{ "schema_version": 1, "query": "", "route_id": "spw-gibbs", "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: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:mixing-certificate", "label": "Gibbs invariance versus mixing", "domains": [ "concept:probability", "concept:operator-theory" ], "tags": [ "Gibbs", "KMS", "Lindblad", "low-temperature", "mixing", "gap" ], "question": "Does the generator only preserve the target, or converge to it with a useful rate?", "formula": "\\mathcal L(\\rho_\\beta)=0\\quad\\not\\Rightarrow\\quad t_{\\rm mix}=\\operatorname{poly}(n,\\beta)", "mechanism": "Connect invariance, reversibility/detailed balance, coercivity and error contraction only through explicit model-specific hypotheses.", "assumptions": [ "Specified Hamiltonian, temperature and target state", "Primitive semigroup and quantitative convergence assumptions when used", "Implementation and mixing error separated" ], "proof_steps": [ "Prove the invariant-state equation.", "Identify the quantitative coercivity/mixing certificate still missing.", "Compose convergence with implemented-channel error." ], "lean_refs": [], "source_ids": [ "ding-li-lin-2025" ], "boundary": "This is a conceptual connection to Samplinglib, not a transfer of classical LSI results to arbitrary quantum generators." } ], "routes": [ { "id": "spw-gibbs", "title": "Low-temperature Gibbs preparation with quantitative mixing", "priority": 3, "status": "research-target", "setting_id": "setting:local-kms-generator", "formula": "\\rho_\\beta=\\frac{e^{-\\beta H}}{\\operatorname{Tr}(e^{-\\beta H})},\\quad \\|\\mathcal E_t(\\rho)-\\rho_\\beta\\|_1\\le\\epsilon", "goal": "For a restricted noncommuting local Hamiltonian family, prove a mixing certificate and implemented-channel error bound.", "motivation": "An invariant Gibbs state does not guarantee that an implementable sampler reaches it quickly at low temperature.", "input_model": "Specified Hamiltonian and KMS generator with charged simulation access, temperature, initial state and convergence norm.", "assumptions": [ "Primitivity/ergodicity or invariant-subspace restriction stated", "Spectral or functional-inequality mixing certificate supplied", "Generator implementation and statistical output contract specified" ], "target_bound": "A model-specific polynomial mixing and total-cost theorem in the promised temperature regime.", "known_boundary": "KMS detailed balance and stationarity are not polynomial mixing. Classical sampling inequalities cannot be transferred to quantum operators without a hypothesis map.", "families": [ "family:mixing-certificate", "family:charged-access", "family:matrix-contract" ], "source_ids": [ "ding-li-lin-2025" ], "lean_refs": [], "steps": [ { "id": "invariance", "target": "Port a finite-dimensional invariant-state identity.", "acceptance": "Exact generator, Gibbs normalizer and adjoint convention." }, { "id": "mixing", "target": "Identify and prove a coercivity or mixing condition for one class.", "acceptance": "An explicit rate with temperature/system-size dependence; no assumed conclusion." }, { "id": "channel-error", "target": "Compose mixing with generator-simulation error.", "acceptance": "A trace-norm output theorem and total implementation budget." } ], "next": "Use a finite-dimensional warm-up and then separate the high-temperature baseline from the intended low-temperature regime.", "lower_bound": { "status": "model-definition-pending", "task": "State the oracle and Hamiltonian class before importing hardness or mixing obstructions.", "comparison_key": "setting:local-kms-generator" }, "benchmarks": [ "Commuting warm-up", "Small noncommuting local families", "Explicit high-temperature comparison baseline" ] } ], "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" }, { "id": "transport:classical-quantum-mixing", "label": "A coercivity mechanism, not an equality of samplers", "tails": [ "concept:probability", "concept:operator-theory" ], "heads": [ "family:mixing-certificate" ], "formula": "\\text{invariance}+\\text{quantitative dissipation}\\Longrightarrow\\text{a declared convergence rate}", "mechanism": "Compare the role of a functional inequality in classical and quantum semigroups.", "hypothesis_map": "Different state spaces, noncommutativity, detailed-balance conventions and divergences remain explicit.", "conclusion_map": "A hypothesis-mapped conceptual mirror with candidate shared matrix/semigroup lemmas.", "failure_boundary": "Classical LSI or reversibility is not automatically a quantum KMS mixing certificate.", "source_ids": [ "ding-li-lin-2025", "samplinglib-protocol" ], "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" ] }