{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-envelope",
  "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: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:spectral-filter",
      "label": "Block extraction and spectral filtering",
      "domains": [
        "concept:operator-theory",
        "concept:block-encoding",
        "concept:state-preparation"
      ],
      "tags": [
        "QSVT",
        "QET",
        "filter",
        "ground",
        "Hamiltonian",
        "success",
        "filling"
      ],
      "question": "What norm, overlap and polynomial approximation determine success?",
      "formula": "U_A|0\\rangle|\\psi\\rangle=|0\\rangle A|\\psi\\rangle/\\alpha+|\\perp\\rangle,\\quad p=\\|A\\psi\\|_2^2/\\alpha^2",
      "mechanism": "A projected block supplies an unnormalized desired branch. Normalize conditionally and explicitly account for amplification and polynomial-approximation errors.",
      "assumptions": [
        "A nonzero accepted branch",
        "Known normalization alpha and available inverses/controls as used",
        "Gap/overlap or filling assumptions for efficiency, not for mere definition"
      ],
      "proof_steps": [
        "Apply the clean projection to identify the accepted branch.",
        "Use Born's rule to derive its norm-squared probability.",
        "Bound approximation and success amplification under the declared access model."
      ],
      "lean_refs": [],
      "source_ids": [
        "mcardle-gilyen-berta-2026",
        "lin-tong-2020"
      ],
      "boundary": "Block encoding does not make normalized state preparation deterministic or uniformly cheap."
    },
    {
      "id": "family:envelope-reweight",
      "label": "Envelopes and coherent rejection",
      "domains": [
        "concept:probability",
        "concept:state-preparation"
      ],
      "tags": [
        "envelope",
        "reference",
        "precondition",
        "rejection",
        "localized",
        "Gaussian",
        "filling"
      ],
      "question": "Is a reference state closer to the target than the uniform state in a controllable sense?",
      "formula": "p_{\\rm succ}=\\frac{\\|f\\|_2^2}{C^2\\|g\\|_2^2}=\\kappa_{\\rm env}^{-2},\\qquad C\\ge\\max_{g_j\\ne0}|f_j/g_j|",
      "mechanism": "Prepare g, apply a controlled contraction f/(Cg) on its support, and retain the success branch. Costs of g, the ratio and amplification must all be paid.",
      "assumptions": [
        "support(f) subset support(g), with f/g defined as zero off support(g)",
        "C>0, nonzero f and g, and a verified pointwise domination bound",
        "Magnitude and complex phase of the ratio implemented coherently"
      ],
      "proof_steps": [
        "Write the reference amplitude as g_j/||g||.",
        "Multiply the accepted branch by f_j/(C g_j); the g_j factor cancels on support.",
        "Sum squared accepted amplitudes to obtain success probability, then normalize."
      ],
      "lean_refs": [],
      "source_ids": [
        "quantum-rejection-2013"
      ],
      "boundary": "Changing from probability sampling to amplitudes requires square roots and phase handling. An envelope with small kappa is not automatically easy to prepare."
    },
    {
      "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-envelope",
      "title": "Preconditioned function preparation with an explicit envelope",
      "priority": 1,
      "status": "research-target",
      "setting_id": "setting:coherent-envelope-ratio",
      "formula": "\\kappa_{\\rm env}=\\frac{C\\|g\\|_2}{\\|f\\|_2},\\quad p_{\\rm succ}=\\kappa_{\\rm env}^{-2},\\quad C\\ge\\max_{g_j\\ne0}|f_j/g_j|",
      "goal": "Construct g for one useful localized function class so that reference preparation, ratio implementation and amplification are all controlled.",
      "motivation": "Uniform-state postselection can have exponentially small filling fraction in dimension; a reference should match localization without hiding an equally hard preparation problem.",
      "input_model": "A clean U_g and U_g-dagger with charged construction; coherent access to a bounded complex ratio; classical or certified scalar C.",
      "assumptions": [
        "support(f) subset support(g); define the ratio as zero off support(g)",
        "f and g nonzero and C>0",
        "A constructive domination certificate, not an empirical maximum on a few grid points"
      ],
      "target_bound": "For a selected class, prove kappa_env and reference/ratio costs polynomial in declared class parameters; add a model-matched obstruction or lower bound.",
      "known_boundary": "Coherent rejection and reweighting are prior art. Uniform g recovers kappa_env=1/F2 where F2=||f||2/(sqrt(M)||f||infinity). A small ratio on average does not imply a valid pointwise envelope.",
      "families": [
        "family:envelope-reweight",
        "family:bounded-memory",
        "family:spectral-filter",
        "family:charged-access"
      ],
      "source_ids": [
        "quantum-rejection-2013",
        "mcardle-gilyen-berta-2026",
        "multivariate-2025"
      ],
      "lean_refs": [],
      "steps": [
        {
          "id": "success-identity",
          "target": "Formalize support-aware coherent reweighting and its exact success probability.",
          "acceptance": "All amplitudes including zero-reference entries and complex phases agree; normalizer nonzero."
        },
        {
          "id": "construct-envelope",
          "target": "Build a certified reference for a restricted localized class.",
          "acceptance": "A concrete supplier for g, a global domination proof and kappa bound."
        },
        {
          "id": "compose-cost",
          "target": "Add robust ratio approximation and amplification.",
          "acceptance": "State error and actual query/primitive costs, including U_g inversion and repeated calls."
        }
      ],
      "next": "First close the support-aware probability identity, then study anisotropic product envelopes and a controlled perturbation class.",
      "lower_bound": {
        "status": "source-audit-pending",
        "task": "Audit the quantum rejection-sampling query lower bound and prove whether its oracle assumptions match this structured envelope model before transferring it.",
        "comparison_key": "setting:coherent-envelope-ratio"
      },
      "benchmarks": [
        "Localized Gaussian profiles",
        "Boundary-layer initial data",
        "Piecewise payoff functions with declared phase convention"
      ]
    }
  ],
  "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: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:be-to-sp",
      "label": "BE + input + overlap gives a state branch",
      "tails": [
        "concept:block-encoding",
        "family:spectral-filter",
        "family:state-action"
      ],
      "heads": [
        "concept:state-preparation"
      ],
      "formula": "\\Pi U_A(|0\\rangle|\\psi\\rangle)=A|\\psi\\rangle/\\alpha",
      "mechanism": "Project the block, retain the accepted branch and account for success amplification.",
      "hypothesis_map": "Nonzero A psi, available input preparation and declared oracle inverses/controls.",
      "conclusion_map": "Normalized A psi conditional on success or an explicitly amplified approximation.",
      "failure_boundary": "Small overlap can dominate the cost; this is not an unconditional cheap reverse conversion.",
      "source_ids": [
        "mcardle-gilyen-berta-2026",
        "lin-tong-2020"
      ],
      "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"
  ]
}