{
  "_generated": "2026-04-09T15:13:48.411250+00:00",
  "_description": "Site-wide index of all glossary terms across all papers in the series.",
  "_source": "Generated by scripts/generate_index.py",
  "total": 56,
  "cross_paper_terms": {
    "global phase": [
      "qqc-001-foundations",
      "qqc-002-canonical-ir",
      "qqc-003-gate-fusion"
    ],
    "unit quaternion": [
      "qqc-001-foundations",
      "qqc-002-canonical-ir"
    ],
    "canonical representative": [
      "qqc-002-canonical-ir",
      "qqc-003-gate-fusion",
      "qqc-004-verification-equivalence"
    ],
    "gate fusion": [
      "qqc-002-canonical-ir",
      "qqc-003-gate-fusion"
    ],
    "single-qubit segment": [
      "qqc-002-canonical-ir",
      "qqc-003-gate-fusion"
    ],
    "u1q": [
      "qqc-002-canonical-ir",
      "qqc-003-gate-fusion",
      "qqc-004-verification-equivalence"
    ]
  },
  "glossary": [
    {
      "term": "axis-angle representation",
      "definition": "A parametrization of a rotation by an axis n-hat in S^2 and an angle theta. In this paper it appears in both SU(2) form and unit-quaternion form.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "unit quaternion",
        "SU(2)",
        "Bloch sphere"
      ],
      "see_also": [
        "Section 4.1"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "Bloch sphere",
      "definition": "The unit 2-sphere S^2 representing pure states of a single qubit after quotienting normalized state vectors by global phase.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "pure state",
        "global phase",
        "CP^1"
      ],
      "see_also": [
        "prop-2"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "double cover",
      "definition": "A two-to-one surjective map between spaces or groups. In this paper SU(2) and Sp(1) double-cover SO(3).",
      "aliases": [
        "two-to-one cover"
      ],
      "paper_specific": false,
      "related_terms": [
        "SU(2)",
        "SO(3)",
        "spinor"
      ],
      "see_also": [
        "prop-7"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "global phase",
      "definition": "A common complex phase factor e^{i chi} multiplying a state vector. For pure states this phase does not change the physical state.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "pure state",
        "CP^1",
        "U(2)"
      ],
      "see_also": [
        "Section 2.2",
        "Section 2.3"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "normalized spinor",
      "definition": "A unit-norm vector in C^2 used as a representative of a single-qubit pure state before quotienting global phase.",
      "aliases": [
        "normalized state vector"
      ],
      "paper_specific": false,
      "related_terms": [
        "spinor",
        "S^3",
        "pure state"
      ],
      "see_also": [
        "prop-1"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "pure quaternion",
      "definition": "A quaternion with zero real part, identified naturally with a vector in R^3.",
      "aliases": [
        "imaginary quaternion"
      ],
      "paper_specific": false,
      "related_terms": [
        "quaternion",
        "SO(3)",
        "double cover"
      ],
      "see_also": [
        "Section 4.3"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "pure state",
      "definition": "A one-dimensional ray in Hilbert space. For a single qubit, pure states form CP^1 and are represented geometrically by the Bloch sphere S^2.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "global phase",
        "Bloch sphere",
        "qubit"
      ],
      "see_also": [
        "prop-2"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "quaternion",
      "definition": "An element of Hamilton's quaternion algebra H of the form a + b i + c j + d k with real coefficients.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "unit quaternion",
        "pure quaternion",
        "SU(2)"
      ],
      "see_also": [
        "Section 2.4"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "qubit",
      "definition": "A two-level quantum system whose state vectors live in C^2 and whose pure states are rays in that space.",
      "aliases": [
        "quantum bit"
      ],
      "paper_specific": false,
      "related_terms": [
        "pure state",
        "Bloch sphere",
        "spinor"
      ],
      "see_also": [
        "Section 2.1"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "RQM framing",
      "definition": "The project-specific choice in this series to use the quaternion model of SU(2) as a primary expository language for single-qubit gate geometry while preserving standard mathematical distinctions.",
      "aliases": [
        "RQM Technologies framing"
      ],
      "paper_specific": true,
      "related_terms": [
        "SU(2)",
        "unit quaternion",
        "single-qubit gate"
      ],
      "see_also": [
        "int-9"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "single-qubit gate",
      "definition": "A unitary operator acting on C^2. Such gates live in U(2), and after removal of overall phase may be represented by elements of SU(2).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "U(2)",
        "SU(2)",
        "unit quaternion"
      ],
      "see_also": [
        "prop-3"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "spinor",
      "definition": "A vector transforming under the spinorial representation of rotations, naturally associated with SU(2) rather than directly with SO(3).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "SU(2)",
        "double cover",
        "normalized spinor"
      ],
      "see_also": [
        "Section 2.1",
        "prop-7"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "SU(2)",
      "definition": "The group of 2 x 2 complex unitary matrices with determinant 1. It provides representatives for single-qubit gates modulo global phase and is isomorphic to the unit quaternions.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "U(2)",
        "SO(3)",
        "unit quaternion"
      ],
      "see_also": [
        "prop-3",
        "prop-5"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "unit quaternion",
      "definition": "A quaternion of norm 1. Unit quaternions form the group Sp(1), which is isomorphic to SU(2).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "quaternion",
        "Sp(1)",
        "axis-angle representation"
      ],
      "see_also": [
        "prop-5"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "term": "canonical representative",
      "definition": "The unique stored unit quaternion chosen from the sign class {q, -q} by normalization together with the hemisphere and equatorial tie-break rules.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "sign canonicalization",
        "shortest-geodesic representative",
        "u1q"
      ],
      "see_also": [
        "prop-8",
        "Section 4.2"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "canonical hemisphere",
      "definition": "The subset of the unit three-sphere selected by w > 0, together with an explicit tie-break on the equator w = 0, used to choose deterministic quaternion representatives.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "shortest-geodesic representative",
        "unit quaternion"
      ],
      "see_also": [
        "Section 4.2"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "compiler IR",
      "definition": "An internal representation used inside a compiler or rewrite pipeline, distinct from frontend syntax and designed to support deterministic transformation and comparison.",
      "aliases": [
        "intermediate representation"
      ],
      "paper_specific": false,
      "related_terms": [
        "named gate syntax",
        "single-qubit segment",
        "gate fusion"
      ],
      "see_also": [
        "Section 1"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "entangling boundary",
      "definition": "A circuit location at which a wire participates in a multi-qubit operation, thereby terminating a single-qubit segment for the purposes of the present IR.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "single-qubit segment",
        "entangling gate",
        "gate fusion"
      ],
      "see_also": [
        "def-12",
        "Section 6"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "gate fusion",
      "definition": "The replacement of a consecutive run of gates by a single equivalent internal representation of the same action on that wire.",
      "aliases": [
        "segment fusion"
      ],
      "paper_specific": false,
      "related_terms": [
        "single-qubit segment",
        "compiler IR",
        "quaternionic gate representation"
      ],
      "see_also": [
        "prop-11"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "global phase",
      "definition": "An overall complex phase factor multiplying a single-qubit gate or state representative. For the present paper, the IR factors out global phase at the gate-action level.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "SU(2) representative",
        "single-qubit gate",
        "sign canonicalization"
      ],
      "see_also": [
        "def-1",
        "prop-2"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "named gate syntax",
      "definition": "Frontend circuit notation expressed in gate names and parameters, such as H, T, or Rz(theta), before translation into a canonical internal representation.",
      "aliases": [
        "frontend gate list"
      ],
      "paper_specific": false,
      "related_terms": [
        "compiler IR",
        "gate fusion",
        "quaternionic gate representation"
      ],
      "see_also": [
        "Section 1",
        "Section 5"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "normalization map",
      "definition": "The map that rescales a nonzero quaternionic quadruple to unit norm so that numerical drift does not move the representation off the unit sphere.",
      "aliases": [
        "Norm"
      ],
      "paper_specific": true,
      "related_terms": [
        "unit quaternion",
        "sign canonicalization",
        "canonical representative"
      ],
      "see_also": [
        "def-6"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "quaternionic gate representation",
      "definition": "Representation of a single-qubit gate action modulo global phase by a unit quaternion corresponding to an SU(2) representative under the fixed map Phi.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "unit quaternion",
        "SU(2) representative",
        "u1q"
      ],
      "see_also": [
        "prop-3",
        "def-5"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "shortest-geodesic representative",
      "definition": "The canonical sign choice for which the quaternion real part satisfies w >= 0, placing the associated axis-angle rotation on the closed interval [0, pi].",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "canonical hemisphere",
        "sign canonicalization",
        "unit quaternion"
      ],
      "see_also": [
        "rem-9"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "sign canonicalization",
      "definition": "The deterministic rule that chooses one representative from the pair {q, -q}, using w > 0 and an explicit equatorial tie-break when w = 0.",
      "aliases": [
        "SignCan"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "shortest-geodesic representative",
        "global phase"
      ],
      "see_also": [
        "def-7"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "single-qubit segment",
      "definition": "A maximal consecutive run of operations acting only on one wire between circuit endpoints or boundaries such as entangling gates that leave the single-qubit unitary model.",
      "aliases": [
        "segment"
      ],
      "paper_specific": true,
      "related_terms": [
        "entangling boundary",
        "gate fusion",
        "u1q"
      ],
      "see_also": [
        "def-12"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "SU(2) representative",
      "definition": "A determinant-one unitary matrix chosen to represent a single-qubit gate action after removal of overall phase from U(2).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "global phase",
        "quaternionic gate representation",
        "unit quaternion"
      ],
      "see_also": [
        "prop-2"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "u1q",
      "definition": "The project-specific name used in this paper for the canonical quaternionic IR object storing a single-qubit gate action modulo global phase as four real coordinates.",
      "aliases": [
        "u1q payload"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "compiler IR",
        "single-qubit segment"
      ],
      "see_also": [
        "def-5",
        "int-13"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "unit quaternion",
      "definition": "A quaternion of norm 1. Under the fixed Phi convention, unit quaternions model SU(2) exactly and therefore represent single-qubit gate action modulo global phase.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "SU(2) representative",
        "quaternionic gate representation",
        "shortest-geodesic representative"
      ],
      "see_also": [
        "prop-3"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "term": "axis-aware reconstruction",
      "definition": "A layered output rule that recovers identity, recognized named gates, exact axis-aligned rotations, or a generic one-qubit primitive from a fused canonical quaternion.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "gate fusion",
        "canonical representative",
        "single-qubit segment"
      ],
      "see_also": [
        "def-3",
        "Section 5"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "baseline flow",
      "definition": "The reference comparison flow used in the paper. Here it is the syntax-preserving baseline that leaves each single-qubit segment unchanged and performs no fusion.",
      "aliases": [
        "syntax-preserving baseline"
      ],
      "paper_specific": true,
      "related_terms": [
        "benchmark methodology",
        "gate count reduction",
        "depth reduction"
      ],
      "see_also": [
        "Section 4",
        "Section 7"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "canonical representative",
      "definition": "The unique stored unit quaternion selected from the sign class {q, -q} by normalization together with the hemisphere and equatorial tie-break rules inherited from qqc-002-canonical-ir.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "sign-canonicalization",
        "shortest geodesic",
        "u1q"
      ],
      "see_also": [
        "qqc-002-canonical-ir:prop-8",
        "Section 2"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "depth reduction",
      "definition": "Decrease in segment-local sequential depth between the syntax-preserving baseline and the reconstructed output of the local fusion pass.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "gate count reduction",
        "baseline flow",
        "single-qubit segment"
      ],
      "see_also": [
        "Section 7",
        "emp-9"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "equivalence preservation",
      "definition": "Property that the optimized output represents exactly the same single-qubit action as the original segment modulo global phase under the authoritative canonical-quaternion equivalence check.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "canonical representative",
        "global phase",
        "gate fusion"
      ],
      "see_also": [
        "prop-4"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "gate count reduction",
      "definition": "Decrease in the number of emitted single-qubit gates after applying the local fusion pass and reconstruction layer relative to the baseline flow.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "depth reduction",
        "baseline flow",
        "gate fusion"
      ],
      "see_also": [
        "Section 7",
        "emp-9"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "gate fusion",
      "definition": "Replacement of a consecutive single-qubit segment by one fused canonical quaternionic representative, optionally followed by reconstruction into an interpretable one-gate or zero-gate output.",
      "aliases": [
        "segment fusion"
      ],
      "paper_specific": false,
      "related_terms": [
        "single-qubit segment",
        "canonical representative",
        "local optimization pass"
      ],
      "see_also": [
        "def-2",
        "cor-6"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "geometry-native optimization pass",
      "definition": "A local optimization pass whose core rewrite rule is the native group law of the chosen representation rather than an external symbolic heuristic over frontend syntax.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "gate fusion",
        "quaternionic representation",
        "local optimization pass"
      ],
      "see_also": [
        "int-10"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "global phase",
      "definition": "An overall complex phase factor that does not change the physical single-qubit action on pure states. The optimization pass preserves segment action only up to this phase, exactly as intended by the inherited SU(2) representative model.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "SU(2) representative",
        "equivalence preservation",
        "sign-canonicalization"
      ],
      "see_also": [
        "Section 2",
        "prop-4"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "local optimization pass",
      "definition": "A compiler transformation restricted to a bounded region of the circuit rather than the full global structure. In this paper the region is a single-qubit segment.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "single-qubit segment",
        "gate fusion",
        "baseline flow"
      ],
      "see_also": [
        "Section 1",
        "Section 3"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "quaternionic gate fusion",
      "definition": "The specific local fusion method studied in this paper: translate a segment into canonical quaternions, multiply in circuit order, canonicalize the product, and reconstruct an output gate form when possible.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "gate fusion",
        "u1q",
        "axis-aware reconstruction"
      ],
      "see_also": [
        "def-2",
        "emp-9"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "shortest geodesic",
      "definition": "The sign choice convention on the unit three-sphere that prefers representatives with nonnegative real part, thereby selecting the shorter closed axis-angle interval when possible.",
      "aliases": [
        "shortest-geodesic representative"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "sign-canonicalization",
        "S^3"
      ],
      "see_also": [
        "qqc-002-canonical-ir:rem-9"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "sign-canonicalization",
      "definition": "The deterministic rule that chooses one representative from the pair {q, -q} using the primary test w > 0 and an explicit tie-break on the equator w = 0.",
      "aliases": [
        "SignCan"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "shortest geodesic",
        "equivalence preservation"
      ],
      "see_also": [
        "qqc-002-canonical-ir:def-7"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "single-qubit segment",
      "definition": "A maximal consecutive run of one-qubit gates on one wire between circuit endpoints or boundaries such as entangling gates, measurement, reset, or classical control.",
      "aliases": [
        "segment"
      ],
      "paper_specific": true,
      "related_terms": [
        "local optimization pass",
        "gate fusion",
        "baseline flow"
      ],
      "see_also": [
        "def-1"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "u1q",
      "definition": "Canonical quaternionic IR object inherited from qqc-002-canonical-ir and used here as the storage format for fused single-qubit segment action modulo global phase.",
      "aliases": [
        "u1q payload"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "quaternionic gate fusion",
        "single-qubit segment"
      ],
      "see_also": [
        "qqc-002-canonical-ir:def-5"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "term": "canonical equivalence",
      "definition": "Equivalence of two single-qubit segments under the authoritative criterion that their normalized and sign-canonicalized quaternion representatives are exactly equal.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "verification procedure",
        "global-phase-aware equivalence"
      ],
      "see_also": [
        "def-2",
        "prop-5"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "canonical representative",
      "definition": "The unique unit quaternion selected from the sign class {q, -q} by normalization together with the fixed hemisphere and equatorial tie-break rules inherited from qqc-002-canonical-ir.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "canonical sign convention",
        "shortest-geodesic representative",
        "u1q"
      ],
      "see_also": [
        "qqc-002-canonical-ir:prop-8"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "canonical sign convention",
      "definition": "The deterministic rule that chooses the representative with w > 0, and when w = 0 chooses the sign for which the first nonzero component among (x, y, z) is positive.",
      "aliases": [
        "sign-canonicalization"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "equatorial tie-break",
        "shortest-geodesic representative"
      ],
      "see_also": [
        "def-1"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "equatorial tie-break",
      "definition": "The secondary sign rule used when the quaternion real part w equals zero, resolving the sign by requiring the first nonzero coordinate among x, y, z to be positive.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "canonical sign convention",
        "shortest-geodesic representative",
        "verification invariant"
      ],
      "see_also": [
        "def-1"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "global-phase-aware equivalence",
      "definition": "Equivalence of single-qubit gate actions after quotienting the overall complex phase in U(2), so that only the represented SU(2) action matters.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "canonical equivalence",
        "SU(2) representative",
        "phase-aware residual"
      ],
      "see_also": [
        "Section 2",
        "def-2"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "interoperability diagnostic",
      "definition": "A matrix-based quantity, such as phase-aware residual or process fidelity, used to compare quaternionic verification outputs with standard matrix-oriented toolchains.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "phase-aware residual",
        "process fidelity",
        "trace overlap"
      ],
      "see_also": [
        "def-5",
        "Section 6"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "phase-aware residual",
      "definition": "The maximum norm difference between U†V and the closest scalar phase times the identity, used here as an interoperability-friendly diagnostic for matrix equivalence up to global phase.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "process fidelity",
        "trace overlap",
        "global-phase-aware equivalence"
      ],
      "see_also": [
        "def-5"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "process fidelity",
      "definition": "The normalized squared trace overlap |Tr(U†V)|^2 / 4 used here as a matrix-based diagnostic for how closely two single-qubit unitaries agree up to global phase.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "phase-aware residual",
        "trace overlap",
        "interoperability diagnostic"
      ],
      "see_also": [
        "def-5"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "trace overlap",
      "definition": "The absolute trace of U†V for two single-qubit representative matrices U and V, used here as an interpretable matrix-based agreement indicator.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": [
        "process fidelity",
        "phase-aware residual",
        "interoperability diagnostic"
      ],
      "see_also": [
        "def-5"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "u1q",
      "definition": "The canonical single-qubit quaternionic intermediate representation defined in qqc-002-canonical-ir, storing a deterministic unit-quaternion payload.",
      "aliases": [
        "u1q payload"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical representative",
        "verification invariant",
        "single-qubit segment"
      ],
      "see_also": [
        "qqc-002-canonical-ir:def-5"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "validation procedure",
      "definition": "A concrete sequence of checks applied to an optimized segment, consisting here of canonical quaternion comparison as the authoritative test and matrix-based diagnostics as interoperable secondary evidence.",
      "aliases": [
        "verification workflow"
      ],
      "paper_specific": true,
      "related_terms": [
        "canonical equivalence",
        "interoperability diagnostic",
        "verification invariant"
      ],
      "see_also": [
        "Section 5",
        "Section 6"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "term": "verification invariant",
      "definition": "A property that must be preserved by a correct quaternionic compilation pass, such as unit norm after normalization, sign-canonical form, and equality of canonical fused representatives for equivalent segments.",
      "aliases": [],
      "paper_specific": true,
      "related_terms": [
        "canonical sign convention",
        "u1q",
        "validation procedure"
      ],
      "see_also": [
        "prop-6",
        "Section 4"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    }
  ]
}