{ "schema_version": 1, "query": "", "route_id": "spw-no-qram", "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: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:prepare-select", "label": "PREPARE–SELECT–unprepare", "domains": [ "concept:state-preparation", "concept:block-encoding" ], "tags": [ "LCU", "SELECT", "block", "operator", "prepare", "sum" ], "question": "Are coefficient preparation and controlled operator access both available?", "formula": "(\\langle0|\\otimes I)(P^†\\otimes I)\\operatorname{SELECT}(U)(P\\otimes I)(|0\\rangle\\otimes I)=\\sum_a w_aU_a/\\alpha", "mechanism": "Prepare coefficients sqrt(w_a/alpha), select the corresponding unitary, and unprepare. Phases for complex coefficients must be assigned explicitly.", "assumptions": [ "Nonnegative weights w_a and alpha=sum w_a>0, or an explicit complex-phase adapter", "Implementable SELECT and adjoint PREPARE, not merely copies of a state", "Compatible data/ancilla layouts and normalization" ], "proof_steps": [ "Expand PREPARE on the clean ancilla.", "Apply SELECT branch by branch.", "Contract with the same coefficient preparation to obtain weighted matrix entries." ], "lean_refs": [], "source_ids": [ "zhang-yuan-2024" ], "boundary": "SP can be a BE ingredient, but one isolated state does not determine a general matrix. This explanatory bridge has no newly certified transport root." }, { "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." } ], "routes": [ { "id": "spw-no-qram", "title": "Data loading without a free QRAM assumption", "priority": 2, "status": "research-target", "setting_id": "setting:explicit-data-charged-access", "formula": "T_{\\rm total}=T_{\\rm build}+Q\\,T_{\\rm query}+T_{\\rm update}+T_{\\rm readout}", "goal": "Identify succinct data classes for which explicit access construction preserves the intended algorithmic advantage.", "motivation": "An algorithm with few abstract queries can still be expensive when the query unitary is built from real data.", "input_model": "Classical stored arrays, QROM, arithmetic formulas and physical QRAM are separate settings with explicit initialization, updates and memory costs.", "assumptions": [ "Fix data bit length, update frequency and reuse count", "Charge controlled/inverse queries when the algorithm uses them", "Compare against a classical algorithm with the same access assumptions" ], "target_bound": "A full cost theorem for one succinct data representation, not universal polylogarithmic loading of an arbitrary array.", "known_boundary": "General sparse-access and block-encoding constructions can require near-linear cost in matrix dimension. Sparsity does not make all entries free.", "families": [ "family:charged-access", "family:prepare-select", "family:matrix-contract" ], "source_ids": [ "zhang-yuan-2024", "yuan-zhang-2023" ], "lean_refs": [], "steps": [ { "id": "access-contract", "target": "Define the exact classical/quantum access interface.", "acceptance": "Separate data construction, coherent calls, memory and update costs." }, { "id": "succinct-supplier", "target": "Compile one formula, sparse-index or repeated-block supplier.", "acceptance": "Basis-action theorem and count of actual access primitives." }, { "id": "application", "target": "Reinsert access costs into a downstream algorithm.", "acceptance": "A comparison using the same input/output model and total cost, not query count alone." } ], "next": "Reuse existing sparse/LCU routes only after auditing their oracle contracts and compiler frontiers.", "lower_bound": { "status": "source-audit-pending", "task": "Port a pinned circuit-capacity/access lower bound with precision and ancilla assumptions intact.", "comparison_key": "setting:explicit-data-charged-access" }, "benchmarks": [ "Repeated structured blocks", "Sparse matrices with formulaic index/value access", "Amortized repeated-query workloads" ] } ], "hyperedges": [ { "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: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: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: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": "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" ] }