{
  "_generated": "2026-04-09T15:13:48.410205+00:00",
  "_description": "Site-wide index of all notation and symbols across all papers in the series.",
  "_source": "Generated by scripts/generate_index.py",
  "total": 80,
  "notation": [
    {
      "symbol": "|ψ⟩",
      "name": "normalized state vector",
      "latex": "\\ket{\\psi}",
      "meaning": "A normalized representative of a single-qubit pure state in C^2.",
      "domain": "quantum mechanics",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": [
        "α",
        "β",
        "CP^1"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "α, β",
      "name": "spinor components",
      "latex": "\\alpha, \\beta",
      "meaning": "Complex amplitudes of a single-qubit state vector |ψ⟩ = (α, β)^T.",
      "domain": "quantum mechanics",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": [
        "|ψ⟩",
        "S^3"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "C^2",
      "name": "two-dimensional complex vector space",
      "latex": "\\C^2",
      "meaning": "Ambient complex vector space for single-qubit state vectors before projectivization.",
      "domain": "linear algebra",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": [
        "|ψ⟩",
        "S^3",
        "R^4"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "S^3",
      "name": "three-sphere",
      "latex": "S^3",
      "meaning": "The unit sphere in R^4; in this paper it appears both as the normalized-spinor sphere and as the manifold underlying the unit quaternions.",
      "domain": "geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": [
        "C^2",
        "CP^1",
        "Sp(1)"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "CP^1",
      "name": "complex projective line",
      "latex": "\\CP^1",
      "meaning": "Projective pure-state space of a single qubit, obtained by quotienting normalized spinors by global phase.",
      "domain": "geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.2",
      "related_symbols": [
        "S^2",
        "|ψ⟩",
        "global phase"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "S^2",
      "name": "Bloch sphere",
      "latex": "S^2",
      "meaning": "The unit 2-sphere used to represent pure single-qubit states after quotienting global phase.",
      "domain": "quantum geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.2",
      "related_symbols": [
        "CP^1",
        "r(ψ)",
        "SO(3)"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "r(ψ)",
      "name": "Bloch vector",
      "latex": "\\mathbf{r}(\\psi)",
      "meaning": "The unit vector in S^2 associated with a normalized qubit state through the Bloch-sphere map.",
      "domain": "quantum geometry",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.2",
      "related_symbols": [
        "|ψ⟩",
        "S^2",
        "σ_x, σ_y, σ_z"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "U(2)",
      "name": "single-qubit unitary group",
      "latex": "\\mathrm{U}(2)",
      "meaning": "Group of 2 x 2 unitary matrices; this is the full group in which single-qubit gates live before quotienting global phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.3",
      "related_symbols": [
        "SU(2)",
        "U",
        "V"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "SU(2)",
      "name": "special unitary group",
      "latex": "\\mathrm{SU}(2)",
      "meaning": "Group of 2 x 2 unitary matrices with determinant 1; used as a representative model for single-qubit gates modulo global phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.3",
      "related_symbols": [
        "U(2)",
        "SO(3)",
        "Sp(1)"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "U, V",
      "name": "single-qubit gate and SU(2) representative",
      "latex": "U, V",
      "meaning": "U denotes a gate in U(2), while V denotes a determinant-1 representative in SU(2) after removing global phase.",
      "domain": "quantum gates",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.3",
      "related_symbols": [
        "U(2)",
        "SU(2)",
        "q"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "H",
      "name": "quaternion algebra",
      "latex": "\\mathbb{H}",
      "meaning": "Hamilton's quaternion algebra over R with basis 1, i, j, k.",
      "domain": "algebra",
      "type": "algebra",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": [
        "q",
        "i, j, k",
        "Sp(1)"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "q",
      "name": "quaternion or unit-quaternion gate representative",
      "latex": "q",
      "meaning": "A quaternion a + b i + c j + d k; when |q| = 1 it models an element of SU(2).",
      "domain": "algebra / quantum gates",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": [
        "H",
        "|q|",
        "Φ"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "i, j, k",
      "name": "quaternion basis units",
      "latex": "\\mathbf{i}, \\mathbf{j}, \\mathbf{k}",
      "meaning": "Imaginary quaternion units satisfying i^2 = j^2 = k^2 = ijk = -1.",
      "domain": "algebra",
      "type": "constant",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": [
        "H",
        "q",
        "σ_x, σ_y, σ_z"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "Sp(1)",
      "name": "unit-quaternion group",
      "latex": "\\mathrm{Sp}(1)",
      "meaning": "Group of unit quaternions; a Lie-group model of SU(2).",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": [
        "H",
        "SU(2)",
        "S^3"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "Φ",
      "name": "chosen quaternion-matrix identification",
      "latex": "\\Phi",
      "meaning": "The explicit map from H to 2 x 2 complex matrices used in the paper, chosen so that i, j, k align with the x, y, z Pauli axes.",
      "domain": "representation theory",
      "type": "map",
      "standard_or_project_specific": "variant",
      "first_use": "Section 3.1",
      "related_symbols": [
        "q",
        "SU(2)",
        "σ_x, σ_y, σ_z"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "σ_x, σ_y, σ_z",
      "name": "Pauli matrices",
      "latex": "\\sigma_x, \\sigma_y, \\sigma_z",
      "meaning": "Standard Pauli matrices used to express axis-angle rotations in SU(2).",
      "domain": "quantum mechanics",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3.1",
      "related_symbols": [
        "SU(2)",
        "Φ",
        "r(ψ)"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "n̂",
      "name": "unit rotation axis",
      "latex": "\\hat{\\mathbf{n}}",
      "meaning": "Unit vector in S^2 specifying a rotation axis in axis-angle form.",
      "domain": "geometry",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.1",
      "related_symbols": [
        "θ",
        "q",
        "S^2"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "θ",
      "name": "rotation angle",
      "latex": "\\theta",
      "meaning": "Rotation angle appearing in both the SU(2) and quaternionic axis-angle formulas.",
      "domain": "geometry",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.1",
      "related_symbols": [
        "n̂",
        "q",
        "SU(2)"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "ρ",
      "name": "double-cover map to SO(3)",
      "latex": "\\rho",
      "meaning": "Homomorphism from Sp(1) to SO(3) defined by conjugation on pure quaternions.",
      "domain": "Lie groups",
      "type": "map",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.3",
      "related_symbols": [
        "Sp(1)",
        "SO(3)",
        "q"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "SO(3)",
      "name": "rotation group in three dimensions",
      "latex": "\\mathrm{SO}(3)",
      "meaning": "Group of proper rotations of R^3; the physical Bloch-sphere action factors through this group.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.3",
      "related_symbols": [
        "SU(2)",
        "Sp(1)",
        "S^2"
      ],
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/"
    },
    {
      "symbol": "u1q(w,x,y,z)",
      "name": "canonical single-qubit IR element",
      "latex": "\\uoneq(w,x,y,z)",
      "meaning": "The compiler-facing quaternionic intermediate-representation object storing a canonical unit quaternion for a single-qubit gate action modulo global phase.",
      "domain": "compiler IR",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 1",
      "related_symbols": [
        "q",
        "(w,x,y,z)",
        "Can"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "(w,x,y,z)",
      "name": "quaternion coordinates",
      "latex": "(w,x,y,z)",
      "meaning": "The four real coordinates of the quaternion q = w + x i + y j + z k used as the IR payload.",
      "domain": "quaternion algebra",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 1",
      "related_symbols": [
        "q",
        "u1q(w,x,y,z)",
        "Norm"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "q",
      "name": "unit quaternion gate representative",
      "latex": "q",
      "meaning": "A quaternion w + x i + y j + z k; when unit norm it represents an SU(2) element and therefore a single-qubit gate action modulo global phase.",
      "domain": "quaternion algebra / quantum gates",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 1",
      "related_symbols": [
        "(w,x,y,z)",
        "Phi",
        "q ~ +/- q"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "U(2)",
      "name": "single-qubit unitary group",
      "latex": "\\mathrm{U}(2)",
      "meaning": "The full group in which single-qubit gates live before removal of overall phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "SU(2)",
        "U ~gp V"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "SU(2)",
      "name": "special unitary representative group",
      "latex": "\\mathrm{SU}(2)",
      "meaning": "The determinant-one unitary group obtained after removing global phase from single-qubit gates.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "U(2)",
        "Phi",
        "Sp(1)"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "Phi",
      "name": "fixed quaternion-SU(2) map",
      "latex": "\\Phi",
      "meaning": "The explicit map from quaternions to 2 x 2 complex matrices inherited from paper 1, with i, j, k aligned to the Pauli x, y, z generators.",
      "domain": "representation theory",
      "type": "map",
      "standard_or_project_specific": "variant",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "SU(2)",
        "sigma_x, sigma_y, sigma_z"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "Sp(1)",
      "name": "unit-quaternion group",
      "latex": "\\mathrm{Sp}(1)",
      "meaning": "The group of unit quaternions, used as the quaternionic model of SU(2).",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "SU(2)",
        "q",
        "S^3"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "U ~gp V",
      "name": "global-phase equivalence",
      "latex": "U \\sim_{\\mathrm{gp}} V",
      "meaning": "Equivalence relation on U(2) defined by multiplication by an overall complex phase.",
      "domain": "quantum gates",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3",
      "related_symbols": [
        "U(2)",
        "SU(2)",
        "q ~ +/- q"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "q ~ +/- q",
      "name": "sign equivalence",
      "latex": "q \\sim_{\\pm} q'",
      "meaning": "Equivalence relation on unit quaternions identifying q and -q as the same single-qubit gate action modulo global phase.",
      "domain": "quaternion algebra / compiler IR",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3",
      "related_symbols": [
        "q",
        "Can",
        "SignCan"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "Norm",
      "name": "normalization map",
      "latex": "\\normalize",
      "meaning": "Map that divides a nonzero quaternionic quadruple by its Euclidean norm to return to the unit sphere.",
      "domain": "numerical representation",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4.1",
      "related_symbols": [
        "(w,x,y,z)",
        "Can",
        "||q||"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "SignCan",
      "name": "sign-canonicalization map",
      "latex": "\\sgncanon",
      "meaning": "Map choosing one representative from {q, -q} by the rule w > 0, with a first-nonzero-positive tie-break on the equator w = 0.",
      "domain": "compiler IR",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4.2",
      "related_symbols": [
        "Can",
        "q ~ +/- q",
        "(w,x,y,z)"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "Can",
      "name": "canonical representative map",
      "latex": "\\canon",
      "meaning": "Composition of normalization and sign-canonicalization producing the deterministic stored IR representative.",
      "domain": "compiler IR",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4.2",
      "related_symbols": [
        "Norm",
        "SignCan",
        "u1q(w,x,y,z)"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "H_can",
      "name": "canonical hemisphere",
      "latex": "\\mathcal{H}_{\\mathrm{can}}",
      "meaning": "Closed hemisphere of S^3 selected by w > 0 together with the equatorial tie-break used for canonical representatives.",
      "domain": "geometry / compiler IR",
      "type": "set",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4.2",
      "related_symbols": [
        "Can",
        "q ~ +/- q",
        "S^3"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "Rx(theta), Ry(theta), Rz(theta)",
      "name": "axis-rotation families",
      "latex": "R_x(\\theta), R_y(\\theta), R_z(\\theta)",
      "meaning": "Standard single-qubit rotation gates whose quaternion representatives have direct axis-angle form.",
      "domain": "quantum gates",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 5",
      "related_symbols": [
        "theta",
        "q",
        "X, Y, Z"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "I, X, Y, Z, H, S, T",
      "name": "named single-qubit gates",
      "latex": "I, X, Y, Z, H, S, T",
      "meaning": "Standard gate names used as frontend syntax examples before translation into canonical quaternion tuples.",
      "domain": "quantum circuit syntax",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 5",
      "related_symbols": [
        "u1q(w,x,y,z)",
        "Rx(theta), Ry(theta), Rz(theta)"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "theta",
      "name": "rotation angle",
      "latex": "\\theta",
      "meaning": "Rotation parameter appearing in axis-rotation gates and in the axis-angle interpretation of unit quaternions.",
      "domain": "geometry",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.2",
      "related_symbols": [
        "Rx(theta), Ry(theta), Rz(theta)",
        "q"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "segment(a)",
      "name": "single-qubit segment on wire a",
      "latex": "\\mathrm{segment}(a)",
      "meaning": "A maximal consecutive run of single-qubit gates on wire a between entangling or other out-of-scope boundaries.",
      "domain": "compiler IR",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "u1q(w,x,y,z)",
        "Can",
        "I, X, Y, Z, H, S, T"
      ],
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/"
    },
    {
      "symbol": "u1q(w,x,y,z)",
      "name": "canonical single-qubit IR payload",
      "latex": "\\uoneq(w,x,y,z)",
      "meaning": "Canonical quaternionic IR object storing one single-qubit gate action modulo global phase as four real coordinates.",
      "domain": "compiler IR",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 1",
      "related_symbols": [
        "q",
        "(w,x,y,z)",
        "Fuse(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "q",
      "name": "unit quaternion representative",
      "latex": "q",
      "meaning": "Unit quaternion representing an SU(2) element and therefore a single-qubit gate action modulo global phase under the fixed Phi convention.",
      "domain": "quaternion algebra / quantum gates",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 1",
      "related_symbols": [
        "u1q(w,x,y,z)",
        "Phi",
        "q_fused"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "(w,x,y,z)",
      "name": "quaternion coordinates",
      "latex": "(w,x,y,z)",
      "meaning": "Real coordinates of the unit quaternion q = w + x i + y j + z k.",
      "domain": "quaternion algebra",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 1",
      "related_symbols": [
        "q",
        "u1q(w,x,y,z)",
        "Norm"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Phi",
      "name": "fixed quaternion-SU(2) map",
      "latex": "\\Phi",
      "meaning": "Inherited map from quaternions to 2 x 2 complex matrices under which unit quaternions model SU(2).",
      "domain": "representation theory",
      "type": "map",
      "standard_or_project_specific": "variant",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "SU(2)",
        "sigma_x, sigma_y, sigma_z"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "U(2)",
      "name": "single-qubit unitary group",
      "latex": "\\mathrm{U}(2)",
      "meaning": "Full group in which single-qubit gates live before removal of overall phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "SU(2)",
        "q ~ +/- q"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "SU(2)",
      "name": "determinant-one representative group",
      "latex": "\\mathrm{SU}(2)",
      "meaning": "Special unitary group used to represent single-qubit gate action after removal of global phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "U(2)",
        "Phi",
        "Sp(1)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Sp(1)",
      "name": "unit-quaternion group",
      "latex": "\\mathrm{Sp}(1)",
      "meaning": "Group of unit quaternions, identified with SU(2) under the fixed map Phi.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "SU(2)",
        "S^3"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "q ~ +/- q",
      "name": "sign equivalence",
      "latex": "q \\sim_{\\pm} q'",
      "meaning": "Equivalence relation identifying a unit quaternion and its negative as the same single-qubit action modulo global phase.",
      "domain": "quaternion algebra / compiler IR",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "Can",
        "S^3"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "seg",
      "name": "single-qubit segment",
      "latex": "\\mathrm{seg}",
      "meaning": "A maximal consecutive run of single-qubit gates on one wire between out-of-scope boundaries.",
      "domain": "compiler IR",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "g_1,...,g_n",
        "Fuse(seg)",
        "G(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "g_1,...,g_n",
      "name": "gates in application order",
      "latex": "g_1,\\dots,g_n",
      "meaning": "Ordered list of gates in a single-qubit segment, written in the order in which they are applied to the state.",
      "domain": "quantum circuit syntax",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3",
      "related_symbols": [
        "seg",
        "q_1,...,q_n",
        "Fuse(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "q_1,...,q_n",
      "name": "quaternion gate representatives",
      "latex": "q_1,\\dots,q_n",
      "meaning": "Canonical quaternion representatives of the gates g_1,...,g_n after translation into the single-qubit IR.",
      "domain": "compiler IR",
      "type": "vector",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "g_1,...,g_n",
        "q_fused",
        "Fuse(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "q_fused",
      "name": "fused quaternion",
      "latex": "q_{\\mathrm{fused}}",
      "meaning": "Quaternion product of a full single-qubit segment after multiplication in circuit order and canonicalization.",
      "domain": "compiler IR",
      "type": "scalar",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "q_1,...,q_n",
        "Fuse(seg)",
        "Recon(q)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Fuse(seg)",
      "name": "fusion operator",
      "latex": "\\Fuse(\\mathrm{seg})",
      "meaning": "Canonical fused quaternion obtained from the product q_n ... q_1 followed by canonicalization.",
      "domain": "compiler IR",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "seg",
        "q_fused",
        "Can"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Norm",
      "name": "normalization map",
      "latex": "\\normalize",
      "meaning": "Map rescaling a nonzero quaternionic tuple back to unit norm before sign-canonicalization.",
      "domain": "numerical representation",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "Can",
        "SignCan",
        "u1q(w,x,y,z)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "SignCan",
      "name": "sign-canonicalization map",
      "latex": "\\sgncanon",
      "meaning": "Map that chooses one representative from the sign pair {q, -q} using the w > 0 rule with an equatorial tie-break.",
      "domain": "compiler IR",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "Can",
        "q ~ +/- q",
        "S^3"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Can",
      "name": "canonicalization map",
      "latex": "\\canon",
      "meaning": "Composition of normalization and sign-canonicalization used to produce the deterministic stored representative.",
      "domain": "compiler IR",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "Norm",
        "SignCan",
        "Fuse(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Recon(q)",
      "name": "reconstruction operator",
      "latex": "\\Recon(q)",
      "meaning": "Layered output rule that emits identity elimination, named-gate recognition, axis-aligned rotation recovery, or a generic one-qubit primitive.",
      "domain": "compiler IR",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 5",
      "related_symbols": [
        "q_fused",
        "Rx(theta), Ry(theta), Rz(theta)",
        "u1q(w,x,y,z)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Rx(theta), Ry(theta), Rz(theta)",
      "name": "axis-aligned rotation families",
      "latex": "R_x(\\theta), R_y(\\theta), R_z(\\theta)",
      "meaning": "Single-qubit rotation families recovered exactly when the fused quaternion lies on one coordinate axis in its imaginary part.",
      "domain": "quantum gates",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 5",
      "related_symbols": [
        "theta",
        "Recon(q)",
        "I, X, Y, Z, H, S, T"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "I, X, Y, Z, H, S, T",
      "name": "recognized named gates",
      "latex": "I, X, Y, Z, H, S, T",
      "meaning": "Canonical named-gate outputs recognized exactly by the reconstruction table when the fused quaternion matches a stored tuple.",
      "domain": "quantum circuit syntax",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 5",
      "related_symbols": [
        "Recon(q)",
        "q_fused",
        "Rx(theta), Ry(theta), Rz(theta)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "G(seg)",
      "name": "segment gate count",
      "latex": "\\GateCount(\\mathrm{seg})",
      "meaning": "Number of emitted single-qubit gates in a segment before or after optimization.",
      "domain": "benchmark metrics",
      "type": "function",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "D(seg)",
        "Base(seg)",
        "Eq(seg,seg')"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "D(seg)",
      "name": "segment-local depth",
      "latex": "\\Depth(\\mathrm{seg})",
      "meaning": "Sequential local depth of a single-qubit segment on one wire.",
      "domain": "benchmark metrics",
      "type": "function",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "G(seg)",
        "Base(seg)",
        "Fuse(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Base(seg)",
      "name": "syntax-preserving baseline",
      "latex": "\\Baseline(\\mathrm{seg})",
      "meaning": "Baseline flow that leaves the original single-qubit segment unchanged and performs no fusion.",
      "domain": "benchmark methodology",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "G(seg)",
        "D(seg)",
        "Fuse(seg)"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "Eq(seg,seg')",
      "name": "authoritative equivalence check",
      "latex": "\\Eq(\\mathrm{seg},\\mathrm{seg}')",
      "meaning": "Canonical-quaternion equality criterion used to certify that two segments represent the same single-qubit action modulo global phase.",
      "domain": "equivalence checking",
      "type": "function",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "Fuse(seg)",
        "Can",
        "q_fused"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "S^3",
      "name": "unit three-sphere",
      "latex": "S^3",
      "meaning": "Underlying manifold of unit quaternions on which the hemisphere sign-canonicalization rule is defined.",
      "domain": "geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 1",
      "related_symbols": [
        "q",
        "q ~ +/- q",
        "Can"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "theta",
      "name": "rotation angle",
      "latex": "\\theta",
      "meaning": "Rotation angle parameter recovered for exact axis-aligned single-qubit outputs.",
      "domain": "geometry",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 5",
      "related_symbols": [
        "Rx(theta), Ry(theta), Rz(theta)",
        "q_fused"
      ],
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/"
    },
    {
      "symbol": "u1q(w,x,y,z)",
      "name": "canonical single-qubit IR element",
      "latex": "\\uoneq(w,x,y,z)",
      "meaning": "Canonical quaternionic IR payload for a single-qubit gate action modulo global phase.",
      "domain": "compiler IR",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "Can",
        "Eq_can"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "q",
      "name": "unit quaternion gate representative",
      "latex": "q",
      "meaning": "Unit quaternion representing a determinant-one single-qubit gate action under the fixed quaternion-SU(2) convention.",
      "domain": "quaternion algebra / quantum gates",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "u1q(w,x,y,z)",
        "Phi",
        "q ~ +/- q"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "Phi",
      "name": "fixed quaternion-SU(2) map",
      "latex": "\\Phi",
      "meaning": "Map from quaternions to 2 x 2 complex matrices inherited from paper 1.",
      "domain": "representation theory",
      "type": "map",
      "standard_or_project_specific": "variant",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "SU(2)",
        "M(seg)"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "Can",
      "name": "canonical representative map",
      "latex": "\\canon",
      "meaning": "Composition of normalization and sign-canonicalization used to choose one deterministic representative from each sign class {q, -q}.",
      "domain": "compiler verification",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 2",
      "related_symbols": [
        "Norm",
        "SignCan",
        "Eq_can"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "Norm",
      "name": "normalization map",
      "latex": "\\normalize",
      "meaning": "Map rescaling a nonzero quaternionic tuple to unit norm before canonical comparison.",
      "domain": "numerical representation",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 2",
      "related_symbols": [
        "Can",
        "||q||"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "SignCan",
      "name": "sign-canonicalization map",
      "latex": "\\sgncanon",
      "meaning": "Map choosing one sign representative using the rule w > 0 and an equatorial tie-break when w = 0.",
      "domain": "compiler verification",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 2",
      "related_symbols": [
        "Can",
        "q ~ +/- q",
        "H_can"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "q ~ +/- q",
      "name": "sign equivalence",
      "latex": "q \\sim_{\\pm} q'",
      "meaning": "Equivalence relation identifying q and -q as the same single-qubit gate action modulo global phase.",
      "domain": "quaternion algebra / quantum gates",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2",
      "related_symbols": [
        "q",
        "Can",
        "Eq_can"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "Eq_can",
      "name": "canonical quaternion equivalence",
      "latex": "\\EqCan",
      "meaning": "Authoritative equivalence criterion declaring two segments equivalent when their canonical fused quaternions are equal after normalization and sign-canonicalization.",
      "domain": "compiler verification",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "Can",
        "q",
        "seg"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "Eq_gp",
      "name": "global-phase-aware matrix equivalence",
      "latex": "\\EqGP",
      "meaning": "Interoperable equivalence criterion declaring two single-qubit segment matrices equivalent when they differ only by a global phase.",
      "domain": "matrix verification",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3",
      "related_symbols": [
        "M(seg)",
        "r(U,V)",
        "F_proc(U,V)"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "seg",
      "name": "single-qubit segment",
      "latex": "\\mathrm{seg}",
      "meaning": "Maximal adjacent run of single-qubit gates on one wire between non-fusible boundaries.",
      "domain": "compiler IR",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "g_1, ..., g_n",
        "M(seg)",
        "Q(seg)"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "g_1, ..., g_n",
      "name": "segment gate sequence",
      "latex": "g_1, \\dots, g_n",
      "meaning": "Input gate list for a single-qubit segment, written in application order.",
      "domain": "quantum circuit syntax",
      "type": "other",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3",
      "related_symbols": [
        "seg",
        "q_1, ..., q_n"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "q_1, ..., q_n",
      "name": "translated quaternion sequence",
      "latex": "q_1, \\dots, q_n",
      "meaning": "Canonical unit-quaternion representatives of the gates g_1, ..., g_n.",
      "domain": "compiler verification",
      "type": "other",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "g_1, ..., g_n",
        "Q(seg)",
        "Can"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "Q(seg)",
      "name": "canonical fused quaternion of a segment",
      "latex": "\\mathcal{Q}(\\mathrm{seg})",
      "meaning": "Canonical fused quaternion computed from a segment by multiplying translated gate quaternions in circuit order and canonicalizing the product.",
      "domain": "compiler verification",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "seg",
        "q_1, ..., q_n",
        "Eq_can"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "M(seg)",
      "name": "matrix image of a segment",
      "latex": "\\mathcal{M}(\\mathrm{seg})",
      "meaning": "2 x 2 complex matrix obtained by multiplying the SU(2) representatives of the segment gates in application order.",
      "domain": "matrix verification",
      "type": "map",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 3",
      "related_symbols": [
        "seg",
        "Eq_gp",
        "tr(U^dagger V)"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "r(U,V)",
      "name": "phase-aware residual",
      "latex": "r(U,V)",
      "meaning": "Residual norm of U^dagger V after removal of the best-fit global phase, used as a diagnostic for near-equivalence in matrix-based toolchains.",
      "domain": "matrix verification",
      "type": "function",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 4",
      "related_symbols": [
        "Eq_gp",
        "F_proc(U,V)",
        "tr(U^dagger V)"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "F_proc(U,V)",
      "name": "process fidelity",
      "latex": "F_{\\mathrm{proc}}(U,V)",
      "meaning": "Single-qubit process fidelity diagnostic computed from the phase-aware overlap of two 2 x 2 unitary matrices.",
      "domain": "matrix verification",
      "type": "function",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4",
      "related_symbols": [
        "r(U,V)",
        "tr(U^dagger V)",
        "Eq_gp"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "tr(U^dagger V)",
      "name": "trace overlap",
      "latex": "\\mathrm{tr}(U^{\\dagger}V)",
      "meaning": "Trace overlap used to estimate best-fit global phase and process fidelity for two single-qubit unitaries.",
      "domain": "matrix verification",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4",
      "related_symbols": [
        "F_proc(U,V)",
        "r(U,V)",
        "Eq_gp"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    },
    {
      "symbol": "H_can",
      "name": "canonical hemisphere",
      "latex": "\\mathcal{H}_{\\mathrm{can}}",
      "meaning": "Closed hemisphere of S^3 selected by the rule w > 0 plus the equatorial tie-break at w = 0.",
      "domain": "geometry / compiler verification",
      "type": "set",
      "standard_or_project_specific": "project_specific",
      "first_use": "Section 2",
      "related_symbols": [
        "Can",
        "SignCan",
        "S^3"
      ],
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/"
    }
  ]
}