{ "schema_version": 1, "query": "", "route_id": "spw-structured", "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: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-structured", "title": "High-dimensional structured function states", "priority": 1, "status": "research-target", "setting_id": "setting:structured-tt-supplied-cores", "formula": "|f\\rangle=\\frac{1}{\\|f\\|_{2,N}}\\sum_{\\mathbf j\\in[N]^D}f(\\mathbf x_{\\mathbf j})|\\mathbf j\\rangle,\\quad N=2^n", "goal": "Prove a constructive cost and state-error theorem for a precisely specified low-rank, sparse-frequency, mixed-smoothness or compositional class; do not equate these assumptions.", "motivation": "Multivariate PDE data, localized scientific functions and conditional distributions need more than generic amplitude loading.", "input_model": "An explicit formula-to-core supplier or a charged sparse coefficient oracle; grid, complex phase and a nonzero norm are part of the input.", "assumptions": [ "D and n specified; rank r and degree q are certified, not numerical fit labels", "Core/coefficients and normalizer are computably supplied", "Approximation error includes model compression, grid, arithmetic and primitive synthesis" ], "target_bound": "A candidate target is poly(D,r,q,n,log(1/epsilon)) cost under stated bit-length and stability promises; this bound is not claimed here.", "known_boundary": "Generic tensor-product degree-q expansions have (q+1)^D coefficients; smoothness alone does not remove dimensionality. Hermite's one-dimensional exact-real compiler is a substrate, not this multivariate theorem.", "families": [ "family:bounded-memory", "family:gram-normalization", "family:local-isometry", "family:charged-access" ], "source_ids": [ "holmes-matsuura-2020", "multivariate-2025", "qkan-2026" ], "lean_refs": [ "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec" ], "steps": [ { "id": "class-contract", "target": "Choose one function class and prove a uniform rank/degree certificate.", "acceptance": "A source-faithful class predicate and formula-to-core action theorem, including phases and supports." }, { "id": "error-supplier", "target": "Bound normalized-state error from core approximation and arithmetic.", "acceptance": "An explicit norm lower bound and a theorem for the exact requested vector norm; no uncharged condition number." }, { "id": "compiler", "target": "Compose the class supplier with the existing clean TT compiler.", "acceptance": "Primitive-list semantics, workspace cleanup, and every cost coordinate under the same model." } ], "next": "Start from the Hermite corridor; add a tensor-product or shallow coupled class with a proved rank bound before claiming general high dimension.", "lower_bound": { "status": "model-definition-pending", "task": "Separate arbitrary-vector counting barriers from lower bounds for the selected structured class. Specify which class parameters enter the hard family.", "comparison_key": "setting:structured-tt-supplied-cores" }, "benchmarks": [ "Product Gaussian with explicitly bounded widths", "Weakly coupled Gaussian with a proved rank estimate", "Hermite-smoothed PDE auxiliary profile" ] } ], "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" }, { "id": "transport:diagonal-filter-state", "label": "Diagonal block + prepared input + success contract", "tails": [ "family:diagonal-postselection", "family:state-action", "family:charged-access" ], "heads": [ "concept:state-preparation" ], "formula": "|\\psi\\rangle\\longmapsto D|\\psi\\rangle/\\|D\\psi\\|,\\quad p=\\|D\\psi\\|^2/\\alpha^2", "mechanism": "Specialize the existing BE-to-SP consumer to a diagonal filter; retain input preparation and successful-branch normalization as AND inputs.", "hypothesis_map": "Literal source dilation, bounded diagonal, known success sector, nonzero output, charged inverse/control and matched error metric.", "conclusion_map": "A conditional state and model-specific space–depth–accuracy/resource comparison.", "failure_boundary": "Neither a diagonal function description nor a compiled nearby fixed-size example certifies a scalable supplier; this is not SP-to-BE inclusion.", "source_ids": [ "zylberman-et-al-2025-diagonal", "lin-wiebe-2026-qasc" ], "lean_refs": [], "status": "proposal", "review": "Source-planning edge; independent conceptual review pending; no local transport theorem" }, { "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" } ], "sources": [ { "id": "zylberman-et-al-2025-diagonal", "title": "Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy trade-offs", "authors": "Julien Zylberman; Ugo Nzongani; Andrea Simonetto; Fabrice Debbasch", "url": "https://arxiv.org/abs/2404.02819v4", "requested_version": "https://arxiv.org/pdf/2404.02819v4", "year": 2025, "status": "primary-text-checked", "anchor": "arXiv v4, 21 January 2025; checked 29 September 2026: II.1, III.1–III.2, IV.1 Eqs. (15)–(16), V.1 Eqs. (19)–(21), VI A, VIII.1 and Tables I–II. Figure 4 signal-sector and III.2/VIII.1 accounting ambiguities remain explicit source obligations.", "scope": "Parallel scheduling trades ancillas for depth, not automatically total size. Table I uses spectral-norm error; Eq. (18) bounds normalized-block error delta, corresponding to total-scale times delta in the unscaled-operator convention. Exact-block success in source notation is ||D psi||^2/(alpha^2 dmax^2); alpha>1 controls the transformed-angle derivative in smooth approximation. No local source theorem or resolution of source ambiguities is claimed.", "formal_status": "planned-formalization" }, { "id": "holmes-matsuura-2020", "title": "Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions", "authors": "Adam Holmes; A. Y. Matsuura", "url": "https://arxiv.org/abs/2005.04351", "requested_version": "https://arxiv.org/html/2005.04351v1", "year": 2020, "status": "primary-metadata-checked", "anchor": "Abstract: explicit function to MPS to circuit; requested v1 HTML was unavailable in this audit", "scope": "Prior art for function-to-MPS preparation. The ASPBE Hermite specialization must not be credited with inventing polynomial-to-MPS or sequential MPS preparation.", "formal_status": "external-reference" }, { "id": "yuan-zhang-2023", "title": "Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits", "authors": "Pei Yuan; Shengyu Zhang", "url": "https://quantum-journal.org/papers/q-2023-03-20-956/", "year": 2023, "status": "primary-metadata-checked", "anchor": "Abstract and publication version notice: published arXiv v2; later v3 exists", "scope": "Baseline for arbitrary and controlled state preparation and ancilla/depth trade-offs. Pin a version and gate model before comparing bounds.", "formal_status": "external-reference" }, { "id": "zhang-yuan-2024", "title": "Circuit complexity of quantum access models for encoding classical data", "authors": "Xiao-Ming Zhang; Xiao Yuan", "url": "https://www.nature.com/articles/s41534-024-00835-8", "year": 2024, "status": "primary-text-checked", "anchor": "Results: circuit complexity lower bound; construction of LCU-based block-encoding; Methods: state preparation", "scope": "Explicit access construction is not a free oracle. PREPARE together with SELECT and uncomputation can supply a block encoding; a single prepared state alone does not determine an arbitrary operator.", "formal_status": "external-reference" }, { "id": "multivariate-2025", "title": "Quantum state preparation for multivariate functions", "authors": "See primary source", "url": "https://quantum-journal.org/papers/q-2025-04-09-1703/", "year": 2025, "status": "primary-metadata-checked", "anchor": "Abstract and article record", "scope": "Multivariate preparation baseline; a tensor-product coefficient count is not automatically polynomial in dimension.", "formal_status": "external-reference" }, { "id": "qkan-2026", "title": "QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation", "authors": "Petr Ivashkov; Po-Wei Huang; Kelvin Koor; Lirandë Pira; Patrick Rebentrost", "url": "https://www.nature.com/articles/s41534-026-01202-5", "year": 2026, "status": "primary-text-checked", "anchor": "Abstract, Introduction and Discussion: shallow compositional state preparation", "scope": "Composition-dependent bounds, not an efficient preparation theorem for every continuous multivariate function.", "formal_status": "external-reference" }, { "id": "quantum-rejection-2013", "title": "Quantum Rejection Sampling", "authors": "Maris Ozols; Martin Roetteler; Jérémie Roland", "url": "https://www.microsoft.com/en-us/research/publication/quantum-rejection-sampling/", "year": 2013, "status": "primary-metadata-checked", "anchor": "Authors' publication page: quantum state generation, query characterization and matching lower bound", "scope": "Prior art for coherent amplitude reweighting. Novelty must be in an explicit structured envelope, guarantees, implementation or model-matched bound, not in rejection sampling itself.", "formal_status": "external-reference" }, { "id": "lin-wiebe-2026-qasc", "title": "Quantum Algorithms for Scientific Computation", "authors": "Lin Lin; Nathan Wiebe", "url": "https://math.berkeley.edu/~linlin/qasc/live_notes_0429.pdf", "year": 2026, "status": "primary-text-checked", "anchor": "29 April 2026 public 447-page edition; PDF SHA256 56825000d53025dea4c2998dd6a061592b8fb2b9a446c3194c5f408183327599. First bounded source slice: Chapter 9 Section 9.1, printed pages 141–143, equations (9.2), (9.9), (9.10), (9.12); standing partial-application convention Definition 2.25, page 37. Parts I–IV remain curriculum anchors, not completed formalisation.", "scope": "Peer Part IV curriculum anchor, reusing existing ASPBE state-preparation/block-encoding APIs. New Chapter 9 adapters, operator-norm approximation, scientific constructions and resource certificates still require separate production admission. No whole-section/book completion is claimed.", "formal_status": "external-reference" } ], "required_handoff": [ "frozen target and access model", "exact reused declarations", "bounded mathematical delta", "assumption differences", "independent round-trip evidence", "graph contribution and residual boundary" ] }