "title": "A small example explains both the power and the limit", "text": "For three qubits, the checks ZZI and IZZ select the span of 000 and 111, using familiar binary labels for basis states. Two independent checks reduce the eight-dimensional state space to a two-dimensional code space, enough for one logical qubit. This repetition-style classroom example can reveal single bit flips. It does not correct every arbitrary single-qubit error: phase errors illustrate why a full quantum code needs a stronger condition." }, { "title": "Different physical errors can have the same effect", "text": "In a degenerate code, distinct physical errors may act identically on the protected information. A decoder need not distinguish them merely because their operator descriptions differ. The stabilizer and coset formulation makes this distinction systematic. It also includes earlier CSS codes while supporting further constructions. That generalization is the contribution being described; the earlier examples are not newly discovered a second time, and not every possible quantum code belongs to this family." }, { "title": "Make the algebra inspectable", "text": "Ask your agent to list the two classroom checks, their commutation relation and the eight basis labels. It can determine which labels have both required signs and then test one bit flip and one phase change. Preserve the exact convention for Pauli phases. Mark this as a finite teaching check rather than a proof for all stabilizer codes. A later participant should be able to reproduce the table without guessing the operator convention." } ], "references": [ { "title": "A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound", "url": "https://arxiv.org/abs/quant-ph/9604038v2", "locator": "Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained." }, { "title": "Quantum Error Correction and Orthogonal Geometry", "url": "https://arxiv.org/abs/quant-ph/9605005v3", "locator": "Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained." }, { "title": "Stabilizer Codes and Quantum Error Correction", "url": "https://arxiv.org/abs/quant-ph/9705052v1", "locator": "Original exact-version statement, construction and limitations; source-specific byline and contribution scope retained." } ] }, "reading_status": "EXPLANATORY_GUIDE" }