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Classical Limits of Spectral Filtering in Quantum Generative Models

Marco Roth

arXiv:2608.14169v1quant-phcs.LG

TL;DR

The paper asks whether coherent spectral filtering in quantum generative models offers more than classical sample post-processing. It compares both operations at matched sampling cost and finds that magnitude filters do not create a quantum-classical separation, while surviving separation comes from input spectral phases.

  • Problem

    The paper examines whether coherent spectral filtering can smooth quantum generative-model outputs while providing an advantage over classical post-processing.

  • Method

    The paper compares Fourier-diagonal filtering with convolution by a symmetric probability kernel, accounting for attenuation’s post-selection overhead through affordability and classicality criteria.

  • Results

    Magnitude filters obey a dichotomy: they either yield a constant-size efficiently classically sampleable Fourier object or widen the passband until they no longer smooth, creating no quantum-classical separation.

  • Takeaways & Limitations

    Within the diagonal family, any surviving separation is inherited from input spectral phases, while pure phase filters remain exempt from the magnitude-filter constraints.

  • Takeaways & Limitations

    The analysis is restricted to Fourier-diagonal filters, symmetric-kernel classical comparison, and distributions in the large-sampling limit.

Abstract

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Spectral filtering has been proposed as a route to regularization in quantum generative models: the quantum Fourier transform exposes the amplitude spectrum of a quantum circuit Born machine, and a diagonal filter suppresses the high frequencies associated with finite-sample noise, an operation whose classical counterpart seemingly requires manipulating an exponentially long amplitude vector. We examine whether this coherent operation produces anything that classical post-processing of samples from the unfiltered model cannot match. Measuring the filter against convolution with a symmetric probability kernel at matched sampling cost, which accounts for the post-selection overhead of attenuation, we derive necessary and sufficient conditions for the gap between the two to vanish. Magnitude (attenuating) filters obey a dichotomy: at a fixed affordability threshold, the filtered output is either a constant-size Fourier object with an efficient classical sampler, or the passband must widen until no fixed frequency is attenuated and the filter no longer smooths. In neither case does the filter create a quantum-classical separation. Whatever separation survives is inherited from the spectral phase of the input state. Numerical experiments on trained circuit Born machines confirm the classification and show that the deciding phases are invisible to the Born-rule training loss and set by the initialization. Within the diagonal family, pure phase filters remain the only spectral operations exempt from these constraints.

I. INTRODUCTION … B. Spectral Filtering

The paper tests whether coherent diagonal spectral filtering of quantum Born machines offers an advantage over classical smoothing of sampled outputs. It formalizes this comparison and identifies distinct constraints for magnitude and phase filters.

  • I. INTRODUCTION: Quantum generative models are motivated by potentially beyond-classical expressivity, representability, and sampling capabilities in exponentially large Hilbert spaces.Circuit Born machines define implicit distributions through the Born rule and are considered candidates for quantum advantage because generic sampling is believed classically intractable.
  • I. INTRODUCTION: The comparison gives a classical competitor only sampling access to the unfiltered distribution, using symmetric-kernel convolution at matched post-selection affordability.The coherent route applies a diagonal filter after the QFT, while the classical route measures first and post-processes samples.
  • I. INTRODUCTION: The framework measures the gap by the smallest total-variation distance between the coherently filtered distribution and any permitted classical smoothing.It proves conditions for equivalence and separates magnitude (attenuating) filters from phase filters.
  • I. INTRODUCTION: Magnitude filters obey a dichotomy: the output is either a constant-size Fourier object reproducible classically, or attenuation cannot smooth without widening the passband until no fixed frequency is attenuated.Any remaining separation is attributable to spectral phase already present in the input state; pure phase filters are exempt within the diagonal family.
  • A. Quantum Generative Models: The framework targets regularization because finite empirical samples spread Fourier weight into high-frequency components associated with noise, motivating their suppression.Quantum models prepare an n-qubit state with N = 2^n amplitudes and produce p(x; θ) = |a_x(θ)|^2 through the Born rule.
  • A. Quantum Generative Models: The analysis assumes access to a trained Born-machine state before measurement, enabling operations directly on the amplitudes before sampling.The model is trained to mimic a target distribution using a suitable loss such as forward Kullback–Leibler divergence.
  • B. Spectral Filtering: The QFT exposes the amplitude spectrum ˆa(k), whereas the distribution spectrum ˆp(m) is its autocorrelation, so coherent filtering acts through shifted pairwise spectral weights.A normalized diagonal filter maps ˆa(k) to g(k)ˆa(k), inserting lag-dependent weights inside the Wiener–Khinchin sum.

III. RESULTS · A. A Classicality Criterion

The paper tests whether coherent spectral filtering yields an operational advantage over classical smoothing by separating phase-induced mode rotation from classical kernel realizability. It finds that exact classicality requires both conditions in Proposition 1, while any remaining gap must be assessed against affordability and spectral complexity.

  • III. RESULTS: The coherent filter differs from its classical counterpart through complex reweighting and post-selection cost.These differences determine whether any apparent advantage survives accounting for sampling cost and spectral complexity.
  • A. A Classicality Criterion: The operational gap Φg is zero exactly when classical smoothing reproduces the coherently filtered output in the large-sampling limit.A positive Φg indicates that no symmetric probability kernel matches the coherent output.
  • A. A Classicality Criterion: The coherent residual measures the difference between coherent filtering and classical post-processing, while its phase-charged component attributes gaps to input spectral phases.A nonvanishing residual is sufficient but not necessary for Φg > 0, and vanishing residuals at every lag are necessary but not sufficient for Φg = 0.
  • A. A Classicality Criterion: Two residual-vanishing regimes are identified: a real ρm-weighted average deterministically, and spectrally incoherent inputs statistically at fixed affordability psucc ≥p⋆.For random phases, Im⟨Wm⟩ρm concentrates at zero and the phase-charged residual vanishes lag by lag in mean square.
  • A. A Classicality Criterion: Proposition 1 states that exact classicality holds if and only if two independent conditions are simultaneously satisfied.The criterion also requires ˆpg(m) = 0 at every degenerate lag and a real transfer function realizable by a symmetric probability kernel at regular lags.
  • A. A Classicality Criterion: The criterion’s first condition constrains the imaginary part of the ρm-averaged pair-weight, because classical smoothing rescales modes but cannot rotate them.The orthogonal component of the filtered mode relative to the input mode is carried entirely by the input phases through ρm.
  • A. A Classicality Criterion: The second condition constrains the real part: a real even envelope may fail to be the transfer function of any nonnegative symmetric kernel.This failure produces Φg > 0 independently of spectral phases, making it a classical realizability constraint.
  • A. A Classicality Criterion: Magnitude filters have real pair-weights, whereas phase filters have unit-modulus pair-weights; positive gap alone establishes separation from smoothing, not classical simulation.The paper therefore evaluates both filter families using affordability and spectral complexity, while noting that useful outputs must avoid efficient Fourier capture.

B. Affordability and Spectral Compactness

For magnitude-filter families, affordability and spectral compactness are governed by the passband scale σ: bounded σ yields constant-size Fourier descriptions, while broadband inputs force σ to diverge. Pure phase filters evade this compactness bound because their overlap profile is not square-integrable and filtering incurs no sampling cost.

  • Spectral compactness: Lemma 1 bounds the filtered output’s total-variation approximation using a number of Fourier modes independent of the grid size N = 2^n.The required mode count is controlled by the decay of the maximum pair weight with lag, formalized through the filter’s overlap profile.
  • Phase filters: Pure phase filters with |g(k)| = 1 have psucc = 1 and are exempt from the magnitude-filter compactness constraint.Their unit norm avoids additional sampling cost, while the non-square-integrable overlap profile permits a non-compact output spectrum.
  • Spectral compactness: A magnitude filter with bounded passband scale σ = O(1) produces an O(1)-mode output uniformly in n, hence a constant-size classical object.The compactness result does not apply to pure phase filters, whose overlap profile s ≡ 1 is not square-integrable.
  • Affordability: For broadband inputs, affordability psucc ≥ p⋆ forces the filter to transmit nonnegligible spectral mass beyond every fixed frequency window.Thus, suppressing the entire tail beyond any fixed cutoff drives psucc → 0, so an affordable magnitude filter cannot maintain an n-independent low-pass cutoff.
  • Affordability: For broadband spectra, affordability therefore requires σ⋆(n) → ∞, because a diverging passband eventually covers every fixed window and competes with regularization.Widening σ increases retained spectral mass and success probability but also widens the overlap range governing spectral compactness.

C. Dichotomy for Magnitude Filters

At matched sampling cost, magnitude filters obey a dichotomy: they either yield a constant-size classically samplable output or widen the passband until they stop smoothing. Any remaining separation is inherited from the input’s spectral phase, not created by the filter.

  • Phase certificate: For magnitude filters, Cg > 0 requires spectral phase in the input, while Cg = 0 identically for phase-trivial inputs because the filter window Wm is real.The certificate depends on the phase of ρm(k), not computational-basis phase; asymmetric probability distributions can have nonzero certificates without computational-basis phases.
  • Phase certificate: Under the realizability and degenerate-lag conditions of Proposition 2(iii), Φg = 0 if and only if Cg = 0.Thus the phase certificate exactly characterizes the gap under those additional conditions.
  • Dichotomy: Theorem 1 splits magnitude-filter families by the affordable passband scale σ⋆(n): bounded scale gives classical reproducibility, while unbounded scale prevents fixed-frequency smoothing.The split is determined by σ⋆(n), which depends on spectral magnitudes, whereas the separation branch is determined by the phase certificate Cg.
  • Classically reproducible: If σ⋆(n) = O(1), then M⋆ = O(1) uniformly in n and the filtered output is reproducible to TV ≤2ϵ by a poly(n)-time sampler from those Fourier coefficients.This holds for arbitrary input phases, regardless of Cg and Φg.
  • No smoothing: If the affordable passband is unbounded, the filter cannot implement a cutoff at any fixed frequency and therefore does not smooth.When Cg does not vanish, the surviving separation is carried by input spectral phase; when Cg vanishes, no single-lag observable separates the filtered output from classical smoothing.
  • Dichotomy: In neither dichotomy cell does the magnitude filter create a separation; it contributes only the real spectral window Wm and post-selection cost psucc.A gap may remain because of the input phase or because the output lies outside the smoothing class, but neither mechanism establishes a filter-induced separation.

D. Transfer to the hypercube

The spectral-filter classification transfers from Z_N to the hypercube because its core harmonic-analysis tools hold on finite abelian groups. For the Hadamard-transform setting, matched-cost magnitude filtering is either a low-degree classical object or provides no fixed-degree smoothing, while phase filters remain exempt.

  • Transfer to the hypercube: The Z_N framework transfers to the hypercube by replacing frequencies with the dual group and lag differences with group differences.Character orthogonality, the convolution theorem, Parseval’s theorem, and Bochner’s characterization remain valid on finite abelian groups.
  • Transfer to the hypercube: On Z_2^n, Hamming weight |k| replaces frequency, bitwise XOR k ⊕ m replaces lag, and g_η(k) = (1 − 2η)^|k| corresponds to an i.i.d. bit-flip kernel.The coherent and classical routes differ only through the window in the coherent expression.
  • Transfer to the hypercube: The phase certificate vanishes identically because hypercube characters are real, leaving no certified branch for either magnitude or phase filters.For every regular lag, R_g(m) = 0 because the relevant Fourier quantities and expectation are real.
  • Transfer to the hypercube: A low-degree input with A_w = 0 for w > w_0 and w_0 = O(1) affords constant flip rate and confines the output to degree |m| ≤ 2w_0.The resulting object has poly(n) coefficients and is samplable in poly(n) time; the confinement is inherited from the input.
  • Transfer to the hypercube: At matched cost, magnitude filtering on the hypercube is either a low-degree classical object or supplies no smoothing at any fixed degree.Any behavior beyond the bit-flip-kernel surrogate is inherited from the input rather than produced by the filter; phase filters |g| ≡ 1 remain the genuinely quantum option.

IV. EXAMPLES … C. Numerical Experiments

The examples show how spectral incoherence, Gaussian low-pass filtering, and input bandwidth determine residual behavior, mode-count scaling, and the applicable branch of the theoretical classification. Numerical experiments are then used to assess the analytical framework.

  • A. Spectrally incoherent input: Random-phase inputs have a flat expected spectrum, with E[|â|2] = 1/N, while success-probability gains require widening the filter passband.This places the example in the growing passband scale case.
  • A. Spectrally incoherent input: The coherent residual vanishes in expectation over random phases, while deviations are fluctuations bounded by pmax = max_x p(x).The bound is informative for delocalized magnitude profiles with pmax → 0.
  • A. Spectrally incoherent input: With fixed psucc ≥ p⋆ and pmax → 0, the phase-charged residual washes out asymptotically, Rg(m) → 0 in mean square at every regular lag.The flat expected spectrum alone does not imply this result; it follows from the magnitude-profile assumption.
  • B. Gaussian low-pass: For Gaussian low-pass filters, the mode count M⋆ depends on n only through σ, matching the scaling established in Sec. III B.The Gaussian profile provides a concrete magnitude-filter family for the preceding statements.
  • B. Gaussian low-pass: For Gaussian input spectra, broadband width l(n) → ∞ forces σ⋆ → ∞ and an unbounded M⋆, placing the family in cell (II) of Theorem 1.The threshold σ⋆ is attained at psucc = p⋆.
  • B. Gaussian low-pass: In the broadband case, phases decide the branch: coherent inputs are certified, phase-trivial inputs are uncertified with a vanishing certificate, and incoherent inputs retain their gap as speckle.These three phase classes correspond to the branches described for cell (II).
  • C. Numerical Experiments: The paper performs numerical experiments to gauge the validity of the analytical framework derived in Sec. III.This introduces the empirical section following the analytical examples.

1. Controlled Input Families · 2. Quantum Born Machines · V. DISCUSSION AND CONCLUSION

Controlled input families and quantum Born-machine experiments realize both classification cells: magnitude filtering either admits classical smoothing or retains a phase-dependent gap. In trained circuits, the gap is tied to spectral phases invisible to the Born-rule loss, while phase filters remain outside the dichotomy.

  • 1. Controlled Input Families: Broad-band Gaussian and spectrally incoherent inputs have σ⋆∝2^n, whereas coherent-chirp and phase-trivial inputs keep σ⋆ bounded.The families share a reflection-symmetric bimodal magnitude profile and differ only in spectral phase, except for the Gaussian-spectrum family.
  • 1. Controlled Input Families: The phase-trivial input has Φg = 0.039 despite an empty certificate, while the Gaussian-spectrum input has Φg = 0 up to solver residual (< 10^-9).The former lies in cell (I), whereas the latter lies in the uncertified branch of cell (II), showing that a zero certificate alone does not determine a vanishing gap.
  • 2. Quantum Born Machines: Trained and untrained quantum circuits both have exponentially growing σ⋆ and belong to the certified branch of cell (II).For trained circuits, the regular-lag certificate is flat in n, with fitted slope −0.04 ± 0.02, while the mode count grows proportionally to 2^n.
  • 2. Quantum Born Machines: Erasing phases collapses a trained n = 12 state to σ⋆ = M⋆ = 1 with certificate 10^-5, whereas restoring the phase profile leaves Φg ≈ 0.31.The phase-restored state remains in cell (II), with passband scale and mode count indistinguishable from the trained state.
  • 2. Quantum Born Machines: The trained filtered output is TV ≈ 0.31 from its target, fifteen times farther than the unfiltered model at ten times the sampling cost.Across identically trained seeds, the gap ranges from 0.14–0.55 at n = 12, reflecting initialization-dependent phase configurations.
  • V. DISCUSSION AND CONCLUSION: The conclusions are limited to Fourier-diagonal filters and convolution with symmetric probability kernels, with the detailed analysis on Z_N and only a sketched transfer to the hypercube.Mode-coupling spectral operations and arbitrary classical post-processing are outside the stated scope.
  • V. DISCUSSION AND CONCLUSION: Within the diagonal family, unit-modulus phase filters are the only spectral operations left outside the magnitude-filter dichotomy.They incur no post-selection cost, need not have a kernel surrogate, and escape the compactness bound; in studied circuits, their deciding phases are invisible to Born-rule training and inherited from initialization.

Appendix A: Framework … Appendix B: Simulability criterion

The appendices formalize symmetric-kernel classical post-processing, derive the filtered-state implementation and sampling overhead, and establish necessary and sufficient conditions for exact classical simulability. The criterion separates phase matching at each Fourier lag from realizability of the required symmetric-kernel transfer function.

  • Appendix A: Framework: The appendix frames the analysis around the framework introduced in Sec. II.It serves as supporting material for the paper’s main framework.
  • 1. Classical Competitor: A classical competitor samples x from p and u from a probability kernel K independently, then returns x + u mod N, implementing p ⋆ K by sample post-processing.The kernels are probability distributions on ZN, so this operation is available classically for every such K.
  • 1. Classical Competitor: Reflection-symmetric kernels define the smoothing baseline: their Fourier transfer functions are real and even, unlike general translation-equivariant post-processings.The appendix notes that symmetric kernels retain symmetric mixtures of deterministic translations and are the baseline used throughout.
  • 1. Classical Competitor: Against the larger translation-equivariant class, the median Φ+_g/Φ_g exceeds 0.93 for every input family with a nonzero gap, with no instance changing cell assignment.This supports restricting the main-text comparison to symmetric smoothing kernels.
  • 2. Derivation of Eq. (5): A diagonal spectral filter is block-encoded with one ancilla, and successful post-selection leaves amplitudes g(k)â(k)/√p_succ.The success probability p_succ is the retained spectral mass and heralding probability; repetition costs 1/p_succ preparations, or O(1/√p_succ) with amplitude amplification.
  • Appendix B: Simulability criterion: The simulability gap is Φ_g = inf_K TV(p_g, p ⋆ K), and compactness ensures that Φ_g = 0 exactly when some symmetric kernel reproduces p_g.The appendix uses norm equivalence on the finite group ZN to connect the total-variation formulation with its L2 analysis.
  • Appendix B: Simulability criterion: Exact simulability requires both zero filtered amplitude at every degenerate lag and real phase residuals at regular lags, together with a symmetric-kernel transfer function matching h⋆ on the support S.These conditions are necessary and sufficient: when they hold, a kernel K⋆ exists with ˆK⋆ = h⋆ on S and the L2 gap vanishes.

Appendix C: Spectral Compactness

The appendix shows that filtered distributions can be approximated using a number of Fourier modes controlled by the filter’s magnitude profile and overlap width. The decay of the filtered spectrum depends purely on the filter, with families g_σ having overlap width approximately 2σ.

  • Spectral compactness: The modulus of the filter bounds the Fourier support of the filtered distribution and determines how many modes describe it.This links spectral compactness directly to the filter’s magnitude profile.
  • Spectral compactness: The required mode count M⋆ is defined by approximating the filtered distribution with a truncated Fourier distribution.The analysis derives bounds for the number of modes needed to achieve the approximation.
  • Spectral compactness: The decay of the filtered spectrum is controlled purely by the filter’s overlap width, approximately 2σ for families gσ.The bound depends on the state only through the success-probability threshold psucc ≥ p⋆.

Appendix D: Proofs of Prop. 2 and Theorem 1

The proofs establish that magnitude filters either admit constant-size classical approximations or lose fixed-frequency attenuation as the passband widens. Any certified separation is instead sourced by the input state’s spectral phase, while vanishing certificates do not establish filter-induced coherence.

  • Assumptions: For stretched magnitude filters, the success probability is non-decreasing with passband scale, making the affordability threshold well posed.The filter modulus has the form |gσ(k)| = G(|k|/σ), with G continuous, non-increasing, and G(0) = 1.
  • Proposition 2: Prop. 2(i) bounds the required Fourier mode count uniformly in grid size and independently of input phases at the affordability threshold.The bound depends on the filter’s modulus overlap profile and the condition psucc ≥ p⋆, but not on N = 2^n.
  • Theorem 1: Cell (I): When σ⋆(n) = O(1), truncation, clipping, and renormalization produce an efficiently samplable distribution with TV(q, pg) ≤ 2ϵ.The retained trigonometric polynomial has O(1) modes, and its clipped regions can be computed in poly(n, log(1/δ)) time.
  • Theorem 1: Cell (II): If σ⋆(n) is unbounded, the stretched filter converges to the identity on every fixed frequency band, so fixed-frequency attenuation disappears.For fixed k and lag m, |gσ⋆(k)| and Wm(k) tend to 1 along the divergent subsequence.
  • Theorem 1: Branches: In the certified branch, the nonzero residual is carried entirely by spectral phases of the input, whereas vanishing certificates either permit classical smoothing or indicate that the filter has left the smoothing class.The filter contributes the real window Wm and post-selection cost, but not the imaginary component needed for a certified gap.

Appendix E: Spectral filtering on the hypercube

On the Boolean cube, the spectral-filtering framework extends with Walsh characters and automatic kernel symmetry, but its classification depends on the input’s degree structure. Low-degree inputs yield efficiently samplable filtered outputs, whereas broadband inputs force affordable filters to attenuate no fixed degree.

  • Hypercube framework: The Fourier identities and washout bound extend to finite abelian groups, with lag differences becoming bitwise XOR on the Boolean cube.On the cube, lag counting differs because distances have many lags, so the degree bound replaces the original filter-decay argument.
  • Hypercube framework: On the cube, kernel symmetry is automatic, and the real paired Walsh sums make the phase-filter certificate empty.For every diagonal filter and input state, the relevant expectation is real, so Rg(m) = 0 at every regular lag; Proposition 2(ii) therefore gives no information.
  • Low-degree inputs: For low-degree inputs with Aw = 0 for w > w0, every filtered output has Walsh degree at most 2w0 and is sampled bit by bit in poly(n) time.When w0 = O(1), any flip rate satisfying λ2w0 ≥p⋆ is affordable, giving η⋆= Ω(1); the exact Fourier confinement requires only O(n2w0) coefficients.
  • Broadband inputs: For broadband inputs, affordability forces η⋆(n) →0 and gη⋆(k) →1 at every fixed degree, so no fixed degree is attenuated.The intermediate regime remains unresolved because bounded η⋆ alone does not bound the degree of the filtered output.

Appendix F: Washout for spectrally incoherent input

For spectrally incoherent inputs with independent uniform phases, magnitude filters wash out the residual in expectation and at each fixed lag in mean square as the input spreads. This does not imply that the overall quantum-classical gap closes, whereas phase filters generally retain a residual.

  • Washout mechanism: Independent uniform input phases make the magnitude-filter residual vanish in expectation.The phase average satisfies E[e^{i(φx−φy)}] = δxy, and real pair-weights then give E[Rg] = 0.
  • Washout mechanism: At every fixed nonzero lag, Rg(m) → 0 in mean square as pmax → 0 with fixed psucc ≥ p⋆.The variance bound is uniform in m, and the m = 0 case is trivial because normalization forces Rg(0) = 0.
  • Scope of the bound: For phase filters, the residual does not generally vanish on the same incoherent input.Their pair-weights Wm(k) are complex rather than real, so the magnitude-filter cancellation does not apply.
  • Scope of the bound: The one-lag washout certificate does not establish that the full quantum-classical gap closes.Summing over Θ(N) regular lags can yield a residual of size N^-1/2 across every lag, while Eq. (23) constrains only one lag at a time.
  • Numerical illustration: At p⋆ = 0.1, the exact gap remained near 0.35 from n = 8 through n = 16 while the mode count increased from 35 to 8684.Across 32 phase seeds per size, Φg was 0.345, 0.355, 0.352, 0.354, and 0.354; the modulus bound fell from 0.115 to 0.013.

Appendix G: Gaussian example · Appendix H: Exact gaps for the numerical experiments

Appendix G derives the Gaussian overlap profile and associated tail-bound setup. Appendix H formulates exact linear programs for the numerical gaps and finds that trained circuits retain substantial gaps despite asymmetric classical kernels, with the reported bounds generally tight above.

  • Appendix G: Gaussian example: For the Gaussian filter g(k) = e−k2/2σ2, the overlap profile is s(u) = e−u2/4, hence s(u)2 = e−u2/2.The maximum overlap occurs at k = m/2, with supk |Wm(k)| = e−m2/4σ2.
  • Appendix G: Gaussian example: The Gaussian overlap expression supplies the input to a Chernoff tail bound valid for M ≥σ and its substitution into Eq. (C3).
  • Appendix H: Exact gaps for the numerical experiments: The exact gap is computed as a linear program because Eq. (9) minimizes a piecewise-linear function of K over a polytope.
  • Appendix H: Exact gaps for the numerical experiments: The programs optimize TV(p ⋆K, pg), with the final constraint selecting K and returning Φg; dropping it selects K+ and returns Φ+g.
  • Appendix H: Exact gaps for the numerical experiments: 1647 instances across n = 8–12 were solved exactly, covering constructed, trained, random-parameter, and phase-control circuit families.The O(N) program was affordable at these sizes.
  • Appendix H: Exact gaps for the numerical experiments: Dropping reflection symmetry yields no meaningful classical advantage, while the phase-trivial input has Φg = Φ+g = 0 within solver residual.The phase-trivial case is listed separately because its ratio is uninformative.
  • Appendix H: Exact gaps for the numerical experiments: Every trained circuit retains Φ+g ≥0.115, and asymmetric optimizers have median 1 2 P x |K(x) −K(−x)| = 0.78.The shift freedom is exercised but does not significantly improve performance.
  • Appendix H: Exact gaps for the numerical experiments: For trained circuits, Cg understates Φg by median factor 3.0, while the Gaussian-kernel bound exceeds it by median 0.06% and at most 3.3%.The chirp is an exception, with Φg = 0.515 against 0.449 and the remaining reported bound.
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