{ "schema_version": 1, "query": "", "route_id": "spw-verification", "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: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." }, { "id": "family:structured-verification", "label": "Structure-aware preparation and verification", "domains": [ "concept:verification", "concept:state-preparation", "concept:tensor-networks" ], "tags": [ "fidelity", "witness", "shadow", "measurement", "hardware", "noise" ], "question": "Can the preparation structure supply a cheap experimental witness?", "formula": "F(\\rho,|\\psi\\rangle)=\\langle\\psi|\\rho|\\psi\\rangle", "mechanism": "Keep kernel-checked ideal-circuit equality separate from noisy-hardware fidelity estimation. A structural witness needs an explicit measurement and sample-complexity theorem.", "assumptions": [ "Measurement access and noise model stated", "IID or non-IID assumptions explicit", "Confidence level, sample count and target description cost charged" ], "proof_steps": [ "State the ideal target and the observed channel separately.", "Derive an observable or witness for the chosen state class.", "Bound estimation error and confidence under the measurement assumptions." ], "lean_refs": [], "source_ids": [ "butterworth-2026-candidate" ], "boundary": "The candidate hardware paper is not primary-verified here; no reported numerical performance is used as a proved bound." } ], "routes": [ { "id": "spw-verification", "title": "Joint design of preparation and experimental verification", "priority": 2, "status": "research-target", "setting_id": "setting:structured-witness-measurements", "formula": "\\Pr[|\\widehat F-F(\\rho,|f\\rangle)|\\le\\epsilon]\\ge1-\\delta", "goal": "Automatically derive a low-sample fidelity witness for a precisely defined structured state class.", "motivation": "Full tomography is not the correct default output task; preparation structure may enable a cheaper witness.", "input_model": "Copies or channels producing rho, a specified measurement family, classical target representation and declared IID/noise assumptions.", "assumptions": [ "Target and implemented noisy state distinguished", "Witness computation and measurement basis-change cost included", "Confidence, bias and non-IID handling not silently omitted" ], "target_bound": "A witness correctness theorem, finite-sample bound and implementation cost for one state family.", "known_boundary": "A Lean ideal-circuit proof does not certify noisy hardware. The user-supplied 2609.08414 reference remains primary-source-unavailable in this ledger.", "families": [ "family:structured-verification", "family:bounded-memory", "family:charged-access" ], "source_ids": [ "butterworth-2026-candidate" ], "lean_refs": [], "steps": [ { "id": "measurement-contract", "target": "Specify the measurement and statistical experiment.", "acceptance": "Outcome law, confidence and noise assumptions are formal objects." }, { "id": "witness", "target": "Derive a structure-specific fidelity witness.", "acceptance": "Unbiasedness or certified one-sided bias and computable observable." }, { "id": "sample-cost", "target": "Prove sample and basis-change complexity.", "acceptance": "End-to-end confidence statement with no uncharged target tomography." } ], "next": "Verify the primary hardware reference first; start formal work from a small exactly analyzable MPS witness.", "lower_bound": { "status": "model-definition-pending", "task": "Distinguish lower bounds for full tomography, fidelity estimation and a promised structured witness.", "comparison_key": "setting:structured-witness-measurements" }, "benchmarks": [ "Product-state witness", "Small exact MPS with prescribed measurement access", "Structured function signals" ] } ], "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: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:prepare-verify", "label": "Preparation structure plus measurement contract", "tails": [ "family:state-action", "family:structured-verification" ], "heads": [ "concept:verification" ], "formula": "\\Pr[|\\widehat F-F|\\le\\epsilon]\\ge1-\\delta", "mechanism": "Use the state description to propose a measurement witness; formal circuit proof and statistical evidence remain separate.", "hypothesis_map": "Experimental measurement access, noise, sample model, epsilon and delta.", "conclusion_map": "Candidate low-sample verification protocol for a specified family.", "failure_boundary": "An ideal Lean proof is not a hardware fidelity certificate.", "source_ids": [ "butterworth-2026-candidate" ], "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" ] }