{ "schema_version": 1, "query": "", "route_id": "spw-walsh", "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:walsh-diagonal-phase", "label": "Walsh characters and diagonal phase synthesis", "domains": [ "concept:harmonic-analysis", "concept:approximation", "concept:state-preparation" ], "tags": [ "Walsh", "WSL", "parity", "diagonal", "zero-mode", "phase", "sparse" ], "question": "Can a short parity expansion replace a full table of diagonal phases?", "formula": "w_j(x)=(-1)^{\\langle j,x\\rangle_2},\\quad F_S=\\sum_{j\\in S}a_jW_j,\\quad e^{-i\\tau F_S}=\\prod_{j\\in S}e^{-i\\tau a_jW_j}", "mechanism": "A Walsh character is a diagonal Pauli-Z string once the source's bit-reversal convention is fixed. All strings commute, so each term becomes parity computation, a Z rotation and uncomputation. This makes a short function description into a phase operator; it does not by itself prepare its amplitude state.", "assumptions": [ "Real sampled function, an explicitly supplied Walsh coefficient list and a fixed dyadic grid/bit convention", "Sparsity and maximum Hamming weight are supplied or proved; differentiability is not sparsity", "Retain j=0: its global phase becomes a relative phase when the operator is controlled", "Approximation, coefficient computation, angle precision, connectivity and work-register cleanup are separately charged" ], "proof_steps": [ "Identify each diagonal entry of W_j with the source Walsh character; prove orthogonality and coefficient reconstruction under the selected ordering.", "Compose parity compute–rotate–uncompute into exp(-i tau a_j W_j), including the empty parity j=0.", "Use diagonal commutation to prove the product formula, then bound truncation in the specified norm.", "Only after phase synthesis closes, compose a separately verified interference/postselection consumer and charge success probability." ], "lean_refs": [ "QuantumBlockEncoding.ConcreteSemantics.applyVec_zeroKet" ], "source_ids": [ "zylberman-debbasch-2024-walsh", "zylberman-et-al-2025-diagonal", "pellini-ferrari-dacrema-2025-comment" ], "boundary": "Planned source formalization, not a local Walsh theorem. Existing UCRY or Python Walsh/Gray multiplexor code is not a proof of the Walsh Series Loader. Sparse Walsh support is not sparse computational-basis support.", "technical_card": { "id": "tl-walsh-diagonal-phase", "source_ids": [ "zylberman-debbasch-2024-walsh", "pellini-ferrari-dacrema-2025-comment" ], "statement": "An explicit real Walsh expansion supplies a commuting diagonal phase circuit; the controlled circuit must include its constant coefficient. The amplitude-loading consumer is a separate obligation.", "lean_decl": [], "lean_status": "obligation", "used_by": [ "spw-walsh", "spw-diagonal", "spw-controlled" ], "dependencies": [ "family:matrix-contract", "family:charged-access" ], "next_action": "Prove a bit-order-aware single Walsh-string matrix action, with j=0 and its controlled version, before proving truncation or loader complexity.", "tags": [ "Walsh", "zero-order", "controlled-phase", "planned-formalization" ], "failure_modes": [ "Deleting a global phase before controlling the circuit changes interference.", "Identifying Walsh integer order with physical little-endian wire order without the source adapter.", "Inferring polynomial loading of arbitrary data from a sparse phase expansion.", "Treating shared Walsh terminology as a checked reuse of the UCRY backend." ] } }, { "id": "family:diagonal-postselection", "label": "Diagonal filters, accepted branches and normalization", "domains": [ "concept:operator-theory", "concept:state-preparation", "concept:block-encoding", "concept:probability" ], "tags": [ "diagonal", "non-unitary", "Walsh", "postselection", "amplification", "space-time-accuracy" ], "question": "What does a cheap diagonal circuit really cost when used to prepare a state?", "formula": "D=\\operatorname{diag}(d_x),\\quad \\Pi U_D(|0\\rangle|\\psi\\rangle)=D|\\psi\\rangle/\\alpha,\\quad p=\\|D\\psi\\|_2^2/\\alpha^2", "mechanism": "A unitary diagonal changes phases but not probabilities. A non-unitary diagonal must be realized as an accepted branch of a larger unitary. Normalize only a nonzero branch, and account for repeated attempts or amplitude amplification. Walsh loading is one interference-based consumer, not identical to a general diagonal block encoding.", "assumptions": [ "Normalized input with a charged preparation/inverse supplier; total scale alpha>0 bounds max |d_x|; it equals source alpha_src times dmax", "Source V.1 covers signed real D; complex D needs the separate modulus–phase extension, a zero-entry phase convention and charged phase synthesis", "Chosen ancilla success sector, complex phases, register order and clean workspace are explicit", "Nonzero D psi and a quantitative lower bound on its norm for efficient normalized output", "Data-copy parallelism means coherent computational-basis fanout followed by uncomputation, not cloning an unknown quantum state", "Exact arbitrary tables can have exponential size; approximation savings need a function-class promise" ], "proof_steps": [ "Specify the exact projected-block identity before discussing normalized output.", "Compute the squared norm of the accepted branch and derive p; distinguish operator error from normalized-state error.", "For WSL, derive the actual interference branch -i(I-exp(-i tau F_S))|+^n>/2 before the small-angle approximation.", "Compile a declared sequential/parallel schedule, count workspace and cleanup, then charge all repetitions, reflections and inverses." ], "lean_refs": [ "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveProgram_cleanEntry" ], "source_ids": [ "zylberman-et-al-2025-diagonal", "zylberman-debbasch-2024-walsh", "lin-wiebe-2026-qasc" ], "boundary": "The linked cubic theorem is a fixed two-data-qubit diagonal witness, not an arbitrary-width Walsh/diagonal compiler or its complexity theorem. Small operator error alone does not control a nearly zero postselected state's error.", "technical_card": { "id": "tl-diagonal-postselection", "source_ids": [ "zylberman-et-al-2025-diagonal", "zylberman-debbasch-2024-walsh" ], "statement": "A diagonal block applied to a specified input yields a conditional normalized state with success ||D psi||^2/alpha^2. Space/depth/accuracy tradeoffs must be compared at matched input, error and success contracts.", "lean_decl": [], "lean_status": "obligation", "used_by": [ "spw-diagonal", "spw-walsh", "spw-envelope", "spw-fault-tolerant" ], "dependencies": [ "family:matrix-contract", "family:state-action", "family:charged-access", "family:walsh-diagonal-phase" ], "next_action": "Prove the generic projected-branch probability identity using existing finite-matrix objects; keep the paper's literal dilation and circuit schedule as subsequent source obligations.", "tags": [ "diagonal", "postselection", "success-cost", "planned-formalization" ], "failure_modes": [ "Quoting per-attempt depth as time to a successful prepared state.", "Confusing state infidelity with vector or operator norm error, or forgetting alpha_total when converting source Eq. (18) normalized-block error to the local unscaled-operator convention.", "Ignoring input-state preparation, reflections or inverse access in amplification.", "Claiming a deterministic system-only non-unitary gate.", "Calling a uniform-input cost bound a bound for every localized input.", "Promoting the source Fig. 4 / Eqs. (19)-(21) / Appendix B signal sector before reconciling its labels.", "Assuming III.2 integer rounding and VIII.1 size versus depth accounting have already been reconciled." ] } }, { "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-walsh", "title": "Walsh-series state preparation: phase is not amplitude", "priority": 2, "status": "research-target", "setting_id": "setting:real-walsh-infidelity-postselection", "formula": "p=\\frac{1}{N}\\sum_x\\sin^2(\\tau f_S(x)/2),\\quad N=2^n", "goal": "Formalize the source Walsh Series Loader, including its constant mode, normalized approximation and success-dependent complexity.", "motivation": "Short Walsh expansions can avoid enumerating all amplitudes, but a shallow phase circuit is only the first part of the loader.", "input_model": "A real sampled function on the source dyadic grid, explicitly supplied Walsh coefficients, n data qubits and a signal ancilla; charge classical coefficient generation and primitive rotations.", "assumptions": [ "Pin Walsh ordering and its map to physical wires", "Declare epsilon as target state infidelity, not operator-norm or vector error", "Retain the zero-order phase under control; f_S and its accepted branch must be nonzero", "Smoothness, Walsh sparsity, Hamming weight and norm/filling assumptions are distinct parameters" ], "target_bound": "Reproduce the pinned source's per-attempt and repeat-until-success/amplified resource bounds with all function-dependent constants and gate-model assumptions visible; no new speedup claimed.", "known_boundary": "The state after postselection is proportional to -i(I-exp(-i tau F_S))|+^n>, not exactly to f_S at finite tau. Increasing fidelity can reduce success probability. The zero-order-term implementation issue must be tested explicitly. Local staging now checks the actual H-controlled-diagonal-H-phase composition, constant relative phase and exact sine-squared accepted mass; independent source/semantic admission and approximation/resources remain open.", "families": [ "family:walsh-diagonal-phase", "family:diagonal-postselection", "family:state-action", "family:charged-access" ], "source_ids": [ "zylberman-debbasch-2024-walsh", "pellini-ferrari-dacrema-2025-comment" ], "lean_refs": [], "steps": [ { "id": "walsh-parity", "target": "Close exact Walsh-character and controlled-phase semantics.", "acceptance": "All basis entries, bit order and j=0 agree with the frozen source; constant functions and constant-plus-one-parity are regression cases." }, { "id": "accepted-state", "target": "Derive the actual ancilla branches and normalized approximation theorem.", "acceptance": "Success probability, zero branches, phases and infidelity/truncation errors are explicit; finite plots are not proofs." }, { "id": "total-cost", "target": "Compose phase synthesis with RUS or amplitude amplification.", "acceptance": "Report per-attempt and successful-preparation gates/depth separately, with coefficient preprocessing, inverses/reflections and finite-gate error charged." } ], "next": "Review the staged literal interference chain and constant-mode discriminator, then supply the Walsh parity/wire map and nonzero normalized/error/resource bridges; do not re-prove the already checked branch identity.", "lower_bound": { "status": "source-audit-pending", "task": "Compare sparse/smooth promised inputs with matched accuracy and success costs, never with a free arbitrary-vector table.", "comparison_key": "setting:real-walsh-infidelity-postselection" }, "benchmarks": [ "Nonzero constant function (zero-mode regression)", "Constant plus a single Walsh character", "Localized smooth function with a small filling fraction", "Bit-reversed two-qubit ordering discriminator" ] } ], "hyperedges": [ { "id": "transport:walsh-loader", "label": "Walsh phases + interference + nonzero branch", "tails": [ "family:walsh-diagonal-phase", "family:diagonal-postselection", "family:state-action", "family:charged-access" ], "heads": [ "concept:state-preparation" ], "formula": "-i(I-e^{-i\\tau F_S})|+^n\\rangle/2\\approx\\tau F_S|+^n\\rangle/2", "mechanism": "Combine a controlled diagonal phase supplier with ancilla interference, postselection and a normalized-error theorem.", "hypothesis_map": "Real function, source bit order, supplied coefficients, retained zero mode, explicit input and inverse access, small-angle/truncation bounds and nonzero accepted branch.", "conclusion_map": "An approximate amplitude state with a declared infidelity and success-dependent total cost, not an unconditional n-independent loader.", "failure_boundary": "Constant-term removal changes the controlled circuit; small filling fraction or poor sparsity can remove the advantage.", "source_ids": [ "zylberman-debbasch-2024-walsh", "pellini-ferrari-dacrema-2025-comment" ], "lean_refs": [], "status": "proposal", "review": "Source-planning edge; independent conceptual review pending; no local transport theorem" }, { "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: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" }, { "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" } ], "sources": [ { "id": "zylberman-debbasch-2024-walsh", "title": "Efficient Quantum State Preparation with Walsh Series", "authors": "Julien Zylberman; Fabrice Debbasch", "url": "https://arxiv.org/abs/2307.08384v3", "requested_version": "https://arxiv.org/pdf/2307.08384v3", "year": 2024, "status": "primary-text-checked", "anchor": "arXiv v3, 9 November 2023; checked 29 September 2026: Theorem 1, Corollary 1, Table I and Appendix B (B14–B19, B28).", "scope": "Walsh Series Loader: epsilon is state infidelity. Smooth fixed-function single-attempt depth O(epsilon^-1/2), size O(n+epsilon^-1/2), success Theta(epsilon); expected depth O(epsilon^-3/2), expected size O((n+epsilon^-1/2)/epsilon). Constants retain function/derivative/nonzero-mass assumptions. Sparse variants require supplied coefficients and Hamming-weight bounds. Amplification also charges reflections and inverses.", "formal_status": "planned-formalization" }, { "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": "pellini-ferrari-dacrema-2025-comment", "title": "Comment on arXiv:2307.08384: Efficient Quantum State Preparation with Walsh Series", "authors": "Riccardo Pellini; Maurizio Ferrari Dacrema", "url": "https://arxiv.org/abs/2502.05193v1", "year": 2025, "status": "primary-text-checked", "anchor": "arXiv v1 full HTML, Sections II.2–II.3, Eq. (11), Fig. 2: controlled zero-order term; checked 29 September 2026.", "scope": "Companion adversarial implementation review, not a refutation of the WSL theory. Retain the constant Walsh coefficient in a controlled phase circuit; use constant and constant-plus-parity regression cases.", "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": "mcardle-gilyen-berta-2026", "title": "Quantum State Preparation without Coherent Arithmetic", "authors": "Sam McArdle; András Gilyén; Mario Berta", "url": "https://journals.aps.org/prl/abstract/10.1103/ntvs-c48s", "year": 2026, "status": "primary-metadata-checked", "anchor": "Abstract; Physical Review Letters 136, 240603, published 18 June 2026", "scope": "QET-based function preparation with few ancillas. Approximation, normalization and success probability remain separate costs.", "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-tong-2020", "title": "Near-optimal ground state preparation", "authors": "Lin Lin; Yu Tong", "url": "https://quantum-journal.org/papers/q-2020-12-14-372/", "year": 2020, "status": "primary-metadata-checked", "anchor": "Abstract: initial overlap, spectral-gap promise, energy information and lower bounds", "scope": "Use the promised overlap/gap model; do not erase these costs in a generic strong-correlation claim.", "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" ] }