{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-no-qram",
  "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:prepare-select",
      "label": "PREPARE–SELECT–unprepare",
      "domains": [
        "concept:state-preparation",
        "concept:block-encoding"
      ],
      "tags": [
        "LCU",
        "SELECT",
        "block",
        "operator",
        "prepare",
        "sum"
      ],
      "question": "Are coefficient preparation and controlled operator access both available?",
      "formula": "(\\langle0|\\otimes I)(P^†\\otimes I)\\operatorname{SELECT}(U)(P\\otimes I)(|0\\rangle\\otimes I)=\\sum_a w_aU_a/\\alpha",
      "mechanism": "Prepare coefficients sqrt(w_a/alpha), select the corresponding unitary, and unprepare. Phases for complex coefficients must be assigned explicitly.",
      "assumptions": [
        "Nonnegative weights w_a and alpha=sum w_a>0, or an explicit complex-phase adapter",
        "Implementable SELECT and adjoint PREPARE, not merely copies of a state",
        "Compatible data/ancilla layouts and normalization"
      ],
      "proof_steps": [
        "Expand PREPARE on the clean ancilla.",
        "Apply SELECT branch by branch.",
        "Contract with the same coefficient preparation to obtain weighted matrix entries."
      ],
      "lean_refs": [],
      "source_ids": [
        "zhang-yuan-2024"
      ],
      "boundary": "SP can be a BE ingredient, but one isolated state does not determine a general matrix. This explanatory bridge has no newly certified transport root."
    },
    {
      "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-no-qram",
      "title": "Data loading without a free QRAM assumption",
      "priority": 2,
      "status": "research-target",
      "setting_id": "setting:explicit-data-charged-access",
      "formula": "T_{\\rm total}=T_{\\rm build}+Q\\,T_{\\rm query}+T_{\\rm update}+T_{\\rm readout}",
      "goal": "Identify succinct data classes for which explicit access construction preserves the intended algorithmic advantage.",
      "motivation": "An algorithm with few abstract queries can still be expensive when the query unitary is built from real data.",
      "input_model": "Classical stored arrays, QROM, arithmetic formulas and physical QRAM are separate settings with explicit initialization, updates and memory costs.",
      "assumptions": [
        "Fix data bit length, update frequency and reuse count",
        "Charge controlled/inverse queries when the algorithm uses them",
        "Compare against a classical algorithm with the same access assumptions"
      ],
      "target_bound": "A full cost theorem for one succinct data representation, not universal polylogarithmic loading of an arbitrary array.",
      "known_boundary": "General sparse-access and block-encoding constructions can require near-linear cost in matrix dimension. Sparsity does not make all entries free.",
      "families": [
        "family:charged-access",
        "family:prepare-select",
        "family:matrix-contract"
      ],
      "source_ids": [
        "zhang-yuan-2024",
        "yuan-zhang-2023"
      ],
      "lean_refs": [],
      "steps": [
        {
          "id": "access-contract",
          "target": "Define the exact classical/quantum access interface.",
          "acceptance": "Separate data construction, coherent calls, memory and update costs."
        },
        {
          "id": "succinct-supplier",
          "target": "Compile one formula, sparse-index or repeated-block supplier.",
          "acceptance": "Basis-action theorem and count of actual access primitives."
        },
        {
          "id": "application",
          "target": "Reinsert access costs into a downstream algorithm.",
          "acceptance": "A comparison using the same input/output model and total cost, not query count alone."
        }
      ],
      "next": "Reuse existing sparse/LCU routes only after auditing their oracle contracts and compiler frontiers.",
      "lower_bound": {
        "status": "source-audit-pending",
        "task": "Port a pinned circuit-capacity/access lower bound with precision and ancilla assumptions intact.",
        "comparison_key": "setting:explicit-data-charged-access"
      },
      "benchmarks": [
        "Repeated structured blocks",
        "Sparse matrices with formulaic index/value access",
        "Amortized repeated-query workloads"
      ]
    }
  ],
  "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:sp-select-be",
      "label": "SP + SELECT gives an LCU block",
      "tails": [
        "family:state-action",
        "family:prepare-select"
      ],
      "heads": [
        "concept:block-encoding"
      ],
      "formula": "P^†\\operatorname{SELECT}(U)P\\rightsquigarrow A/\\alpha",
      "mechanism": "The AND node explicitly includes controlled operator access and unpreparation.",
      "hypothesis_map": "Square-root coefficient amplitudes, SELECT unitaries, adjoint access and phase convention.",
      "conclusion_map": "A specified projected matrix block, not merely one state.",
      "failure_boundary": "No arrow from an isolated prepared state to arbitrary A; state copies do not supply controlled U or U-dagger.",
      "source_ids": [
        "zhang-yuan-2024"
      ],
      "lean_refs": [],
      "status": "proposal",
      "review": "independent review pending"
    },
    {
      "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"
  ]
}