{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-controlled",
  "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:state-action",
      "label": "State-action and coherent preparation contracts",
      "domains": [
        "concept:state-preparation"
      ],
      "tags": [
        "state",
        "amplitude",
        "controlled",
        "family",
        "phase",
        "ancilla"
      ],
      "question": "Is the requested object one state, a coherent family, or a mixed state?",
      "formula": "U|0^m\\rangle|0^q\\rangle=|\\psi\\rangle|0^q\\rangle",
      "mechanism": "Fix all output sectors. A family also fixes relative phases between parameters; density-matrix trace error is a separate contract.",
      "assumptions": [
        "Target normalization and nonzero source",
        "Relative phase convention for controlled families",
        "Success probability and garbage register behavior declared"
      ],
      "proof_steps": [
        "Read the exact first-column predicate.",
        "Prove each target amplitude and every non-clean sector.",
        "For controlled families, prove a single coherent map rather than per-parameter rays."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_columns"
      ],
      "source_ids": [
        "yuan-zhang-2023"
      ],
      "boundary": "Correctness up to a parameter-dependent global phase is not sufficient for a coherent family."
    },
    {
      "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-controlled",
      "title": "Coherent parameter-dependent families",
      "priority": 1,
      "status": "research-target",
      "setting_id": "setting:joint-parameter-data-supplier",
      "formula": "U:\\sum_a c_a|a\\rangle|0\\rangle\\longmapsto\\sum_a c_a|a\\rangle|f_a\\rangle",
      "goal": "Produce one coherent template for jointly structured parameters and data with a uniform error bound.",
      "motivation": "Parameterized PDEs, source histories and Bayesian conditional states require superpositions of inputs, not separate classically compiled circuits.",
      "input_model": "Parameter a is a quantum register; one shared joint coefficient/core supplier is provided with a uniform norm bound.",
      "assumptions": [
        "Relative phases are fixed across a",
        "Norms stay bounded away from zero on the promised parameter domain",
        "Branch selection and numerical rank changes are controlled coherently or avoided"
      ],
      "target_bound": "Polynomial dependence on the joint structural parameters and parameter bit length, rather than a table of independent preparations.",
      "known_boundary": "A family of individually easy states need not have an easy coherent supplier. Per-parameter correctness up to phase is insufficient.",
      "families": [
        "family:state-action",
        "family:bounded-memory",
        "family:charged-access"
      ],
      "source_ids": [
        "yuan-zhang-2023",
        "qkan-2026"
      ],
      "lean_refs": [],
      "steps": [
        {
          "id": "phase-contract",
          "target": "Define the coherent family contract including phases and clean workspace.",
          "acceptance": "One matrix action for arbitrary parameter superpositions."
        },
        {
          "id": "joint-structure",
          "target": "Derive a joint tensor/polynomial representation.",
          "acceptance": "Uniform rank/degree and nonzero normalization proof over all allowed parameters."
        },
        {
          "id": "uniform-error",
          "target": "Compile with uniform error and cost.",
          "acceptance": "Operator-action error on the whole parameter-data subspace, not only sampled parameter values."
        }
      ],
      "next": "Use an affine parameterized Hermite or Gaussian family as a small exact model before time-dependent PDE source histories.",
      "lower_bound": {
        "status": "model-definition-pending",
        "task": "Keep arbitrary CQSP lower bounds separate from the restricted joint-structure promise.",
        "comparison_key": "setting:joint-parameter-data-supplier"
      },
      "benchmarks": [
        "Width-dependent Gaussians",
        "Parameter-dependent Hermite smoothing",
        "Time-dependent source terms"
      ]
    }
  ],
  "hyperedges": [
    {
      "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: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: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": "qkan-2026",
      "title": "QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation",
      "authors": "Petr Ivashkov; Po-Wei Huang; Kelvin Koor; Lirandë Pira; Patrick Rebentrost",
      "url": "https://www.nature.com/articles/s41534-026-01202-5",
      "year": 2026,
      "status": "primary-text-checked",
      "anchor": "Abstract, Introduction and Discussion: shallow compositional state preparation",
      "scope": "Composition-dependent bounds, not an efficient preparation theorem for every continuous multivariate function.",
      "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"
  ]
}