{ "schema_version": 1, "query": "", "route_id": "spw-controlled", "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:state-action", "label": "State-action and coherent preparation contracts", "domains": [ "concept:state-preparation" ], "tags": [ "state", "amplitude", "controlled", "family", "phase", "ancilla" ], "question": "Is the requested object one state, a coherent family, or a mixed state?", "formula": "U|0^m\\rangle|0^q\\rangle=|\\psi\\rangle|0^q\\rangle", "mechanism": "Fix all output sectors. A family also fixes relative phases between parameters; density-matrix trace error is a separate contract.", "assumptions": [ "Target normalization and nonzero source", "Relative phase convention for controlled families", "Success probability and garbage register behavior declared" ], "proof_steps": [ "Read the exact first-column predicate.", "Prove each target amplitude and every non-clean sector.", "For controlled families, prove a single coherent map rather than per-parameter rays." ], "lean_refs": [ "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_columns" ], "source_ids": [ "yuan-zhang-2023" ], "boundary": "Correctness up to a parameter-dependent global phase is not sufficient for a coherent family." }, { "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-controlled", "title": "Coherent parameter-dependent families", "priority": 1, "status": "research-target", "setting_id": "setting:joint-parameter-data-supplier", "formula": "U:\\sum_a c_a|a\\rangle|0\\rangle\\longmapsto\\sum_a c_a|a\\rangle|f_a\\rangle", "goal": "Produce one coherent template for jointly structured parameters and data with a uniform error bound.", "motivation": "Parameterized PDEs, source histories and Bayesian conditional states require superpositions of inputs, not separate classically compiled circuits.", "input_model": "Parameter a is a quantum register; one shared joint coefficient/core supplier is provided with a uniform norm bound.", "assumptions": [ "Relative phases are fixed across a", "Norms stay bounded away from zero on the promised parameter domain", "Branch selection and numerical rank changes are controlled coherently or avoided" ], "target_bound": "Polynomial dependence on the joint structural parameters and parameter bit length, rather than a table of independent preparations.", "known_boundary": "A family of individually easy states need not have an easy coherent supplier. Per-parameter correctness up to phase is insufficient.", "families": [ "family:state-action", "family:bounded-memory", "family:charged-access" ], "source_ids": [ "yuan-zhang-2023", "qkan-2026" ], "lean_refs": [], "steps": [ { "id": "phase-contract", "target": "Define the coherent family contract including phases and clean workspace.", "acceptance": "One matrix action for arbitrary parameter superpositions." }, { "id": "joint-structure", "target": "Derive a joint tensor/polynomial representation.", "acceptance": "Uniform rank/degree and nonzero normalization proof over all allowed parameters." }, { "id": "uniform-error", "target": "Compile with uniform error and cost.", "acceptance": "Operator-action error on the whole parameter-data subspace, not only sampled parameter values." } ], "next": "Use an affine parameterized Hermite or Gaussian family as a small exact model before time-dependent PDE source histories.", "lower_bound": { "status": "model-definition-pending", "task": "Keep arbitrary CQSP lower bounds separate from the restricted joint-structure promise.", "comparison_key": "setting:joint-parameter-data-supplier" }, "benchmarks": [ "Width-dependent Gaussians", "Parameter-dependent Hermite smoothing", "Time-dependent source terms" ] } ], "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: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: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" } ], "required_handoff": [ "frozen target and access model", "exact reused declarations", "bounded mathematical delta", "assumption differences", "independent round-trip evidence", "graph contribution and residual boundary" ] }