{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-cvdv",
  "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:matrix-contract",
      "label": "Finite matrices, norms and registers",
      "domains": [
        "concept:matrix-analysis"
      ],
      "tags": [
        "norm",
        "adjoint",
        "matrix",
        "register",
        "normalization"
      ],
      "question": "Are dimensions, basis order, scalar field and selected subspace fixed?",
      "formula": "U^† U=I,\\qquad \\Pi U\\Pi^†=A/\\alpha",
      "mechanism": "Use matrix-entry and state-action contracts before optimizing circuits. A first column and an entire projected block are different targets.",
      "assumptions": [
        "Finite dimensions and explicit little-/big-endian conversion",
        "Positive normalizer; selected input and output ancilla states",
        "Named vector/operator norm and exact versus approximate tolerance"
      ],
      "proof_steps": [
        "Name the basis equivalence and the clean projector.",
        "Prove unitarity independently of the target action.",
        "Evaluate all requested entries, not only a finite diagnostic subset."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.VerifiedStatePreparation"
      ],
      "source_ids": [],
      "boundary": "A structure accepting a proposition is an interface, not an unconditional construction theorem."
    },
    {
      "id": "family:hermite-bernstein",
      "label": "Hermite matching and Bernstein subdivision",
      "domains": [
        "concept:approximation"
      ],
      "tags": [
        "smooth",
        "polynomial",
        "Hermite",
        "Bernstein",
        "de Casteljau",
        "piecewise"
      ],
      "question": "Does a known function admit an exact low-degree local update?",
      "formula": "P_{2k+1}(t-1)=\\sum_{r=0}^{2k+1}c_r\\binom{2k+1}{r}t^r(1-t)^{2k+1-r}",
      "mechanism": "Preserve the literal Hermite polynomial while changing basis. Binary interval subdivision updates a fixed-size coefficient row, rather than enumerating samples.",
      "assumptions": [
        "Degree d=2k+1 and all endpoint jets are fixed",
        "Subdivision coordinate and polynomial coordinate must be reconciled",
        "Positivity on [0,1] is separate from the identity on all real coordinates"
      ],
      "proof_steps": [
        "Match derivatives at both endpoints; uniqueness follows because the difference has 2k+2 zeros counted with multiplicity.",
        "Expand the endpoint factors in the Bernstein basis and identify the coefficients.",
        "Apply de Casteljau restrictions along a bit path and reconcile its coordinate with the sample index."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.HermiteBernstein.sourceInterpolant_bernstein",
        "QuantumBlockEncoding.HermiteBernstein.sourceInterpolant_subdivision_readout"
      ],
      "source_ids": [
        "holmes-matsuura-2020"
      ],
      "boundary": "This is exact representation of the chosen polynomial, not automatic spectral convergence of a PDE solver. The general function-to-MPS idea predates ASPBE."
    },
    {
      "id": "family:harmonic-lift",
      "label": "Fourier multipliers and Schrödingerisation",
      "domains": [
        "concept:harmonic-analysis",
        "concept:operator-theory"
      ],
      "tags": [
        "Fourier",
        "QFT",
        "PDE",
        "Schrodingerisation",
        "derivative",
        "smooth"
      ],
      "question": "Can a nonunitary evolution be represented as transport in an auxiliary coordinate?",
      "formula": "w_t=-A_1\\partial_pw+iA_2w\\quad\\longmapsto\\quad i\\partial_t\\widehat w=(\\eta A_1-A_2)\\widehat w",
      "mechanism": "A Fourier basis converts transport into a real-frequency Hermitian multiplier. Smooth extension improves a separate approximation problem; its sampled preparation is a reusable input supplier.",
      "assumptions": [
        "A_1 and A_2 Hermitian with declared Fourier sign and register order",
        "Domain/boundary or finite discretization fixed",
        "Safe recovery region and truncation error are explicit"
      ],
      "proof_steps": [
        "Split A into its Hermitian and anti-Hermitian parts.",
        "Differentiate the warped profile on its recovery region.",
        "Specify the Fourier transform and identify the derivative multiplier.",
        "Separate initial-state, discretization, simulation and recovery errors."
      ],
      "lean_refs": [],
      "source_ids": [],
      "boundary": "This is a candidate mathematical route, not an end-to-end PDE theorem inferred from Hermite state preparation."
    },
    {
      "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-cvdv",
      "title": "CV–DV function preparation and non-Gaussian resources",
      "priority": 2,
      "status": "research-target",
      "setting_id": "setting:finite-energy-cv-dv-embedding",
      "formula": "\\|V_N|f_N\\rangle-|f\\rangle_{L^2}\\|\\le\\epsilon_{\\rm trunc}+\\epsilon_{\\rm grid}+\\epsilon_{\\rm prep}",
      "goal": "Relate a declared finite-energy oscillator encoding to qubit-grid preparation with a rigorous embedding and separate resource budgets.",
      "motivation": "Schrödingerisation uses auxiliary profiles in both continuous-variable and qubit implementations, but their resources are not interchangeable.",
      "input_model": "An explicit isometric embedding V_N, cutoff, grid or Fock truncation, energy/squeezing and non-Gaussian gate model.",
      "assumptions": [
        "Finite-energy/domain conditions and target normalization",
        "The embedding and quadrature weights are fixed",
        "CV noise, squeezing and non-Gaussian resources are not counted as free qubits"
      ],
      "target_bound": "A model-specific conversion, approximation and preparation theorem, not a universal equivalence of CV and DV costs.",
      "known_boundary": "The finite Hermite state is a useful first example. No CV–DV equivalence or GKP preparation theorem is currently supplied by this route.",
      "families": [
        "family:harmonic-lift",
        "family:hermite-bernstein",
        "family:matrix-contract",
        "family:charged-access"
      ],
      "source_ids": [],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec"
      ],
      "steps": [
        {
          "id": "embedding",
          "target": "Define and prove an isometric finite-to-continuous embedding.",
          "acceptance": "Inner products and normalization match the stated quadrature/Fock convention."
        },
        {
          "id": "truncation",
          "target": "Prove energy-tail and discretization error for a fixed profile family.",
          "acceptance": "All cutoff, domain and regularity hypotheses explicit."
        },
        {
          "id": "resource-map",
          "target": "Compose preparation and conversion costs.",
          "acceptance": "Separate DV gates and CV physical resources plus total state error."
        }
      ],
      "next": "Choose a finite-energy Hermite-smoothed profile and one embedding before comparing grid, oscillator or GKP implementations.",
      "lower_bound": {
        "status": "source-survey-pending",
        "task": "First specify the energy and non-Gaussian resource model; no cross-model lower-bound transfer is asserted.",
        "comparison_key": "setting:finite-energy-cv-dv-embedding"
      },
      "benchmarks": [
        "Hermite-smoothed auxiliary profiles",
        "Gaussian plus controlled non-Gaussian perturbation"
      ]
    }
  ],
  "hyperedges": [
    {
      "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:fourier-pde",
      "label": "Harmonic analysis to quantum evolution",
      "tails": [
        "family:harmonic-lift",
        "family:matrix-contract",
        "family:state-action"
      ],
      "heads": [
        "concept:block-encoding"
      ],
      "formula": "H_{\\rm Sch}=D_p\\otimes A_1-I\\otimes A_2",
      "mechanism": "The prepared auxiliary profile is one supplier; operator access and recovery remain separate.",
      "hypothesis_map": "Hermitian components, Fourier sign, finite grid and norm/recovery budget.",
      "conclusion_map": "Candidate Hamiltonian-access route for Schrödingerisation.",
      "failure_boundary": "SP certification alone proves neither the Hamiltonian block encoding nor end-to-end PDE accuracy.",
      "source_ids": [],
      "lean_refs": [],
      "status": "proposal",
      "review": "independent review pending"
    },
    {
      "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"
    }
  ],
  "required_handoff": [
    "frozen target and access model",
    "exact reused declarations",
    "bounded mathematical delta",
    "assumption differences",
    "independent round-trip evidence",
    "graph contribution and residual boundary"
  ]
}