{ "schema_version": 1, "title": "Mathematical mechanism atlas", "policy": "A curated mechanism graph over one generated Lean inventory. No conceptual edge is a Lean implication or certified functor. Repository graph additions are not automatic mathematical novelty.", "domains": [ { "id": "concept:matrix-analysis", "label": "Matrix analysis", "question": "What is preserved by a map? Norm, rank, adjoint and singular values." }, { "id": "concept:approximation", "label": "Approximation theory", "question": "Which basis or partition exposes finite-dimensional structure?" }, { "id": "concept:harmonic-analysis", "label": "Harmonic analysis", "question": "Can a derivative or convolution become a diagonal multiplier?" }, { "id": "concept:tensor-networks", "label": "Tensor networks", "question": "How much information must pass across each bit cut?" }, { "id": "concept:probability", "label": "Probability and sampling", "question": "What normalizer, overlap, envelope or mixing certificate is available?" }, { "id": "concept:complexity", "label": "Oracle and circuit complexity", "question": "Which input model, gate set, precision and resource are actually charged?" }, { "id": "concept:operator-theory", "label": "Operator and semigroup theory", "question": "What are the domains, invariant state and convergence assumptions?" }, { "id": "concept:state-preparation", "label": "State Preparation", "question": "Prepare a normalized state, coherently and with declared garbage/success sectors." }, { "id": "concept:block-encoding", "label": "Block Encoding", "question": "Realize a scaled operator as a specified clean projected block." }, { "id": "concept:verification", "label": "Verification and metrology", "question": "Separate formal circuit correctness from experimental state certification." } ], "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:hermite-bernstein", "label": "Hermite matching and Bernstein subdivision", "domains": [ "concept:approximation" ], "tags": [ "smooth", "polynomial", "Hermite", "Bernstein", "de Casteljau", "piecewise" ], "question": "Does a known function admit an exact low-degree local update?", "formula": "P_{2k+1}(t-1)=\\sum_{r=0}^{2k+1}c_r\\binom{2k+1}{r}t^r(1-t)^{2k+1-r}", "mechanism": "Preserve the literal Hermite polynomial while changing basis. Binary interval subdivision updates a fixed-size coefficient row, rather than enumerating samples.", "assumptions": [ "Degree d=2k+1 and all endpoint jets are fixed", "Subdivision coordinate and polynomial coordinate must be reconciled", "Positivity on [0,1] is separate from the identity on all real coordinates" ], "proof_steps": [ "Match derivatives at both endpoints; uniqueness follows because the difference has 2k+2 zeros counted with multiplicity.", "Expand the endpoint factors in the Bernstein basis and identify the coefficients.", "Apply de Casteljau restrictions along a bit path and reconcile its coordinate with the sample index." ], "lean_refs": [ "QuantumBlockEncoding.HermiteBernstein.sourceInterpolant_bernstein", "QuantumBlockEncoding.HermiteBernstein.sourceInterpolant_subdivision_readout" ], "source_ids": [ "holmes-matsuura-2020" ], "boundary": "This is exact representation of the chosen polynomial, not automatic spectral convergence of a PDE solver. The general function-to-MPS idea predates ASPBE." }, { "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: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: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:spectral-filter", "label": "Block extraction and spectral filtering", "domains": [ "concept:operator-theory", "concept:block-encoding", "concept:state-preparation" ], "tags": [ "QSVT", "QET", "filter", "ground", "Hamiltonian", "success", "filling" ], "question": "What norm, overlap and polynomial approximation determine success?", "formula": "U_A|0\\rangle|\\psi\\rangle=|0\\rangle A|\\psi\\rangle/\\alpha+|\\perp\\rangle,\\quad p=\\|A\\psi\\|_2^2/\\alpha^2", "mechanism": "A projected block supplies an unnormalized desired branch. Normalize conditionally and explicitly account for amplification and polynomial-approximation errors.", "assumptions": [ "A nonzero accepted branch", "Known normalization alpha and available inverses/controls as used", "Gap/overlap or filling assumptions for efficiency, not for mere definition" ], "proof_steps": [ "Apply the clean projection to identify the accepted branch.", "Use Born's rule to derive its norm-squared probability.", "Bound approximation and success amplification under the declared access model." ], "lean_refs": [], "source_ids": [ "mcardle-gilyen-berta-2026", "lin-tong-2020" ], "boundary": "Block encoding does not make normalized state preparation deterministic or uniformly cheap." }, { "id": "family:harmonic-lift", "label": "Fourier multipliers and Schrödingerisation", "domains": [ "concept:harmonic-analysis", "concept:operator-theory" ], "tags": [ "Fourier", "QFT", "PDE", "Schrodingerisation", "derivative", "smooth" ], "question": "Can a nonunitary evolution be represented as transport in an auxiliary coordinate?", "formula": "w_t=-A_1\\partial_pw+iA_2w\\quad\\longmapsto\\quad i\\partial_t\\widehat w=(\\eta A_1-A_2)\\widehat w", "mechanism": "A Fourier basis converts transport into a real-frequency Hermitian multiplier. Smooth extension improves a separate approximation problem; its sampled preparation is a reusable input supplier.", "assumptions": [ "A_1 and A_2 Hermitian with declared Fourier sign and register order", "Domain/boundary or finite discretization fixed", "Safe recovery region and truncation error are explicit" ], "proof_steps": [ "Split A into its Hermitian and anti-Hermitian parts.", "Differentiate the warped profile on its recovery region.", "Specify the Fourier transform and identify the derivative multiplier.", "Separate initial-state, discretization, simulation and recovery errors." ], "lean_refs": [], "source_ids": [], "boundary": "This is a candidate mathematical route, not an end-to-end PDE theorem inferred from Hermite state preparation." }, { "id": "family:envelope-reweight", "label": "Envelopes and coherent rejection", "domains": [ "concept:probability", "concept:state-preparation" ], "tags": [ "envelope", "reference", "precondition", "rejection", "localized", "Gaussian", "filling" ], "question": "Is a reference state closer to the target than the uniform state in a controllable sense?", "formula": "p_{\\rm succ}=\\frac{\\|f\\|_2^2}{C^2\\|g\\|_2^2}=\\kappa_{\\rm env}^{-2},\\qquad C\\ge\\max_{g_j\\ne0}|f_j/g_j|", "mechanism": "Prepare g, apply a controlled contraction f/(Cg) on its support, and retain the success branch. Costs of g, the ratio and amplification must all be paid.", "assumptions": [ "support(f) subset support(g), with f/g defined as zero off support(g)", "C>0, nonzero f and g, and a verified pointwise domination bound", "Magnitude and complex phase of the ratio implemented coherently" ], "proof_steps": [ "Write the reference amplitude as g_j/||g||.", "Multiply the accepted branch by f_j/(C g_j); the g_j factor cancels on support.", "Sum squared accepted amplitudes to obtain success probability, then normalize." ], "lean_refs": [], "source_ids": [ "quantum-rejection-2013" ], "boundary": "Changing from probability sampling to amplitudes requires square roots and phase handling. An envelope with small kappa is not automatically easy to prepare." }, { "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." }, { "id": "family:structured-verification", "label": "Structure-aware preparation and verification", "domains": [ "concept:verification", "concept:state-preparation", "concept:tensor-networks" ], "tags": [ "fidelity", "witness", "shadow", "measurement", "hardware", "noise" ], "question": "Can the preparation structure supply a cheap experimental witness?", "formula": "F(\\rho,|\\psi\\rangle)=\\langle\\psi|\\rho|\\psi\\rangle", "mechanism": "Keep kernel-checked ideal-circuit equality separate from noisy-hardware fidelity estimation. A structural witness needs an explicit measurement and sample-complexity theorem.", "assumptions": [ "Measurement access and noise model stated", "IID or non-IID assumptions explicit", "Confidence level, sample count and target description cost charged" ], "proof_steps": [ "State the ideal target and the observed channel separately.", "Derive an observable or witness for the chosen state class.", "Bound estimation error and confidence under the measurement assumptions." ], "lean_refs": [], "source_ids": [ "butterworth-2026-candidate" ], "boundary": "The candidate hardware paper is not primary-verified here; no reported numerical performance is used as a proved bound." } ], "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" }, { "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" }, { "id": "transport:prepare-verify", "label": "Preparation structure plus measurement contract", "tails": [ "family:state-action", "family:structured-verification" ], "heads": [ "concept:verification" ], "formula": "\\Pr[|\\widehat F-F|\\le\\epsilon]\\ge1-\\delta", "mechanism": "Use the state description to propose a measurement witness; formal circuit proof and statistical evidence remain separate.", "hypothesis_map": "Experimental measurement access, noise, sample model, epsilon and delta.", "conclusion_map": "Candidate low-sample verification protocol for a specified family.", "failure_boundary": "An ideal Lean proof is not a hardware fidelity certificate.", "source_ids": [ "butterworth-2026-candidate" ], "lean_refs": [], "status": "proposal", "review": "independent review pending" } ], "contributions": [ { "id": "contribution:hermite-polynomial", "label": "Hermite: dense tree to bounded-memory construction", "baseline": "129bc2ad38c8e82d259459607ef0aae601af3212", "head": "d638f725b7ed07fc471c25730cdf3bcfebe00510", "classes": [ "add-node", "shortcut", "reorganisation", "bridge" ], "families": [ "family:hermite-bernstein", "family:bounded-memory", "family:gram-normalization", "family:local-isometry" ], "before": "Generic amplitude tree retains one independent branch per sample; exponential gate count in the data-qubit count, not exponential ancillas.", "after": "Preserve the formula through Bernstein subdivision and bounded TT cores, then reuse a generic local-isometry compiler.", "invariant": "Same discrete normalized Hermite state and clean-output convention; exact-real primitive gate model.", "formula": "R=2k+6,\\quad q=\\lceil\\log_2R\\rceil,\\quad G\\le48mR^3", "lean_refs": [ "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec" ], "novelty_boundary": "Curated repository contribution. Function-to-MPS and sequential preparation are prior art; the metadata does not certify global novelty, optimality or a finite-bit speedup.", "review": "curated; independent topology/novelty review pending" } ] }