{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-verification",
  "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:structured-verification",
      "label": "Structure-aware preparation and verification",
      "domains": [
        "concept:verification",
        "concept:state-preparation",
        "concept:tensor-networks"
      ],
      "tags": [
        "fidelity",
        "witness",
        "shadow",
        "measurement",
        "hardware",
        "noise"
      ],
      "question": "Can the preparation structure supply a cheap experimental witness?",
      "formula": "F(\\rho,|\\psi\\rangle)=\\langle\\psi|\\rho|\\psi\\rangle",
      "mechanism": "Keep kernel-checked ideal-circuit equality separate from noisy-hardware fidelity estimation. A structural witness needs an explicit measurement and sample-complexity theorem.",
      "assumptions": [
        "Measurement access and noise model stated",
        "IID or non-IID assumptions explicit",
        "Confidence level, sample count and target description cost charged"
      ],
      "proof_steps": [
        "State the ideal target and the observed channel separately.",
        "Derive an observable or witness for the chosen state class.",
        "Bound estimation error and confidence under the measurement assumptions."
      ],
      "lean_refs": [],
      "source_ids": [
        "butterworth-2026-candidate"
      ],
      "boundary": "The candidate hardware paper is not primary-verified here; no reported numerical performance is used as a proved bound."
    },
    {
      "id": "family:bounded-memory",
      "label": "Bounded-memory function representations",
      "domains": [
        "concept:approximation",
        "concept:tensor-networks"
      ],
      "tags": [
        "MPS",
        "TT",
        "QTT",
        "rank",
        "bond",
        "function",
        "high-dimensional",
        "boundary"
      ],
      "question": "Can each bit update a small state instead of selecting a table entry?",
      "formula": "f(b_1,\\ldots,b_m)=\\ell^\\top G_1[b_1]\\cdots G_m[b_m]r,\\quad R\\le 2k+6",
      "mechanism": "Exponentials factor over bits; polynomial coefficients evolve in a degree-bounded space; one unresolved partition boundary is routed through a small additional state.",
      "assumptions": [
        "Explicit cores are supplied from formulas, not selected from an existence theorem",
        "TT rank is bounded across every chosen bit cut",
        "Branch masks, phase information and endpoints are preserved"
      ],
      "proof_steps": [
        "Prove each local core computes its intended bit update.",
        "Induct on the remaining word to identify the contraction with the literal function value.",
        "Count stored core entries and compare with a dense amplitude table."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.HermiteBoundaryInjection.hermiteKernel_eq_sample",
        "QuantumBlockEncoding.HermiteFiniteChain.sourceChain_contract",
        "QuantumBlockEncoding.HermiteFiniteChain.sourceChain_storage"
      ],
      "source_ids": [
        "holmes-matsuura-2020",
        "multivariate-2025"
      ],
      "boundary": "Smoothness or a symbolic formula alone does not guarantee low TT rank or a cheap core supplier in arbitrary dimension."
    },
    {
      "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."
    }
  ],
  "routes": [
    {
      "id": "spw-verification",
      "title": "Joint design of preparation and experimental verification",
      "priority": 2,
      "status": "research-target",
      "setting_id": "setting:structured-witness-measurements",
      "formula": "\\Pr[|\\widehat F-F(\\rho,|f\\rangle)|\\le\\epsilon]\\ge1-\\delta",
      "goal": "Automatically derive a low-sample fidelity witness for a precisely defined structured state class.",
      "motivation": "Full tomography is not the correct default output task; preparation structure may enable a cheaper witness.",
      "input_model": "Copies or channels producing rho, a specified measurement family, classical target representation and declared IID/noise assumptions.",
      "assumptions": [
        "Target and implemented noisy state distinguished",
        "Witness computation and measurement basis-change cost included",
        "Confidence, bias and non-IID handling not silently omitted"
      ],
      "target_bound": "A witness correctness theorem, finite-sample bound and implementation cost for one state family.",
      "known_boundary": "A Lean ideal-circuit proof does not certify noisy hardware. The user-supplied 2609.08414 reference remains primary-source-unavailable in this ledger.",
      "families": [
        "family:structured-verification",
        "family:bounded-memory",
        "family:charged-access"
      ],
      "source_ids": [
        "butterworth-2026-candidate"
      ],
      "lean_refs": [],
      "steps": [
        {
          "id": "measurement-contract",
          "target": "Specify the measurement and statistical experiment.",
          "acceptance": "Outcome law, confidence and noise assumptions are formal objects."
        },
        {
          "id": "witness",
          "target": "Derive a structure-specific fidelity witness.",
          "acceptance": "Unbiasedness or certified one-sided bias and computable observable."
        },
        {
          "id": "sample-cost",
          "target": "Prove sample and basis-change complexity.",
          "acceptance": "End-to-end confidence statement with no uncharged target tomography."
        }
      ],
      "next": "Verify the primary hardware reference first; start formal work from a small exactly analyzable MPS witness.",
      "lower_bound": {
        "status": "model-definition-pending",
        "task": "Distinguish lower bounds for full tomography, fidelity estimation and a promised structured witness.",
        "comparison_key": "setting:structured-witness-measurements"
      },
      "benchmarks": [
        "Product-state witness",
        "Small exact MPS with prescribed measurement access",
        "Structured function signals"
      ]
    }
  ],
  "hyperedges": [
    {
      "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: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:hermite-compression",
      "label": "Exact function structure to bounded memory",
      "tails": [
        "family:hermite-bernstein",
        "family:matrix-contract"
      ],
      "heads": [
        "family:bounded-memory"
      ],
      "formula": "\\{\\text{degree and subdivision},\\text{bit/branch contract}\\}\\Longrightarrow R\\le2k+6",
      "mechanism": "Keep local polynomial updates and branch routing together; neither alone proves the sampled source.",
      "hypothesis_map": "Endpoint jets, grid coordinate, boundary ownership and bit order agree.",
      "conclusion_map": "The finite core contraction equals the literal sampled Hermite function.",
      "failure_boundary": "A degree statement alone does not account for an arbitrary number of pieces.",
      "source_ids": [
        "holmes-matsuura-2020"
      ],
      "lean_refs": [
        "QuantumBlockEncoding.HermiteFiniteChain.sourceChain_contract"
      ],
      "status": "curated-transport",
      "review": "independent conceptual review pending; local Lean roots have their own build evidence"
    },
    {
      "id": "transport:prepare-verify",
      "label": "Preparation structure plus measurement contract",
      "tails": [
        "family:state-action",
        "family:structured-verification"
      ],
      "heads": [
        "concept:verification"
      ],
      "formula": "\\Pr[|\\widehat F-F|\\le\\epsilon]\\ge1-\\delta",
      "mechanism": "Use the state description to propose a measurement witness; formal circuit proof and statistical evidence remain separate.",
      "hypothesis_map": "Experimental measurement access, noise, sample model, epsilon and delta.",
      "conclusion_map": "Candidate low-sample verification protocol for a specified family.",
      "failure_boundary": "An ideal Lean proof is not a hardware fidelity certificate.",
      "source_ids": [
        "butterworth-2026-candidate"
      ],
      "lean_refs": [],
      "status": "proposal",
      "review": "independent review pending"
    }
  ],
  "sources": [
    {
      "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": "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": "butterworth-2026-candidate",
      "title": "Efficient quantum state preparation on Quantinuum hardware (candidate reference)",
      "authors": "Unverified in a primary source during this audit",
      "url": "https://arxiv.org/abs/2609.08414",
      "year": 2026,
      "status": "primary-source-unavailable",
      "anchor": "User-supplied arXiv identifier; primary fetch failed; do not use reported hardware numbers as evidence",
      "scope": "Retained in the candidate queue rather than silently promoted from secondary indexing.",
      "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"
  ]
}