Classical Limits of Spectral Filtering in Quantum Generative Models

Marco Roth

quant-phAug 14, 2026

Classical Limits of Spectral Filtering in Quantum Generative Models

Authors: Marco Roth

Organizations: Fraunhofer Institute for Industrial Engineering IAO, Nobelstr. 12, D-70569 Stuttgart, Germany

Abstract

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.

Figures & tables

Figure 1: Overview of this work. (a) The two routes from the same quantum generative model Eq. ( 1 ). Coherent route (left): a QFT exposes the amplitude spectrum, a spectral filter attenuates it,

\hat{a}(k)\mapsto g(k)\hat{a}(k)

(dashed: Gaussian low-pass at matched cost

p_{\rm succ}\geq p^{\star}

), and measuring after the inverse QFT yields the filtered distribution

p_{g}

. Classical route (right): measuring first yields samples of

p(x)=\absolutevalue{a_x}^{2}

, which are convolved with a classical kernel

K

. The gap

\Phi_{g}

, Eq. ( 9 ), measures what the coherent route achieves beyond classical smoothing. The example shows the optimal kernel (

\mathrm{TV}=0.17

). (b) The resulting classification. At matched cost, a magnitude filter (

\absolutevalue{g(k)}\leq 1

) either returns an

O(1)

-mode object with an explicit kernel surrogate, or leaves a separation that is inherited from the spectral phase of the input rather than produced by the filter. Genuinely quantum spectral operations are phase filters (

\absolutevalue{g(k)}=1

), which incur no sampling cost but admit no kernel surrogate.

Figure 2: (a) Geometry of the coherent residual $R_{g}$ [Eq. ( 10 )] at a fixed lag $m$ . For a given input distribution $\hat{p}(m)$ , the coherent filter can rotate the modes $p_{g}(m)$ [Eq. ( 5 )], while classical smoothing can only rescale each mode by the real factor $\hat{K}(m)$ , resulting in $\hat{p}_{G}(m)$ [Eq. ( 4 )]. All modes reachable by classical smoothing are thus on a line $L_{m}=\{h\hat{p}(m):h\in\mathbb{R}\}$ through the origin. The best classical value $h^{\star}(m)\hat{p}(m)$ is the orthogonal projection of $\hat{p}_{g}(m)$ onto the line $L_{m}$ . The coherent residual $R_{g}$ is the perpendicular component, which cannot be removed by a kernel. The rotation of $\hat{p}_{g}(m)$ off $L_{m}$ is set by $\arg\langle W_{m}\rangle_{\rho_{m}}$ , with $R_{g}(m)\propto\mathrm{Im}\langle W_{m}\rangle_{\rho_{m}}$ . (b) Relationship between the filter Fourier transformed modes $\hat{a}(k)$ , the filter $g(k;\sigma)$ and the relevant probabilities $p_{\rm succ}$ and $p^{\star}$ . In this example, we have chosen a low-pass filter $|g(k;\sigma)|^{2}=\exp(-k^2/\sigma^2)$ (blue, solid) applied to a flat-topped spectrum of half-width $W$ , (black). Filtering the spectrum retains the fraction $p_{\mathrm{succ}}(\sigma)$ [Eq. ( 6 )] of the unit spectral mass $|\hat{a}(k)|^{2}$ , drawn as the shaded area under the window (blue). The affordable width $\sigma^{\star}$ [Eq. ( 13 )] is obtained as the smallest $\sigma$ with $p_{\mathrm{succ}}(\sigma^{\star})=p^{\star}$ . The resulting filter is shown in grey (dashed). Its window then captures exactly $p^{\star}$ (grey area). In the shown example $\sigma^{\star}>\sigma$ , such that $p_{\mathrm{succ}}<p^{\star}$ such that the filter is classified as unaffordable.

Figure 3: Input families for a Gaussian low-pass filter Eq. ( 24 ) at the affordable passband scale

\sigma^{\star}(n)

[Eq. ( 13 )] at

p^{\star}=0.1

. (a) The affordable passband scale for the constructed families and quantum circuits [cf. Sec. II.1 ] with random and with trained parameters. The closed form solution for Gaussian input spectra Eq. ( 27 ) is shown as dotted (black). (b) The phase certificate

C_{g}

[Eq. ( 19 )] as a function of the modes

M^{\star}

[Eq. ( 14 )] required to achieve an error of

\epsilon=0.05

. Each family is a track parametrised by

n

. The arrows point towards large

n

. For the coherent chirp and the spectrally incoherent input, the Gaussian-kernel upper bound on

\Phi_{g}

is shown as a dashed line with the enclosed bracket indicated as shaded areas. The spectrally incoherent family and the circuits are shown as median over seeds (solid) and the error bars are the interquartile range. The visual classification into cells is schematic and not rigorous. The dashed arrow (dark blue) marks the phase erasure control experiment and the dotted track with the hollow markers is the complementary experiment described in the main text.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

		$\Phi_{g}^{+}/\Phi_{g}$
input	instances	median	min
phase-trivial	$5^{\dagger}$	$1.000$	$1.000$
spectrally incoherent	$160$	$0.997$	$0.947$
coherent chirp	$5^{\dagger}$	$0.966$	$0.966$
Gaussian spectrum	$5^{\dagger}$	$\Phi_{g}=\Phi_{g}^{+}=0$
random-parameter circuits	$320$	$0.931$	$0.201$

Appendix

Table 1: Cost of dropping reflection symmetry, by input family, over all instances at

8\leq n\leq 12

. A dagger marks a deterministic family, with one instance per size.

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