{
  "schema_version": 1,
  "query": "",
  "route_id": "spw-fault-tolerant",
  "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:gram-normalization",
      "label": "Local Gram normalization",
      "domains": [
        "concept:matrix-analysis",
        "concept:tensor-networks"
      ],
      "tags": [
        "normalizer",
        "Gram",
        "environment",
        "preprocessing",
        "norm"
      ],
      "question": "Can normalization be computed without summing exponentially many amplitudes?",
      "formula": "E_i=\\sum_{b=0}^1G_i[b]E_{i+1}G_i[b]^\\top,\\qquad Z=\\ell^\\top E_1\\ell",
      "mechanism": "Distribute the sum of squared contractions into local matrix products; compute the same finite norm from right to left.",
      "assumptions": [
        "Real cores in this local Hermite route; use conjugate transpose for complex cores",
        "A nonzero raw state and compatible boundary dimensions",
        "Core construction and finite-bit arithmetic are separately charged"
      ],
      "proof_steps": [
        "Expand the squared scalar contraction.",
        "Exchange finite sums and collect the two values of the next bit.",
        "Inductively identify the Gram environment with the exact suffix norm and take the positive square root."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.HermiteFiniteNorm.localSampleNorm_eq_sampleNorm",
        "QuantumBlockEncoding.HermiteFiniteNorm.norm_arithmetic_budget"
      ],
      "source_ids": [],
      "boundary": "An exact-real operation count is not a bit-complexity bound or a floating-point stability theorem."
    },
    {
      "id": "family:local-isometry",
      "label": "Canonicalization, completion and local compilation",
      "domains": [
        "concept:matrix-analysis",
        "concept:tensor-networks",
        "concept:state-preparation"
      ],
      "tags": [
        "QR",
        "LQ",
        "Givens",
        "isometry",
        "canonical",
        "unitary",
        "compile",
        "clean"
      ],
      "question": "Can a bounded-rank description be turned into an actual primitive list?",
      "formula": "V_i^† V_i=I\\quad\\Longrightarrow\\quad U_i(|a\\rangle|0\\rangle)=\\sum_{b,a'}(V_i)_{b,a',a}|a'\\rangle|b\\rangle",
      "mechanism": "Canonicalize without changing the contraction, absorb the signed scalar boundary, complete each local isometry, compile its actual orthogonal matrix and assemble physical wires.",
      "assumptions": [
        "Normalized scalar-boundary real TT",
        "Rank-deficient factors and signed boundary cases are handled",
        "All non-clean output sectors vanish at the final stage"
      ],
      "proof_steps": [
        "Factor local cores and pass the residual factor to the neighboring core.",
        "Prove preservation of every contracted amplitude and the local isometry identity.",
        "Complete to an orthogonal matrix, decompose into plane rotations and compile to RY/CX.",
        "Compose the stages and prove both clean output and primitive resource bounds."
      ],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveTensorTrainCompiler.compile_spec",
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec"
      ],
      "source_ids": [
        "holmes-matsuura-2020"
      ],
      "boundary": "The current symbolic resource theorem does not include a full stable finite-precision T-gate compiler."
    },
    {
      "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-fault-tolerant",
      "title": "Fault-tolerant resource trade-offs for structured preparation",
      "priority": 2,
      "status": "research-target",
      "setting_id": "setting:structured-clifford-t-connectivity",
      "formula": "\\mathcal R=(T,\\operatorname{Toffoli},d,q_{\\rm anc},Q_{\\rm oracle},\\epsilon)",
      "goal": "Compile the exact-real structured route into a finite gate set with a proved error budget and declared connectivity.",
      "motivation": "A small number of arbitrary real rotations is not a complete fault-tolerant resource estimate.",
      "input_model": "Finite-bit source data and a fixed Clifford+T gate set, connectivity graph, tolerance and ancilla policy.",
      "assumptions": [
        "Input/separation and conditioning promises exposed",
        "Each angle approximation and synthesis error budget explicit",
        "Count or bound the actual emitted primitive list"
      ],
      "target_bound": "A source-to-finite-bit theorem and a model-matched resource frontier; do not claim simultaneous global optimality of every coordinate.",
      "known_boundary": "The Hermite exact-real primitive theorem is already a useful substrate. Full bit complexity and stable angle generation are additional obligations.",
      "families": [
        "family:charged-access",
        "family:local-isometry",
        "family:gram-normalization"
      ],
      "source_ids": [
        "zhang-yuan-2024",
        "yuan-zhang-2023"
      ],
      "lean_refs": [
        "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec"
      ],
      "steps": [
        {
          "id": "input-bits",
          "target": "Freeze the finite-bit input and numerical-stability contract.",
          "acceptance": "All arbitrary-real comparisons and small pivots have explicit handling."
        },
        {
          "id": "stable-angles",
          "target": "Bound coefficient, canonicalization and angle errors.",
          "acceptance": "A normalized-state error theorem tied to supplied bit precision."
        },
        {
          "id": "logical-compiler",
          "target": "Compose finite-gate synthesis and connectivity routing.",
          "acceptance": "T/Toffoli/depth/ancilla accounting plus final state error and cleanup."
        }
      ],
      "next": "Treat the current Hermite bit-complexity boundary as the first local target rather than reopening its exact-real correctness proof.",
      "lower_bound": {
        "status": "source-audit-pending",
        "task": "Compare ancilla/depth trade-offs only in the same finite gate and connectivity model.",
        "comparison_key": "setting:structured-clifford-t-connectivity"
      },
      "benchmarks": [
        "Hermite baseline versus structured route at matched error",
        "Connectivity-constrained local TT compilation"
      ]
    }
  ],
  "hyperedges": [
    {
      "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: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"
  ]
}