{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-fault-tolerant",
  "truth_boundary": "Curated retrieval packet, not an execution result or Lean implication. Hyperedge tails are conjunctive. Check exact source, input/oracle/phase/norm/resource contracts before reuse.",
  "families": [
    {
      "id": "family:charged-access",
      "label": "Charged access and finite-precision compilation",
      "domains": [
        "concept:complexity",
        "concept:matrix-analysis"
      ],
      "tags": [
        "QRAM",
        "QROM",
        "oracle",
        "T-count",
        "precision",
        "lower-bound",
        "cost",
        "classical"
      ],
      "question": "Does a query or symbolic gate hide the dominant work?",
      "formula": "T_{\\rm total}=T_{\\rm preprocessing}+Q\\,T_{\\rm oracle}+T_{\\rm synthesis}+T_{\\rm readout}",
      "mechanism": "Separate descriptions, arithmetic operations, bit operations, logical rotations, T/Toffoli gates, depth, connectivity and ancillas. Compare only matching models.",
      "assumptions": [
        "Input encoding and precision specified",
        "One cost model for both upper and lower bounds",
        "Initialization, maintenance, amplification and measurements charged"
      ],
      "proof_steps": [
        "Freeze the access model before optimizing.",
        "Bind costs to actual data producers and primitive lists.",
        "Budget each numerical error and then sum compositional errors."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.HermiteFiniteNorm.norm_arithmetic_budget"
      ],
      "source_ids": [
        "zhang-yuan-2024",
        "yuan-zhang-2023"
      ],
      "boundary": "The local norm theorem is only a substrate. It is not the entire displayed end-to-end cost decomposition certified in Lean."
    },
    {
      "id": "family:local-isometry",
      "label": "Canonicalization, completion and local compilation",
      "domains": [
        "concept:matrix-analysis",
        "concept:tensor-networks",
        "concept:state-preparation"
      ],
      "tags": [
        "QR",
        "LQ",
        "Givens",
        "isometry",
        "canonical",
        "unitary",
        "compile",
        "clean"
      ],
      "question": "Can a bounded-rank description be turned into an actual primitive list?",
      "formula": "V_i^† V_i=I\\quad\\Longrightarrow\\quad U_i(|a\\rangle|0\\rangle)=\\sum_{b,a'}(V_i)_{b,a',a}|a'\\rangle|b\\rangle",
      "mechanism": "Canonicalize without changing the contraction, absorb the signed scalar boundary, complete each local isometry, compile its actual orthogonal matrix and assemble physical wires.",
      "assumptions": [
        "Normalized scalar-boundary real TT",
        "Rank-deficient factors and signed boundary cases are handled",
        "All non-clean output sectors vanish at the final stage"
      ],
      "proof_steps": [
        "Factor local cores and pass the residual factor to the neighboring core.",
        "Prove preservation of every contracted amplitude and the local isometry identity.",
        "Complete to an orthogonal matrix, decompose into plane rotations and compile to RY/CX.",
        "Compose the stages and prove both clean output and primitive resource bounds."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveTensorTrainCompiler.compile_spec",
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec"
      ],
      "source_ids": [
        "holmes-matsuura-2020"
      ],
      "boundary": "The current symbolic resource theorem does not include a full stable finite-precision T-gate compiler."
    },
    {
      "id": "family:gram-normalization",
      "label": "Local Gram normalization",
      "domains": [
        "concept:matrix-analysis",
        "concept:tensor-networks"
      ],
      "tags": [
        "normalizer",
        "Gram",
        "environment",
        "preprocessing",
        "norm"
      ],
      "question": "Can normalization be computed without summing exponentially many amplitudes?",
      "formula": "E_i=\\sum_{b=0}^1G_i[b]E_{i+1}G_i[b]^\\top,\\qquad Z=\\ell^\\top E_1\\ell",
      "mechanism": "Distribute the sum of squared contractions into local matrix products; compute the same finite norm from right to left.",
      "assumptions": [
        "Real cores in this local Hermite route; use conjugate transpose for complex cores",
        "A nonzero raw state and compatible boundary dimensions",
        "Core construction and finite-bit arithmetic are separately charged"
      ],
      "proof_steps": [
        "Expand the squared scalar contraction.",
        "Exchange finite sums and collect the two values of the next bit.",
        "Inductively identify the Gram environment with the exact suffix norm and take the positive square root."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.HermiteFiniteNorm.localSampleNorm_eq_sampleNorm",
        "QuantumBlockEncoding.HermiteFiniteNorm.norm_arithmetic_budget"
      ],
      "source_ids": [],
      "boundary": "An exact-real operation count is not a bit-complexity bound or a floating-point stability theorem."
    }
  ],
  "routes": [
    {
      "id": "spw-fault-tolerant",
      "title": "Fault-tolerant resource trade-offs for structured preparation",
      "priority": 2,
      "status": "research-target",
      "setting_id": "setting:structured-clifford-t-connectivity",
      "formula": "\\mathcal R=(T,\\operatorname{Toffoli},d,q_{\\rm anc},Q_{\\rm oracle},\\epsilon)",
      "goal": "Compile the exact-real structured route into a finite gate set with a proved error budget and declared connectivity.",
      "motivation": "A small number of arbitrary real rotations is not a complete fault-tolerant resource estimate.",
      "input_model": "Finite-bit source data and a fixed Clifford+T gate set, connectivity graph, tolerance and ancilla policy.",
      "assumptions": [
        "Input/separation and conditioning promises exposed",
        "Each angle approximation and synthesis error budget explicit",
        "Count or bound the actual emitted primitive list"
      ],
      "target_bound": "A source-to-finite-bit theorem and a model-matched resource frontier; do not claim simultaneous global optimality of every coordinate.",
      "known_boundary": "The Hermite exact-real primitive theorem is already a useful substrate. Full bit complexity and stable angle generation are additional obligations.",
      "families": [
        "family:charged-access",
        "family:local-isometry",
        "family:gram-normalization"
      ],
      "source_ids": [
        "zhang-yuan-2024",
        "yuan-zhang-2023"
      ],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec"
      ],
      "steps": [
        {
          "id": "input-bits",
          "target": "Freeze the finite-bit input and numerical-stability contract.",
          "acceptance": "All arbitrary-real comparisons and small pivots have explicit handling."
        },
        {
          "id": "stable-angles",
          "target": "Bound coefficient, canonicalization and angle errors.",
          "acceptance": "A normalized-state error theorem tied to supplied bit precision."
        },
        {
          "id": "logical-compiler",
          "target": "Compose finite-gate synthesis and connectivity routing.",
          "acceptance": "T/Toffoli/depth/ancilla accounting plus final state error and cleanup."
        }
      ],
      "next": "Treat the current Hermite bit-complexity boundary as the first local target rather than reopening its exact-real correctness proof.",
      "lower_bound": {
        "status": "source-audit-pending",
        "task": "Compare ancilla/depth trade-offs only in the same finite gate and connectivity model.",
        "comparison_key": "setting:structured-clifford-t-connectivity"
      },
      "benchmarks": [
        "Hermite baseline versus structured route at matched error",
        "Connectivity-constrained local TT compilation"
      ]
    }
  ],
  "hyperedges": [
    {
      "id": "transport:tt-to-sp",
      "label": "Normalize AND compile AND clean",
      "tails": [
        "family:bounded-memory",
        "family:gram-normalization",
        "family:local-isometry"
      ],
      "heads": [
        "family:state-action"
      ],
      "formula": "\\{\\text{explicit TT},\\ Z>0,\\text{local compiler}\\}\\Longrightarrow U|0^{m+q}\\rangle=|g_k\\rangle|0^q\\rangle",
      "mechanism": "Compose the source, norm and primitive-circuit interfaces.",
      "hypothesis_map": "Real normalized scalar-boundary chain, rank bound, padded register layout and signed residual boundary.",
      "conclusion_map": "All target amplitudes and all non-clean sectors, plus actual primitive-list count.",
      "failure_boundary": "An existence-only TT representation or unknown normalizer is not a data-producing compiler.",
      "source_ids": [
        "holmes-matsuura-2020"
      ],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec"
      ],
      "status": "curated-transport",
      "review": "independent conceptual review pending; not a certified functor"
    },
    {
      "id": "transport:diagonal-filter-state",
      "label": "Diagonal block + prepared input + success contract",
      "tails": [
        "family:diagonal-postselection",
        "family:state-action",
        "family:charged-access"
      ],
      "heads": [
        "concept:state-preparation"
      ],
      "formula": "|\\psi\\rangle\\longmapsto D|\\psi\\rangle/\\|D\\psi\\|,\\quad p=\\|D\\psi\\|^2/\\alpha^2",
      "mechanism": "Specialize the existing BE-to-SP consumer to a diagonal filter; retain input preparation and successful-branch normalization as AND inputs.",
      "hypothesis_map": "Literal source dilation, bounded diagonal, known success sector, nonzero output, charged inverse/control and matched error metric.",
      "conclusion_map": "A conditional state and model-specific space–depth–accuracy/resource comparison.",
      "failure_boundary": "Neither a diagonal function description nor a compiled nearby fixed-size example certifies a scalable supplier; this is not SP-to-BE inclusion.",
      "source_ids": [
        "zylberman-et-al-2025-diagonal",
        "lin-wiebe-2026-qasc"
      ],
      "lean_refs": [],
      "status": "proposal",
      "review": "Source-planning edge; independent conceptual review pending; no local transport theorem"
    },
    {
      "id": "transport:envelope-structured",
      "label": "Sampling envelope meets function structure",
      "tails": [
        "family:envelope-reweight",
        "family:bounded-memory",
        "family:charged-access"
      ],
      "heads": [
        "concept:state-preparation"
      ],
      "formula": "\\kappa_{\\rm env}=C\\|g\\|_2/\\|f\\|_2",
      "mechanism": "Search for an envelope with both a provable ratio bound and a constructive small representation.",
      "hypothesis_map": "Support domination, ratio degree/rank, phase access and charged reference preparation.",
      "conclusion_map": "A model-specific success and end-to-end cost target.",
      "failure_boundary": "A good classical envelope need not have a cheap coherent preparation; no universal cure for dimensionality.",
      "source_ids": [
        "quantum-rejection-2013",
        "multivariate-2025"
      ],
      "lean_refs": [],
      "status": "proposal",
      "review": "independent review pending"
    },
    {
      "id": "transport:walsh-loader",
      "label": "Walsh phases + interference + nonzero branch",
      "tails": [
        "family:walsh-diagonal-phase",
        "family:diagonal-postselection",
        "family:state-action",
        "family:charged-access"
      ],
      "heads": [
        "concept:state-preparation"
      ],
      "formula": "-i(I-e^{-i\\tau F_S})|+^n\\rangle/2\\approx\\tau F_S|+^n\\rangle/2",
      "mechanism": "Combine a controlled diagonal phase supplier with ancilla interference, postselection and a normalized-error theorem.",
      "hypothesis_map": "Real function, source bit order, supplied coefficients, retained zero mode, explicit input and inverse access, small-angle/truncation bounds and nonzero accepted branch.",
      "conclusion_map": "An approximate amplitude state with a declared infidelity and success-dependent total cost, not an unconditional n-independent loader.",
      "failure_boundary": "Constant-term removal changes the controlled circuit; small filling fraction or poor sparsity can remove the advantage.",
      "source_ids": [
        "zylberman-debbasch-2024-walsh",
        "pellini-ferrari-dacrema-2025-comment"
      ],
      "lean_refs": [],
      "status": "proposal",
      "review": "Source-planning edge; independent conceptual review pending; no local transport theorem"
    }
  ],
  "sources": [
    {
      "id": "zylberman-debbasch-2024-walsh",
      "title": "Efficient Quantum State Preparation with Walsh Series",
      "authors": "Julien Zylberman; Fabrice Debbasch",
      "url": "https://arxiv.org/abs/2307.08384v3",
      "requested_version": "https://arxiv.org/pdf/2307.08384v3",
      "year": 2024,
      "status": "primary-text-checked",
      "anchor": "arXiv v3, 9 November 2023; checked 29 September 2026: Theorem 1, Corollary 1, Table I and Appendix B (B14–B19, B28).",
      "scope": "Walsh Series Loader: epsilon is state infidelity. Smooth fixed-function single-attempt depth O(epsilon^-1/2), size O(n+epsilon^-1/2), success Theta(epsilon); expected depth O(epsilon^-3/2), expected size O((n+epsilon^-1/2)/epsilon). Constants retain function/derivative/nonzero-mass assumptions. Sparse variants require supplied coefficients and Hamming-weight bounds. Amplification also charges reflections and inverses.",
      "formal_status": "planned-formalization"
    },
    {
      "id": "zylberman-et-al-2025-diagonal",
      "title": "Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy trade-offs",
      "authors": "Julien Zylberman; Ugo Nzongani; Andrea Simonetto; Fabrice Debbasch",
      "url": "https://arxiv.org/abs/2404.02819v4",
      "requested_version": "https://arxiv.org/pdf/2404.02819v4",
      "year": 2025,
      "status": "primary-text-checked",
      "anchor": "arXiv v4, 21 January 2025; checked 29 September 2026: II.1, III.1–III.2, IV.1 Eqs. (15)–(16), V.1 Eqs. (19)–(21), VI A, VIII.1 and Tables I–II. Figure 4 signal-sector and III.2/VIII.1 accounting ambiguities remain explicit source obligations.",
      "scope": "Parallel scheduling trades ancillas for depth, not automatically total size. Table I uses spectral-norm error; Eq. (18) bounds normalized-block error delta, corresponding to total-scale times delta in the unscaled-operator convention. Exact-block success in source notation is ||D psi||^2/(alpha^2 dmax^2); alpha>1 controls the transformed-angle derivative in smooth approximation. No local source theorem or resolution of source ambiguities is claimed.",
      "formal_status": "planned-formalization"
    },
    {
      "id": "pellini-ferrari-dacrema-2025-comment",
      "title": "Comment on arXiv:2307.08384: Efficient Quantum State Preparation with Walsh Series",
      "authors": "Riccardo Pellini; Maurizio Ferrari Dacrema",
      "url": "https://arxiv.org/abs/2502.05193v1",
      "year": 2025,
      "status": "primary-text-checked",
      "anchor": "arXiv v1 full HTML, Sections II.2–II.3, Eq. (11), Fig. 2: controlled zero-order term; checked 29 September 2026.",
      "scope": "Companion adversarial implementation review, not a refutation of the WSL theory. Retain the constant Walsh coefficient in a controlled phase circuit; use constant and constant-plus-parity regression cases.",
      "formal_status": "external-reference"
    },
    {
      "id": "holmes-matsuura-2020",
      "title": "Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions",
      "authors": "Adam Holmes; A. Y. Matsuura",
      "url": "https://arxiv.org/abs/2005.04351",
      "requested_version": "https://arxiv.org/html/2005.04351v1",
      "year": 2020,
      "status": "primary-metadata-checked",
      "anchor": "Abstract: explicit function to MPS to circuit; requested v1 HTML was unavailable in this audit",
      "scope": "Prior art for function-to-MPS preparation. The ASPBE Hermite specialization must not be credited with inventing polynomial-to-MPS or sequential MPS preparation.",
      "formal_status": "external-reference"
    },
    {
      "id": "yuan-zhang-2023",
      "title": "Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits",
      "authors": "Pei Yuan; Shengyu Zhang",
      "url": "https://quantum-journal.org/papers/q-2023-03-20-956/",
      "year": 2023,
      "status": "primary-metadata-checked",
      "anchor": "Abstract and publication version notice: published arXiv v2; later v3 exists",
      "scope": "Baseline for arbitrary and controlled state preparation and ancilla/depth trade-offs. Pin a version and gate model before comparing bounds.",
      "formal_status": "external-reference"
    },
    {
      "id": "zhang-yuan-2024",
      "title": "Circuit complexity of quantum access models for encoding classical data",
      "authors": "Xiao-Ming Zhang; Xiao Yuan",
      "url": "https://www.nature.com/articles/s41534-024-00835-8",
      "year": 2024,
      "status": "primary-text-checked",
      "anchor": "Results: circuit complexity lower bound; construction of LCU-based block-encoding; Methods: state preparation",
      "scope": "Explicit access construction is not a free oracle. PREPARE together with SELECT and uncomputation can supply a block encoding; a single prepared state alone does not determine an arbitrary operator.",
      "formal_status": "external-reference"
    },
    {
      "id": "multivariate-2025",
      "title": "Quantum state preparation for multivariate functions",
      "authors": "See primary source",
      "url": "https://quantum-journal.org/papers/q-2025-04-09-1703/",
      "year": 2025,
      "status": "primary-metadata-checked",
      "anchor": "Abstract and article record",
      "scope": "Multivariate preparation baseline; a tensor-product coefficient count is not automatically polynomial in dimension.",
      "formal_status": "external-reference"
    },
    {
      "id": "quantum-rejection-2013",
      "title": "Quantum Rejection Sampling",
      "authors": "Maris Ozols; Martin Roetteler; Jérémie Roland",
      "url": "https://www.microsoft.com/en-us/research/publication/quantum-rejection-sampling/",
      "year": 2013,
      "status": "primary-metadata-checked",
      "anchor": "Authors' publication page: quantum state generation, query characterization and matching lower bound",
      "scope": "Prior art for coherent amplitude reweighting. Novelty must be in an explicit structured envelope, guarantees, implementation or model-matched bound, not in rejection sampling itself.",
      "formal_status": "external-reference"
    },
    {
      "id": "lin-wiebe-2026-qasc",
      "title": "Quantum Algorithms for Scientific Computation",
      "authors": "Lin Lin; Nathan Wiebe",
      "url": "https://math.berkeley.edu/~linlin/qasc/live_notes_0429.pdf",
      "year": 2026,
      "status": "primary-text-checked",
      "anchor": "29 April 2026 public 447-page edition; PDF SHA256 56825000d53025dea4c2998dd6a061592b8fb2b9a446c3194c5f408183327599. First bounded source slice: Chapter 9 Section 9.1, printed pages 141–143, equations (9.2), (9.9), (9.10), (9.12); standing partial-application convention Definition 2.25, page 37. Parts I–IV remain curriculum anchors, not completed formalisation.",
      "scope": "Peer Part IV curriculum anchor, reusing existing ASPBE state-preparation/block-encoding APIs. New Chapter 9 adapters, operator-norm approximation, scientific constructions and resource certificates still require separate production admission. No whole-section/book completion is claimed.",
      "formal_status": "external-reference"
    }
  ],
  "required_handoff": [
    "frozen target and access model",
    "exact reused declarations",
    "bounded mathematical delta",
    "assumption differences",
    "independent round-trip evidence",
    "graph contribution and residual boundary"
  ]
}