Classical Limits of Spectral Filtering in Quantum Generative Models

Classical Limits of Spectral Filtering in Quantum Generative Models Marco Roth

arXiv:2608.14169v1 [quant-ph] 14 Aug 2026

Fraunhofer Institute for Industrial Engineering IAO, Nobelstr. 12, D-70569 Stuttgart, Germany∗ (Dated: August 17, 2026) 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

Using quantum computers to perform machine learning tasks is an intriguing idea that has received increasing attention in recent years [1–4]. Motivated by the potential beyond-classical capabilities that an exponentially sized Hilbert space offers, a lot of focus has been on trying to harness the non-classical properties of such a state space in terms of expressivity and representability [5–7]. While some clear signatures of quantum advantage have been identified [8–11], the vastness of the Hilbert space and its associated objects has proven to be a double-edged sword, and the very same properties that spawned the original interest lead to a curseof-dimensionality problem that obstructs the training of many quantum machine learning (QML) models of interest [12–14]. Recently, a line of research has focused on exploiting properties beyond expressiveness as a road to a potential quantum advantage. In this context, the quantum Fourier transform (QFT) has received renewed attention as a tool to design ML models [15–17]. It has been argued that the QFT enables spectral design operations whose classical counterparts appear to require manipulating an exponentially long amplitude vector and are therefore suggested to be classically prohibitive [18]. Viewed through the lens of Fourier analysis, the simplicity biases that underpin generalization acquire a concrete spectral form: smooth distributions have decaying Fourier spectra, so suppressing the high-frequency content of an empirical distribution acts as a regularizer that discards finite-sample artifacts [cf. Fig. 1 (a) for an illustration]. The natural model class for such spectral design

∗ marco.roth@iao.fraunhofer.de

is the quantum circuit Born machine [19, 20]. In these models, a quantum state is prepared by a (typically parameterized) circuit that defines an implicit generative model through the Born rule from which one can draw samples but not evaluate likelihoods. Born machines are among the leading candidates for a quantum advantage in generative modeling, since sampling from generic circuit families is believed to be classically intractable [21– 23]. Notably, the output distribution depends only on the magnitudes, while the phases are invisible to it. This is the basis of a recent proposal to design quantum generative models by filtering their spectrum coherently, before measurement [18]. In this work, we explore the necessary conditions for such an operation to bear a potential quantum advantage for generative quantum models. A key insight is that a classical competitor never needs the amplitude vector, only the measured distribution, to which it has sampling access. We make the comparison precise for diagonal spectral filters acting on the Fourier-transformed state. We measure the coherent filter against a classical competitor with the same interface, namely post-processing of samples from the unfiltered model by convolution with a probability kernel. We take the operational reading throughout. Two models are the same if no sampler can tell them apart at the same sampling cost, and a filter earns its coherence only if it produces an output that no classical post-processing of the unfiltered samples can match. We quantify the difference by a single quantity: the smallest total-variation distance between the coherently filtered distribution and any such classical smoothing. We prove the conditions under which a coherent filter operation is equivalent to classical smoothing. More generally, we find that it is instructive to separate magnitude (attenuating) filters from phase filters in less restricted

2 (a)

(b) magnitude filter at matched cost psucc ≥ p? : the input decides the cell P

(I) x ax |xi

QFT

measure

Cg 6→ 0

|ψi =

(II) arg â(k)

vacuous gap

−N/2

0

N/2

â(k) → g(k)â(k) †

p(x) = |ax |2

QFT measure

pg (x)

Cg → 0

(I)

inherited phase

(II)

pg = p ? K

uncertified

M ? = O(1)

M ? grows

?K

(p ? K)(x)

phase filters (|g| = 1): no passband, no washout, psucc = 1

Φg = inf K TV(p ? K, pg )

FIG. 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, â(k) 7→ g(k)â(k) (dashed: Gaussian low-pass at matched cost psucc ≥ p⋆ ), and measuring after the inverse QFT yields the filtered distribution pg . Classical route (right): measuring first yields samples of p(x) = |ax |2 , which are convolved with a classical kernel K. The gap Φg , Eq. (9), measures what the coherent route achieves beyond classical smoothing. The example shows the optimal kernel (TV = 0.17). (b) The resulting classification. At matched cost, a magnitude filter (|g(k)| ≤ 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 (|g(k)| = 1), which incur no sampling cost but admit no kernel surrogate.

settings. We show that magnitude filters are fundamentally restricted by a dichotomy. Either the filtered output is a constant-size Fourier object, or any quantumclassical separation is traceable to spectral phase already present in the input state. In the former case, the output distribution can be reproduced by a classical model directly. In the latter case, the filter merely windows at a post-selection cost. As a consequence, we find that a magnitude filter cannot perform smoothing at an affordable cost while leaving the spectral separation intact. Phase filters are the only spectral operations left that are genuinely quantum. The classification is illustrated in Fig. 1 (b).

In the remainder of this work, Sec. II sets up quantum generative models and coherent spectral filtering, and Sec. III develops a classicality criterion based on the coherent residual, the compactness of the filtered output, and the resulting dichotomy for magnitude filters. Section IV illustrates the two cells with spectrally incoherent input, a Gaussian low-pass, and numerical experiments on random and trained Born machines. We conclude in Sec. V. Proofs and technical details are deferred to the appendices.

II.

FRAMEWORK

In this section, we briefly introduce the concepts used throughout the manuscript. We start by explaining quantum generative models as the object of interest and discuss how quantum operations that shape the amplitudes of such models can be used to design the output distributions.

A.

Quantum Generative Models

Consider generative machine learning models, i.e., models that allow drawing samples from a distribution p(x), where the models are trained to mimic a target distribution ptarget . Creating models that learn the underlying distribution instead of overfitting to particular examples is one of the core problems in machine learning. Fundamentally, this results in a bias-variance tradeoff [24]. A standard approach to addressing this trade-off is to introduce a simplicity bias into the model by artificially constraining the hypothesis space [25]. In the following, we focus on achieving this regularization through manipulation of a model in Fourier space rather than operating on the feature space directly. This is motivated by the following insight: in practical scenar-

3 ios, we can only approximate the true target distribution ptarget using an empirical distribution that is obtained by collecting samples. The empirical distribution of a finite training set is typically sparse, concentrating all of its mass on the observed samples. This results in a Fourier spectrum that spreads across the entire frequency range, with appreciable weight on high-order components that reflect finite-sample noise rather than some underlying structure. Suppressing these high frequencies with a lowpass filter therefore acts as a smoothness prior by discarding the data-specific detail and returning a distribution that interpolates between the training points [18]. For quantum machine learning models, this is often taken to be promising, since the QFT allows for efficient access to an exponentially sized amplitude vector. Concretely, we consider a generative quantum machine learning model that is obtained by preparing an n-qubit quantum state |ψ(θ)⟩ =

N −1 X

ax (θ) |x⟩ = U (θ) |0⟩ ,

(1)

x=0

using a parametrized quantum circuit U (θ). Here ax ∈ C are the amplitudes of |ψ⟩ in the basis {|x⟩} with x ∈ {0, . . . , N − 1} and N = 2n is the number of computational basis states of |ψ⟩. From Eq. (1), we obtain the model distribution through the Born rule 2 p(x; θ) = |ax (θ)| . The resulting models are thus called quantum Born machines [19]. Learning the target distribution can be operationalized as minimizing a suitable loss function such as the forward Kullback–Leibler divergence L(θ) = DKL (ptarget ∥ p(· ; θ)). This is achieved by optimizing the parameters θ, e.g., using gradient descent. In this work, we assume access to a (trained) Born machine before measurement such that we can perform operations on the state of Eq. (1) directly. This will be made precise in the subsequent section.

to the spectrum of the amplitudes, â(k). The two are related by the Wiener–Khinchin theorem, which in the above conventions is free of prefactors, X p̂(m) = ρm (k) , (3) k

with the lag-m amplitude product ρm (k) ≡ â(k)â∗ (k − m). The spectrum of the distribution is thus the autocorrelation of the spectrum of the amplitudes, with the lag m playing the role of the shift between the two copies. For m = 0 this returns the normalization P 2 p̂(0) = k |â(k)| = 1. We are interested in delineating two scenarios: coherent manipulation of the amplitudes of Eq. (2) before measurement [left track in Fig. 1 (a)] and classical postprocessing operating on the output distribution p(x) obtained through sampling [right track in Fig. 1 (a)]. To decide whether the additional freedom of coherent access has any operational consequences, we must weigh it against what an experimenter with the same interface, i.e., sampling access to p(x), could already achieve. We thus first fix that classical baseline and then turn to its coherent counterpart. To realize the regularization motivated in Sec. II A, we model classical smoothing by linear, probabilitypreserving maps GK that are invariant under translation and reflection. The latter two symmetries define a smoothing filter without shifting. We argue why restricting the classical competition model to this class is appropriate in Appendix A. Under these requirements GK acts on the output distribution p(x) as a convolution with a symmetric probability kernel K, GK : p 7→ p ⋆ K [26], with a real, even transfer function K̂(m). By the convolution theorem this acts on the spectrum of p as a pointwise multiplication, so the filtered distribution has Fourier coefficients p̂G (m) = K̂(m) p̂(m) ,

B.

(4)

Spectral Filtering

In the following, we study spectral operations on the quantum Born machine. To this end, we apply a QFT to the state of Eq. (1), which yields the transformed amplitudes 1 X â(k) = √ ax ω −kx , (2) N x where ω ≡ exp(i2π/N ). Here we have omitted the dependence on the parameters θ for clarity. Note that for distributions we use the unnormalized transform fˆ(m) = √ P −mx , so that p̂(0) = 1. The 1/ N in Eq. (2) x f (x)ω is reserved for amplitudes, where the transform must be unitary. Two different spectra are now in play, and it is worth separating them at the outset. The regularization argument of Sec. II A is about the spectrum of the distribution, p̂(m), whereas the QFT of Eq. (2) gives access

i.e., GK rescales each mode m by the real, even factor K̂(m). Equation (4) is the result of classical post-processing. The coherent quantum analogues of convolution filters on the output distribution as given in Eq. (4) are spectral filters g : {−N/2, . . . , N/2 − 1} → C that are diagonal in the Fourier basis, i.e., g : â(k) 7→ g(k)â(k). In the following, we assume a normalized filter ∥g∥∞ = 1. As shown in Appendix A, the modes obtained after applying a filter, followed by an inverse QFT and subsequent measurement of the amplitudes, can be factorized as p̂g (m) =

1 psucc

X

Wm (k)ρm (k) ,

where Wm (k) ≡ g(k)g ∗ (k − m), and X 2 2 psucc = |g(k)| |â(k)| . k

(5)

k

(6)

4 Equation (5) reveals that the filter modifies the distribution via pairwise spectral weights Wm (k), the lag-m overlap of the filter with its own shifted copy. These spectral weights do not act on the amplitudes directly but rather on their autocorrelation ρm (k). The implications can be understood by reading Eq. (3) against Eq. (5): ρm (k) is the summand of the Wiener–Khinchin sum, and the filter inserts the weight Wm (k) inside that sum. Because the insertion happens under the sum and not in front of it, the induced action on p̂ is an average of Wm over the terms of the autocorrelation, rather than the pointwise multiplication of Eq. (4). In the following, we call a lag regular if p̂(m) ̸= 0 and degenerate if p̂(m) = 0, and we collect the regular lags in the set S ≡ {m : p̂(m) ̸= 0} .

(7)

We can use the previous definitions to write down the coherent equivalent of Eq. (4) for regular lags m ∈ S p̂g (m) =

⟨Wm ⟩ρm p̂(m) . psucc

(8)

P P Here, ⟨Wm ⟩ρm = = k Wm (k)ρm (k)/( k ρm (k)) P [W (k)ρ (k)]/p̂(m) is the ρ -weighted average of m m m k the pair weight Wm (k), the second equality being Eq. (3). Comparing Eq. (4) to Eq. (8) reveals two important distinctions between coherent manipulation of the spectrum and filtering after measurement in classical postprocessing: classical post-processing modifies the spectrum by a multiplication with a real scalar, whereas ⟨Wm ⟩ρm in Eq. (5) is complex, which allows for additional modification of the phases of p̂(m). The appearance of the normalization psucc in Eq. (6) has an operational interpretation: spectral attenuation (e.g., a lowpass filter) requires |g(k)| < 1 for some k. This is a nonunitary operation that requires ancillary measurements with post-selection on measurement outcomes with success probability psucc (see Appendix A). As a consequence, compared to sampling the non-filtered distribution with a given measurement budget of Nshots , filtering with |g(k)| < 1 and subsequent sampling results in only psucc Nshots effective samples. Note that more sophisticated schemes such as amplitude amplification [27] could increase the efficiency of the post-selection. To make the following analysis independent of the specific postselection method, we define an affordability threshold p⋆ in the following discussions (cf. Sec. III B). III.

RESULTS

These two differences, a complex rather than real reweighting and a post-selection cost, are the entire gap between the coherent filter and its classical counterpart. The rest of the paper is intended to make the implications of this difference concrete: first, whether the complex reweighting ever produces an output that no classical smoothing can reach, and second, whether any such

advantage survives once its cost and spectral complexity are accounted for. A.

A Classicality Criterion

To determine whether the additional freedom granted by the coherent filter has operational consequences, we introduce the gap
Φg ≡ inf TV p ⋆ K, pg , (9) K∈K

′ the P smallest ′total-variation distance, TV(z, z ) = 1 |z(x) − z (x)|, between the coherently filtered disx 2 tribution p̂g and any classical smoothing GK of the unfiltered samples. The minimization is carried out over the space K of symmetric probability kernels. In the following, we assume a large sampling limit in which shot noise is not a dominant factor that decides whether Φg vanishes or not. In this limit, we have Φg = 0 exactly when classical smoothing reproduces the coherent output. In this case the filter confers no advantage. A positive gap, however, witnesses an output no classical smoothing of the samples can reach. To identify the regimes with Φg > 0, we consider the coherent residual

Rg (m) = p̂g (m) − h⋆ (m)p̂(m) ,

(10)

i.e., the difference between the modes obtained through a coherent filter and classical post-processing. Here Φg is the operational quantity, deciding whether a classical smoothing exists, while Rg (m) is the per-lag diagnostic that certifies a gap and attributes it to the input phases. A nonvanishing residual is sufficient but not necessary for Φg > 0, and a residual vanishing at every lag is necessary but not sufficient for Φg = 0. Here, we define 2 h⋆ (m) = arg minh∈R |p̂g (m) − hp̂(m)| on regular lags m ∈ S as the classical weight that minimizes the squared difference between the coherently filtered mode and the mode obtained through classical post-processing. On degenerate lags, we define h⋆ (m)p̂(m) ≡ 0. As derived in Appendix B, the minimum can be computed in closed form, which for m ∈ S yields Rg (m) = i Im⟨Wm ⟩ρm

p̂(m) . psucc

(11)

From this we determine two cases where the coherent residual can vanish. First, deterministically, when the ρm -weighted average ⟨Wm ⟩ρm is real. Second, statistically, when the input is spectrally incoherent: there Im⟨Wm ⟩ρm concentrates at zero over the random phases and the phase-charged residual vanishes lag by lag in mean square at fixed affordability psucc ≥ p⋆ . We formalize the first case in the following proposition and return to the second case later. Proposition 1 (Simulability criterion). The coherent filter is exactly classical smoothing and Φg = 0, if and only if two independent conditions both hold:

5

(a)

(b)

arghWm iρm

Squared Magnitude (a.u.)

Im p̂g (m) Rg (m) p̂(m) h=1

Lm

Re p̂G (m) = h p̂(m) ?

−N 2

|â(k)|2 |g(k; σ ? )|2 |g(k; σ)|2 psucc p?

−W

0

Frequency k

W

N 2

FIG. 2. (a) Geometry of the coherent residual Rg [Eq. (10)] at a fixed lag m. For a given input distribution p̂(m), the coherent filter can rotate the modes pg (m) [Eq. (5)], while classical smoothing can only rescale each mode by the real factor K̂(m), resulting in p̂G (m) [Eq. (4)]. All modes reachable by classical smoothing are thus on a line Lm = {hp̂(m) : h ∈ R} through the origin. The best classical value h⋆ (m)p̂(m) is the orthogonal projection of p̂g (m) onto the line Lm . The coherent residual Rg is the perpendicular component, which cannot be removed by a kernel. The rotation of p̂g (m) off Lm is set by arg⟨Wm ⟩ρm , with Rg (m) ∝ Im⟨Wm ⟩ρm . (b) Relationship between the filter Fourier transformed modes â(k), the filter
g(k; σ) and the relevant probabilities psucc and p⋆ . In this example, we have chosen a low-pass filter |g(k; σ)|2 = exp −k2 /σ 2 (blue, solid) applied to a flat-topped spectrum of half-width W , (black). Filtering the spectrum retains the fraction psucc (σ) [Eq. (6)] of the unit spectral mass |â(k)|2 , drawn as the shaded area under the window (blue). The affordable width σ ⋆ [Eq. (13)] is obtained as the smallest σ with psucc (σ ⋆ ) = p⋆ . The resulting filter is shown in grey (dashed). Its window then captures exactly p⋆ (grey area). In the shown example σ ⋆ > σ, such that psucc < p⋆ such that the filter is classified as unaffordable.

(a) Phase alignment. Rg (m) = 0 at every lag. Equivalently ⟨Wm ⟩ρm ∈ R at every regular lag and p̂g (m) = 0 at every degenerate lag. (b) Envelope realizability. The real envelope h⋆ (m) = Re⟨Wm ⟩ρm /psucc is the transfer function of a symmetric probability kernel P for regular lags, i.e. there exists K ≥ 0 with x K(x) = 1 and K̂(m) = h⋆ (m) , ∀m ∈ S. Proposition 1 locates the entire quantum content of a spectral filter in a single scalar per lag, the ρm -averaged pair-weight ⟨Wm ⟩ρm . The two conditions constrain different parts of ⟨Wm ⟩ρm . Condition (a) constrains its imaginary part: the classical side can rescale p̂(m) but not rotate it, so by Eq. (11), the residual is the component of p̂g (m) orthogonal to p̂(m), carried entirely by the input phases through ρm . Figure 2 (a) visualizes the relationship between the filtered distributions and the coherent residual. Condition (b) constrains its real part: a real even envelope normalized to h⋆ (0) = 1 need not arise from any nonnegative kernel, and when it does not, Φg > 0 even though (a) holds and every residual vanishes. This is a purely classical realizability constraint with no dependence on the phases. The elementary classical case of the low-pass filter (idealized constant on suppρm ) satis-

fies both: a ρm -constant pair-weight Wm (k) ≡ w(m) ∈ R on the support of ρm aligns every lag, and the canonical low-pass profiles induce a positive-definite h⋆ . To turn Prop. 1 into a classification of actual filters, we write the pair-weight in polar form using g(k) = |g(k)|eiθ(k) , which results in Wm (k) = |g(k)| |g(k − m)| ei[θ(k)−θ(k−m)] .

(12)

Based on this decomposition, we define two archetypes: magnitude filters (θ ≡ 0), whose pair-weight is real, and phase filters (|g| ≡ 1), whose pair-weight has unit modulus. A positive gap establishes separation from smoothing but not from classical simulation. Additionally, a useful quantum filter should be affordable and its output should not be captured by, e.g., efficient Fourier analysis. In the following, we discuss both the affordability and the spectral complexity of magnitude and phase filters. B.

Affordability and Spectral Compactness

To make the following discussion more concrete, we focus on filters that straightforwardly achieve a highfrequency regularization as discussed in Sec. II B: oneparameter magnitude-filter families gσ , with low-pass

6 profiles in which a single scale σ > 0 sets how aggressively high frequencies are attenuated. We call σ the passband scale, where the passband is the band of frequencies transmitted with appreciable weight. We have |gσ (k)| of order unity for |k| ≲ σ and |gσ (k)| decaying for |k| ≳ σ. As discussed in Sec. II B, magnitude filters with |g(k)| < 1 result in a reduced success probability psucc < 1.1 As this implicitly imposes a cost on the number of measurements needed to create samples, we compare filters only at a fixed affordability floor psucc ≥ p⋆ , in the following denoted as matched-cost constraint, which puts the quantum filter and its classical competitor on the same sampling budget. For the filtering families gσ , we define the affordable passband scale σ ⋆ (n) = min{ σ : psucc (σ) ≥ p⋆ } .

(13)

It is the smallest σ with psucc ≥ p⋆ . A visualization of the involved quantities is shown in Fig. 2 (b). Given a family of filters gσ as defined above, we now show that the mode count of the filtered output is set by the passband scale and not by the grid size N = 2n . Whether this results in a classically substitutable object is decided by how fast σ must grow with n for the filter to remain affordable. To this end, we define the smallest number of modes required to reproduce p̂g to accuracy ϵ as 
M ⋆ (pg ; ϵ) = min M : TV pg , p≤M ≤ ϵ . (14) g Here, we have introduced the filtered output truncated to its 2M + 1 lowest frequencies p≤M (x) = g

1 X p̂g (m) ω mx . N

(15)

|m|≤M

Using these definitions, we can state a bound on M ⋆ formalized in the following Lemma. Lemma 1 (Spectral compactness collapse). Let {gσ } be a family with passband scale σ, i.e., a filter family whose
overlap obeys supk |gσ (k)| |gσ (k − m)| ≤ s(|m|/σ) for all m, with a fixed, non-increasing, square-integrable overlap profile s : [0, ∞) → [0, 1]. Then for every input state with psucc ≥ p⋆ , the filtered output pg can be truncated to its |m| ≤ M ⋆ Fourier modes with total-variation error at most ϵ, where  2(p⋆ ϵ)2  M ⋆ (pg ; ϵ) ≤ σ T −1 , σ

T (a) :=

Z ∞

s(u)2 du,

The proof is in Appendix C and relies on bounding |p̂g (m)| by the supremum of the pair-weight Wm (k) using Cauchy-Schwarz followed by converting the high-lag tail into a TV bound using Parseval’s theorem. The bound for M ⋆ is made explicit for the example of a Gaussian low-pass filter in Sec. IV B. How many modes are needed to approximate the filtered distribution to an error ϵ is governed entirely by how fast the maximum pair weight decays with the lag m. For a family gσ this decay is not arbitrary: enlarging σ simply stretches the filter along the frequency axis, so the overlap at lag m and passband scale σ depends only on the ratio m/σ. In the hypothesis of Lemma 1 this is captured by the overlap-profile s as a single, σindependent quantity which is the universal shape of the overlap when lag is measured in units of the passband scale. This factors out σ and isolates the scale from the shape. From Lemma 1 it is apparent that a magnitude filter with a bounded passband scale (σ = O(1)) produces an O(1)-mode output uniformly in n, a constant-size classical object. A pure phase filter (|g| ≡ 1) has overlap profile s ≡ 1, which is not square-integrable. Consequently, the bound does not apply, there is no passband restriction, and the output spectrum can be non-compact. Widening the passband raises the success probability Eq. (6), since more spectral mass is retained, but it also widens the overlap width, i.e., the range of lags m on which gσ (k) and gσ (k − m) are simultaneously appreciable. Affordability therefore pushes σ up while compactness pushes it down. We can make this tension more concrete for the important class of broadband spectra. Call a family of inputs indexed by n broadband if its spectral mass escapes every fixed window, i.e. X 2 |â(k)| → 0 (n → ∞) , (16) |k|≤L

for each fixed L. Splitting the sum Eq. (6) at |k| = L, P 2 and using |g| ≤ 1 inside the window and k |â(k)| = 1 outside it, gives X 2 2 psucc ≤ |â(k)| + sup |g(k)| . (17) The first term is all the filter can collect from the fixed window, and on a broadband input it vanishes: the retained mass must come from outside the window. Affordability psucc ≥ p⋆ therefore forces

a

sup |g(k)|

n

|k|>L

is a bound independent of the grid size N = 2 .

1 In contrast, pure phase filters with unit norm |g(k)| = 1, p

succ = 1 such that the filtering operation does not incur an additional sampling cost.

|k|>L

|k|≤L

2

≥ p⋆ − o(1) ,

(18)

for every fixed L. That is, however wide the window, some √ frequency beyond it is still transmitted with amplitude p⋆ up to o(1). An affordable magnitude filter on a broadband input can thus not be a low-pass with an n-independent cutoff: suppressing the whole tail |k| > L to o(1) at fixed L sends psucc → 0. For a filter family gσ

7 the same conclusion is read off σ ⋆ (n) → ∞ directly, since a diverging passband scale eventually covers any fixed window. Regularization and affordability are competing for broadband spectra.

C.

Dichotomy for Magnitude Filters

Two independent mechanisms are now in place: Lemma 1 bounds the size of the filtered output through |g| alone, and the coherent residual Eq. (11) fixes the direction in which that output leaves the classical set. In this section, we combine these two with the affordability threshold set by the affordable scale σ ⋆ (n) of Eq. (13), to establish a classification for magnitude filters. To this end, it is useful to observe that the gap Φg is lower bounded by the coherent residual as Φg ≥ Cg ≡ 21 max |Rg (m)| . m∈S

(19)

We call the quantity Cg the phase certificate, the derivation is shown in (Appendix D) and requires no hypothesis on g, on the input, or on the realizability of any envelope. For a magnitude filter Wm is real, so by Eq. (11) the certificate is nonzero only if ρm (k) = â(k)â∗ (k − m) carries phase. It is therefore a statement about the phases of the input states rather than about the filter. We thus turn our attention to the input phases. Definition 1 (Input phases). Fix a magnitude-filter family at matched cost. A family of inputs indexed by n is • phase-trivial if â(k) = eiα ω tk rk for some α ∈ R, t ∈ 21 ZN and real rk , i.e. if the spectrum is real up to a global phase and a (possibly half-integer) translation of the state. p • spectrally incoherent if ax = p(x) eiφx with φx independent and uniform. • spectrally coherent if Cg ̸→ 0. We stress that phase-triviality constrains arg â(k) and not arg ax : computational-basis phase erasure is not spectral phase erasure. Real nonnegative amplitudes only make the spectrum Hermitian, â(−k) = â(k)∗ , which is real up to a translation precisely when p is reflection-symmetric. For an asymmetric p the certificate is generically nonzero even though the state carries no computational-basis phase at all. Using these definitions, we formulate the following proposition. Proposition 2 (Phases and magnitudes at matched cost). Fix an accuracy ϵ > 0, an affordability threshold p⋆ ∈ (0, 1), and a magnitude-filter family {gσ } [θ ≡ 0 in Eq. (12)] with an overlap profile in the sense of Lemma 1. Let σ ⋆ (n) be the affordable passband scale Eq. (13) and let pg be the output of gσ⋆ (n) .

(i) σ ⋆ (n) is a functional of |â(k)| alone, and the mode count bound Eq. (14) is independent of N = 2n and of the input phases. In particular σ ⋆ (n) = O(1) forces M ⋆ = O(1) uniformly in n. (ii) At every regular lag m ∈ S, Rg (m) ̸= 0 ⇐⇒ Im⟨Wm ⟩ρm ̸= 0, and Wm being real, Im⟨Wm ⟩ρm is sourced solely by the phase of ρm (k). Hence Cg > 0 only if the input carries spectral phase, and Cg = 0 identically on phase-trivial inputs. (iii) If in addition the induced envelope h⋆ = Re⟨Wm ⟩ρm /psucc is realizable [condition (b) of Prop. 1] and p̂g (m) = 0 at every degenerate lag, then X 1/2 2 Cg ≤ Φg ≤ 12 |Rg (m)| , (20) m∈S

and in particular Φg = 0 if and only if Cg = 0. The three parts are proved in Appendix D: (i) is Lemma 1 evaluated at σ ⋆ , (ii) follows from Eq. (11) together with the reality of Wm , and (iii) evaluates the exact L2 decomposition of Appendix B at the kernel realizing h⋆ . Using this proposition, we arrive at the following classification for magnitude filters. Theorem 1 (Dichotomy). Assume the hypotheses of Prop. 2. Every family of inputs falls into exactly one of the following two cells, according to whether the affordable passband scale σ ⋆ (n) remains bounded. (I) Classically reproducible. If σ ⋆ (n) = O(1), then M ⋆ (pg ; ϵ) = O(1) uniformly in n and for arbitrary input phases, and pg is reproduced to TV ≤ 2ϵ by a distribution samplable in poly(n) time from those O(1) Fourier coefficients. The model is classically simulable regardless of the values of Cg and Φg . (II) No affordable cutoff. If σ ⋆ (n) is unbounded, then by Eq. (18) the filter cannot implement a cutoff at any fixed frequency and hence does not smooth. Distinguish two cases (a) certified, if Cg ̸→ 0, the separation is carried by the spectral phase of the input. (b) uncertified, if Cg → 0 then no single-lag Fourier observable separates pg from classical smoothing. In neither cell is a separation attributable to the filter, which only contributes the real window Wm and the postselection cost psucc . The theorem asserts that at matched cost a magnitude filter either returns a constant-size classical object or stops smoothing altogether, and in neither case does it create the separation, it only inherits whatever phase the input already carried. The cell is decided by σ ⋆ (n), a functional of the magnitudes |â(k)| alone, the branch by Cg , a functional of the phases arg ρm alone (Prop. 2). A gap surviving in cell (I) is self-undermining: it places pg outside the smoothing class {p ⋆ K}, but so does the

8 constant-size sampler that reproduces pg . The split at σ ⋆ = O(1) is chosen for simplicity and is not the boundary of classical simulability. If the overlap profile decays polynomially, the bound of Lemma 1 gives M ⋆ = poly(n) whenever σ ⋆ (n) = poly(n), so the cell (I) construction extends verbatim to that larger class and cell (II) is correspondingly smaller. We state the bounded case, which is the one the examples of Sec. IV realize, and note that nothing in the argument turns on the distinction. For cell (II), in branch (a) the certificate propagates to the gap, Φg ≥ Cg by Eq. (19), and by Prop. 2 (ii) it requires Im⟨Wm ⟩ρm ̸= 0 at a regular lag, which the real window Wm cannot supply. Branch (b) constrains the witness rather than the gap. Cg is an ℓ∞ quantity in√the lag whereas Φg is an ℓ1 one, so a residual of size 1/ N at each of Θ(N ) lags contributes to Φg while Cg → 0, and the phase-free channels of Prop. 2 (iii) remain open, detected without any phase hypothesis by
Φg ≥ 12 max |p̂g (m)| − |p̂(m)| , (21) m

valid since |K̂(m)| ≤ 1 for every probability kernel. A gap witnessed only this way places pg outside the smoothing class but attests nothing a classical sampler could not do. Under the hypotheses of Prop. 2 (iii) both channels close and the classification by Cg is one by Φg at fixed n. We return to both points in Sec. IV. D.

Transfer to the hypercube

The classification in Sec. III C is aimed at filters operating on ZN . However, the treatment uses the cyclic structure of the group only through character orthogonality, the convolution theorem, Parseval’s theorem, and Bochner’s characterization of realizable envelopes, all of which hold on any finite abelian group, so the framework transfers with the dual group supplying the frequencies and the group difference replacing the lag k − m. The one exception is Lemma 1, whose mode count orders the lags along a line; on the hypercube its role is played by a degree bound (Appendix E). We sketch the instance relevant to the didactic example that motivates this work [18], the hypercube Zn2 . P Consider a spectrum â(k) = 2−n/2 x ax (−1)k·x reached by a layer of Hadamard gates. This is the Fourier transform on Zn2 . The equivalent of the frequency in the ZN case is the Hamming weight |k|, and the lag difference is the bitwise XOR k ⊕ m. We use a low-pass, gη (k) = (1 − 2η)|k| with flip rate 0 ≤ η ≤ 1/2. In the taxonomy of Eq. (12) this is a magnitude filter and its pair-weight factorizes  |k\m| Wm (k) = (1 − 2η)|m| (1 − 2η)2 , (22) | {z } | {z } = K̂η (m)

window

where k and m are identified with their supports in [n], so |k \ m| counts the ones of k at positions where m vanishes. The lag factor is the transfer function of the i.i.d.

bit-flip kernel Kη (z) = η |z| (1 − η)n−|z| [28]. That this kernel reproduces the filter applied to the distribution is already noted in Ref. [18, Sec. II E]. The coherent route differs from it only through the window inside Eq. (5). Two group properties then decide the classification. First, the phase channel is closed: the characters are real, so p̂(m) and p̂g (m) are real for every input and every diagonal filter, hence so is ⟨Wm ⟩ρm , and Rg (m) = 0 at every regular lag (Lemma 2 in Appendix E). The certificate is thus identically zero, here not only for magnitude filters but for phase filters as well. As a result, the certified branch of cell (II) is empty, and any surviving gap runs through the phase-free channels of the uncertified branch [cf. Eq. (21)]. Second, matched P 2 cost is set by the weights Aw = |k|=w |â(k)| through P 2w psucc = w (1 − 2η) Aw , so the affordable flip rate is η ⋆ (n) = max{η : psucc (η) ≥ p⋆ }, the counterpart of the affordable passband scale with small η playing the role of large σ. A low-degree input (Aw = 0 for w > w0 with w0 = O(1)) affords a constant flip rate and confines the output to degree |m| ≤ 2w0 , which is a poly(n)-coefficient object samplable in poly(n) time. This is cell (I) with “O(1) modes” replaced by “O(1) degree”, with the caveat that the confinement is inherited from the input, which already restricts the unfiltered p̂ to the same degrees, rather than imposed by the filter. A broadband input, whose spectral mass escapes every fixed degree, instead forces the affordable flip rate to vanish, η ⋆ (n) → 0, so no fixed degree is attenuated. This is the analogue of a diverging σ ⋆ (n). On these two input classes, a magnitude filter on Zn2 at matched cost is therefore either a low-degree classical object or no smoothing at any fixed degree, with no phase branch in either case, and whatever escapes the kernel surrogate is inherited from the input rather than produced by the filter. Genuinely quantum spectral operations must consequently be phase filters (|g| ≡ 1), the direction Ref. [18] itself points to in suggesting a deterministic phase mask on the amplitudes. These incur no post-selection cost, attenuate no degree here, and escape the compactness bound of Lemma 1 on ZN . On the cube the certificate is blind to them as well, so what such a filter buys there would have to be established by something other than Cg .

IV.

EXAMPLES

In this section, we probe the implications of the previous sections. In particular, we show that a spectrally incoherent input results in a coherent residual that vanishes in expectation at every lag, while the gap itself survives as phase speckle. As a canonical example of a low-pass filter, we specify the properties of a Gaussian magnitude filter and analyse a collection of quantum Born machines with random parameters and contrast them by trained counterparts.

9 A.

Spectrally incoherent input

In the following, we investigate the canonical example of an ensemble with i.i.d. random phases. In particular, consider p a magnitude filter (Wm real) with input states ax = p(x) exp(iφx ) with φx independent and uniform. The spectral distribution of the input isPflat in expec2 2 tation, E[|â| ] = 1/N such that psucc ≈ k |gσ (k)| /N . The only way psucc can increase to reduce the cost is by widening the passband. This qualifies the scenario as an example for the growing passband scale case. As shown in Appendix F, the coherent residual Eq. (10) vanishes in expectation (w.r.t the phases) and thus any deviation of the residual is a fluctuation around this vanishing mean. The fluctuations can be shown to be bounded by the largest probability pmax = maxx p(x) such that at every regular lag m  pmax 2 E |Rg (m)| ≤ 2 . psucc

(23)

The bound is therefore informative for input families whose magnitude profile delocalizes, pmax → 0, which we assume in the following. Note that this is a hypothesis on the profile p(x) and not a consequence of the flat expected 2 spectrum E[|â| ] = 1/N , which the random phases produce for any p. With psucc ≥ p⋆ fixed, the phase-charged residual then washes out asymptotically, Rg (m) → 0 in mean square at every regular lag. In particular, note that the vanishing coherent residual is not caused by a vanishing success probability since a smaller psucc would result in a larger Rg [cf. Eq. (11)]. The vanishing residual is thus caused by a vanishing Im⟨Wm ⟩ρm through phase concentration at fixed affordability. We stress that the same incoherent ensemble fed to a phase filter does not wash out: there Wm is complex, Rg ̸→ 0, and the certificate survives at every lag. From Eq. (23) we see that the certificate Eq. (19) vanishes and the family occupies the uncertified branch of cell (II). The bound Eq. (23) is per lag, however, and does not control Φg (Appendix F). B.

Gaussian low-pass

To make the statements in Sec. III concrete, we now fix a family of magnitude filters. As a prototypical example, we consider a low-pass filter with a Gaussian profile
g(k) = exp −k 2 /2σ 2 , (24) such that    2 (k − m m2 2) Wm (k) = exp − 2 exp − , 4σ σ2

(25)

with g, Wm ∈ R. A straightforward calculation (Appendix G) yields the mode count Eq. (14) p
(26) M ⋆ (σ; ϵ) = O σ ln(1/ϵ) .

In particular, M ⋆ depends on n only through σ, consistent with the results of Sec. III B. For the Gaussian filter, the success probability Eq. (6)
P 2 evaluates to psucc (σ) = k exp −k 2 /σ 2 |â(k)| increases for increasing σ. The classification in terms of Theorem 1 thus depends on the input spectrum. We can make this concrete for a Gaussian input √ spectrum of
width l with 2 magnitudes |â(k)| = (1/l π) exp −k 2 /l2 . As shown in Appendix G, for sufficiently large l, we can turn the sum in psucc into an integral and solve Eq. (13) for σ ⋆ , which is attained at psucc = p⋆ σ⋆ = p

lp⋆ 1 − p⋆2

(27)

From this, we can treat two separate cases. For broadband input, we have l(n) → ∞ and thus σ ⋆ → ∞. In this case the number of modes M ⋆ is unbounded, placing the family in cell (II) of Theorem 1, and the branch is decided by the phases: a coherent input occupies the certified branch, a phase-trivial input the uncertified branch with a vanishing certificate, and an incoherent input the uncertified branch with its gap surviving as speckle (see previous section). A narrow input spectrum, however, keeps σ ⋆ and thus M ⋆ bounded and the output is a constantsize classical object according to cell (I).

C.

Numerical Experiments

To gauge the validity of the analytical framework derived in Sec III, we perform numerical experiments.

1.

Controlled Input Families

Figure 3 (a) shows the affordable passband scale σ ⋆ Eq. (13) of a Gaussian low-pass filter Eq. (24) for a fixed affordability threshold p⋆ = 0.1 as a function of the qubit number for different inputs. Three families share the same n-independent bimodal magnitude profile 2 p(x) = |ax | as input. The profile is reflection-symmetric about x0 = 0 on the torus, p(−x) = p(x). p The families differ only in argâ(k): phase-trivial (ax = p(x); light blue), spectrally incoherent (i.i.d. uniform phases; yellow; median over 128 seeds), and coherent chirp (arg ax = πγ(x/N )2 with fixed γ = 48; pink). A fourth family real2 2 2 izes the Gaussian input spectrum |â(k)| ∝ e−k /l with l = N/8 (green). It can be seen that the broad-band input of the Gaussian spectrum and the spectrally incoherent inputs result in an unbounded σ ⋆ ∝ 2n , whereas the coherent chirp and the phase-trivial input stay bounded. For the Gaussian spectrum the measured σ ⋆ coincides with the closed form Eq. (27) (dotted) to a relative deviation of 10−8 . Figure 3 (b) shows the phase certificate Cg Eq. (19) against the mode count M ⋆ for each family. The two axes are the two functionals Theorem 1. Here, M ⋆ reads

10 phase-trivial spectrally incoherent coherent: chirp Gaussian spectrum (l = N/8) (a)

random circuit (d = 4) trained circuit (d = 4) √ trained phase, exact p closed form 100

104

10

Cg

10 102

(II) certified

(I) M = O(1)

10−2

103 σ ? (n)

−1

(b)

erase phase √ qθ : (II)→(I)

−3

10−4 (II) uncertified: Cg → 0

10−5 101 10−16 10−17

100

10−18 8

10

12

14

16

18

100

101

102

103

104

M ? (pg ; )

n

FIG. 3. Input families for a Gaussian low-pass filter Eq. (24) at the affordable passband scale σ ⋆ (n) [Eq. (13)] at p⋆ = 0.1. (a) The affordable passband scale for the constructed families and quantum circuits [cf. Sec. II A] 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 Cg [Eq. (19)] as a function of the modes M ⋆ [Eq. (14)] required to achieve an error of ϵ = 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 Φg 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.

the magnitudes through σ ⋆ , and Cg , a maximum over the regular lags alone, reads the phases. To calculate M ⋆ we fix an approximation error of ϵ = 0.05. For the spectrally incoherent family and the coherent chirp, we can in addition to the bounds Eq. (19) and Eq. (21), obtain a Gaussian-kernel upper bound by minimizing TV(p ⋆ Kσ , pg ) over the bandwidth σ (dashed; for the chirp the minimizer is the degenerate σ → 0 kernel, p ⋆ K = p), which together with Cg encloses the true gap Φg . Each family traces a short track as n grows, and since cell membership is asymptotic it is read off the direction of that track (arrows) rather than off any single point. The cell is set by σ ⋆ . A track running right witnesses cell (II), but a bounded M ⋆ is consistent with either cell, since Lemma 1 has no converse. Within cell (II) a height that holds marks the certified branch and one that falls the uncertified one. Since Cg ≤ Φg , a persistent height is also a persistent gap. Both the phase-trivial and the Gaussian-spectrum input have Rg ≡ 0, so the shown certificate is 0 at machineprecision. However, they land in different cells, since membership is set by σ ⋆ . The phase-trivial input keeps σ ⋆ bounded and falls in cell (I). Its empty certificate is not a vanishing gap as condition (b) of Prop. 1 fails. To make this more concrete, we run a linear program over the full kernel class K (Appendix H) which returns Φg = 0.039.

The Gaussian-spectrum input is broadband, so σ ⋆ ∝ 2n places it in cell (II), in the uncertified branch since its real spectrum empties the certificate. There both conditions hold and the gap vanishes, Φg = 0 up to the solver residual (< 10−9 ), so classical smoothing reproduces the output. The incoherent track runs to the right and downwards. Its certificate decays as 2−0.49n (fitted slope −0.489 ± 0.004 over 8 ≤ n ≤ 18, 128 seeds), matching the 2−n/2 scale Eq. (23) sets, while the Gaussiankernel upper bound for the same family plateaus. The best kernel recovers the envelope of pg and is left with its speckle. The family descends into the uncertified branch of cell (II), its persistent gap carried by the phase-free channel. The chirp keeps σ ⋆ bounded and is therefore in cell (I), yet its gap persists, the self-undermining case noted after Theorem 1.

2.

Quantum Born Machines

Having numerically tested the analytical regimes of Sec III, we now analyze distributions obtained from quantum circuits. To this end, we train a hardware-efficient ansatz (depth-d Ry Rz rotation layers with a CZ ring) to reproduce the fixed bimodal target p(x) of Fig. 3 using the Kullback–Leibler divergence L of Sec. II A as the

11 loss function. In particular, L only sees the distribution 2 obtained through the Born rule pθ (x) = |ax (θ)| and is thus independent of any phases. We train circuits on 32 random parameter initializations with a depth of d = 4. In Fig. 3, the untrained circuits are shown in orange and the trained circuits in dark blue. In Figure 3 (a), both trained and untrained circuits have an exponentially growing affordable passband scale σ ⋆ , which places both families in cell (II) of Theorem 1. The regular-lag certificate Cg of the trained circuits is flat in n (fitted slope −0.04 ± 0.02). Hence Cg ̸→ 0 and the circuits occupy the certified branch of cell (II). The mode count inherits the ∝ 2n growth of the broadband circuit spectrum, so the track runs to the right, while its height stays at the level of the same circuit with random parameters, a persistent certified gap at a growing M ⋆ . Training barely displaces the track. The classification is not an artifact of an imperfect fit. The depth-4 models reproduce the target to TV ≈ 0.02, flat in n (median over seeds between 0.020 and 0.025 for 8 ≤ n ≤ 16). The mechanism is the one anticipated in Prop. 2: the loss is a functional of only pθ , so it fixes the spectral phase only through the component the profile itself dictates and neither needs nor removes the rest, which is inherited at the initialization. Trained circuits are therefore coherent inputs in the sense of Def. 1, and the certified branch of cell (II) is reached generically rather than by construction. To isolate which half of the input carries the certified branch, we perform two control experiments that cut a trained n = 12 state along the magnitude/phase seam. √ First, erasing the phases yields the twin pθ , with an identical output distribution. It collapses into cell (I) at the narrowest admissible passband, σ ⋆ = M ⋆ = 1 [dashed arrow in Fig. 3 (b)], its certificate falling to 10−5 (nonzero only through the residual fit error breaking the target’s reflection symmetry) and its exact gap to Φg ≈ 0.038 (flat in n and evaluated at n ≤ 10, the sizes at which the linear program for this control is affordable; Appendix H), the phase-free floor of the profile and an order of magnitude below the ≈ 0.31 of the state it was built from. Second, the converse surgery [dotted track, hollow markers]: keeping the trained phases p but substituting the exact target magnitudes, ax = p(x) exp(i arg ax (θ)), removes the residual misfit. The remaining profile is reflection-symmetric and hence phase-trivial on its own. The gap survives intact, Φg ≈ 0.31, within 0.1% of the trained value in the median, at a passband scale and mode count indistinguishable from the trained state, so the track stays in cell (II). The two controls bracket the classification from either side. It is only decided by the phases. This finding does not promote the magnitude filter into a quantum resource. By Eq. (9), Φg is the distance of the filter’s output from its purpose of smoothing, so for a magnitude filter working as smoothing and having a gap are mutually exclusive. On the trained models the filter does not smooth. The filtered output lands at

TV ≈ 0.31 from the target the model was trained to represent. Moreover, it lands fifteen times further from that target than the unfiltered model does, at ten times the sampling cost, and the gap scatters across identically trained seeds (0.14–0.55 at n = 12), reflecting the phase configuration the initialization happened to supply rather than any structure related to the task. Both readings of what the filter is for lead to the same place. If the goal is smoothing, the filter delivers it only in the regime Φg → 0 which is the regime in which a classical kernel applied to the unfiltered samples delivers it as well, and does so without paying the post-selection cost. If the goal is instead the filtered distribution pg in its own right, then claiming an advantage means claiming that pg is hard to sample classically. That is a statement about the input state family, which can be prepared and sampled without ever applying the filter. The magnitude filter adds only the real window Wm and the cost psucc . Either way, the quantum candidate is the input state, not the filter.

V.

DISCUSSION AND CONCLUSION

In this work we studied spectral filtering of generative quantum machine learning models. Spectral filtering is proposed as a regularizer, its job is to smooth the output distribution of a generative model without destroying whatever makes that model worth running on a quantum device. We have investigated whether it can do both, and whether the coherent operation produces anything a classical competitor with the same interface cannot. We made the comparison explicit for quantum circuit Born machines, whose output distributions are blind to the phases of the amplitudes that generate them, measuring coherent filtering against convolution of the unfiltered samples with a symmetric probability kernel at matched sampling cost. For magnitude filters, the coherent counterpart of classical smoothing, the two requirements are in tension. At a fixed affordability threshold, either the filtered output is already a classical object, reproducible by smoothing the unfiltered samples or specified by a constant number of Fourier modes and efficiently samplable from them, or the filter provides no attenuation at any fixed frequency: applied to the broadband spectra that trained circuits actually produce, it is affordable only with a passband so wide that it attenuates nothing at any fixed frequency. Regularization, affordability, and any separation from classical simulation cannot be achieved together. Where a separation does survive the operation, it is not manufactured by the filter. The residual that carries it is sourced entirely by the spectral phase of the input state, and replacing that phase by a linear one while leaving the output distribution untouched, which for the reflection-symmetric target studied here is simply erasing the computational-basis phases, collapses a trained model into the classically reproducible cell. Numerical experiments on controlled input families and on trained and untrained Born machines illustrate both cells, and

12 on the trained circuits the point is visible without any simulability argument at all, since the filtered model sits fifteen times further from its own training target than the unfiltered one at ten times the sampling cost. Several limitations qualify these conclusions. We work out the cyclic group ZN in detail and only sketch the transfer to the hypercube Zn2 , even though every ingredient we use is available on any finite abelian group. The analysis is restricted to filters diagonal in the Fourier basis, so spectral operations that couple modes are outside its scope. The classical competitor is convolution with a symmetric probability kernel, which is the operation the filter advertises, rather than arbitrary classical postprocessing of the samples. Dropping reflection symmetry costs the competitor almost nothing on every instance we solve (Appendix H), but a gap measured against K is a separation from smoothing and not, on its own, from classical simulation. Finally, the gap Eq. (9) is a statement about distributions in the large-sampling limit. At a finite shot budget Φg must be resolved against shot noise, at a rate itself reduced by the post-selection cost psucc , and how the classification degrades there we leave open. What the dichotomy delineates is what is not a quantum resource, which is also where it points. Within the diagonal family, phase filters with unit modulus are the only spectral operations left. They incur no postselection cost, need not admit a kernel surrogate, and escape the compactness bound of Lemma 1. Whether

they can be turned into a design principle with a useful inductive bias, rather than merely an unsimulable one, is the natural next question. A second direction follows from the mechanism of Sec. IV C 2: the spectral phases that decide the classification are, on the circuits studied here, invisible to a Born-rule loss and are inherited from the initialization. This is an empirical statement about trained circuits rather than a structural one, since the component of arg â(k) sourced by the asymmetry of the profile is a functional of pθ and hence is seen by such a loss; our controls show that component to be negligible here, 10−5 against a gap of 0.31. Training objectives that see the phase, rather than only the output distribution, would make that degree of freedom a design variable instead of an accident.

ACKNOWLEDGMENTS

This work has been supported by the Dieter Schwarz Foundation through the Fraunhofer Heilbronn Research and Innovation Center Applied Quantum AI. Large language models were used in the preparation of this manuscript for literature research, editorial work on the text, code prototyping, and as a verification device on the technical arguments. All results were derived and verified by the author, who takes full responsibility for the content.

Appendix A: Framework

This appendix supports the framework described in Sec. II

1.

Classical Competitor

We define the classical competitor as smoothing operations induced by a symmetric probability kernel as follows. Definition 2 (Classical post-processing kernels). A probability kernel on ZN is a map K : ZN → R with K(x) ≥ 0 P and x K(x) = 1, acting on the model as GK : p 7→ p ⋆ K. Operationally GK draws x ∼ p and u ∼ K independently and returns x + u mod N , so it is classical post-processing of the samples for every such K. We write n o X K+ ≡ K K ≥ 0, K(x) = 1 , (A1) x  K ≡ K ∈ K+ K(−x) = K(x) , for the translation-equivariant post-processings and for their reflection-symmetric subclass, the smoothing kernels. Within K+ the transfer function K̂ is real and even within K. Reflection symmetry is what separates smoothing from shifting: the extreme points of K+ are the deterministic translations x 7→ x + x0 , and K retains only their symmetric mixtures. We take K as the classical baseline throughout, since it is the class to which the filter’s advertised semantics Eq. (4) belongs. Since K ⊂ K+ gives Φg ≥ Φ+ g ≡ inf K∈K+ TV(p ⋆ K, pg ), every classicality conclusion drawn against K holds a fortiori against K+ , whereas a positive gap must be checked against the larger class. On all instances reported here the median Φ+ g /Φg exceeds 0.93 for every input family with a nonzero gap, and no instance changes its cell assignment (Appendix H). This justifies the restriction to K in the main text.

13 2.

Derivation of Eq. (5)

P We use the inverse of Eq. (2) to write the filtered amplitudes ag (x) = √1N k âg (k) ω kx . The filtered modes then have the form X X X ′ 1 X 2 p̂g (m) = |ag (x)| ω −mx = âg (k)â∗g (k ′ ) ω (k−k −m)x = âg (k)â∗g (k − m) , (A2) N ′ x x k

k,k

P ′ where we have used x ω (k−k −m)x = N [ k − k ′ − m ≡ 0 ] in the last equality. For a diagonal spectral filter as introduced in Sec. II B, normalized to ∥g∥∞ = 1, the map â 7→ gâ can be realized P on a single ancilla. We write Dg = k g(k) |k⟩ ⟨k| for the corresponding operator, diagonal in the Fourier basis. Since 2 |g(k)| ≤ 1, the filter admits a dilation V |0⟩a |k⟩ = g(k) |0⟩a |k⟩ + (1 − |g(k)| )1/2 |1⟩a |k⟩ which block-encodes the filter

⟨0|a ⊗ I V |0⟩a ⊗ I = Dg , (A3) 2

so the outcome |0⟩a occurs with probability ∥gâ∥ = psucc [Eq. (6)] and leaves the normalized amplitudes âg (k) = √ g(k)â(k)/ psucc . This gives psucc its two readings, the retained spectral mass and the heralding probability, with psucc = 1 exactly when |g(k)| = 1 on the support of â. Repeating until success costs 1/psucc state preparations, √ which amplitude amplification [27] reduces to O(1/ psucc ). No result depends on which, since only psucc ≥ p⋆ at n-independent p⋆ is used. Substituting âg into Eq. (A2) results in Eq. (5) with the definitions of Wm (k) and ρm (k) as given in the main text.

Appendix B: Simulability criterion

In this appendix, we derive the simulability criterion of Prop. 1. The coherent residual Eq. (10) vanishes if the classical filter matches the quantum filter. Expanding h⋆ and identifying the zero point of its derivative gives   2 2 ∂h |p̂g (m)| − 2h Re(p̂g (m)p̂∗ (m)) + h2 |p̂(m)| = 0 (B1)   p̂g (m) ⇐⇒ h⋆ (m) = Re . (B2) p̂(m) Substituting this into Eq. (10) results in Eq. (11). Using this equation, we study the gap as defined in Eq. (9), which can be formulated as Φg = infK ∥pg − p ⋆ K∥1 /2. The kernel set K is a closed and bounded, hence compact, subset of RN , and K 7→ TV(p ⋆ K, pg ) is continuous, so the infimum is attained and Φg = 0 holds exactly when some K ∈ K √ reproduces pg . On the finite group ZN , the two norms are equivalent, ∥f ∥2 ≤ ∥f ∥1 ≤ N ∥f ∥2 , so they vanish √ 2 together and ∥pg − p ⋆ K∥2 characterizes Φg = 0 in both directions. The equivalence constant is N ; hence the L2 route says nothing quantitative or asymptotic about Φg itself (see the scope paragraph of App. F). Working in L2 allows us to use Parseval’s theorem with the convention X x

2

|f (x)| =

2 1 X ˆ f (m) . N m

(B3)

This yields 2

∥pg − p ⋆ K∥2 =

2 2 1 X 1 X p̂g (m) − p\ ⋆ K(m) = p̂g (m) − K̂(m)p̂(m) , N m N m

(B4)

where K̂(m) ∈ R is real and even for symmetric kernels K, and varies with the lag m; the scalar h of Eq. (B2) is the minimization variable at a fixed lag. Rearranging Eq. (10) and subtracting K̂(m)p̂(m) from both sides gives
p̂g (m) − K̂(m)p̂(m) = Rg (m) + h⋆ (m) − K̂(m) p̂(m) , (B5) such that p̂g (m) − K̂(m)p̂(m)

2

2 2
2 2 = Rg (m) + h⋆ (m) − K̂(m) p̂(m) = |Rg (m)| + h⋆ (m) − K̂(m) |p̂(m)| .

(B6)

14 Here, the cross terms in the second equality vanish since the first term is an imaginary multiple of p̂(m) by Eq. (11), and the second term is a real multiple of p̂(m). We thus have 2

∥pg − p ⋆ K∥2 =

2 1 X ⋆ 1 X 2 2 h (m) − K̂(m) |p̂(m)| . |Rg (m)| + N m N m

(B7)

2

At a degenerate lag, the second term is absent, and the first contributes |p̂g (m)| for every K. Hence, Φg = 0 requires p̂g (m) = 0 at every degenerate lag, which is the degenerate half of condition (a) of Prop. 1. For regular lags m ∈ S [Eq. (7)], a vanishing gap requires both terms to vanish. Since p̂(m) ̸= 0 and psucc > 0, the first term vanishes exactly when ⟨Wm ⟩ρm ∈ R by Eq. (11), which is the regular half of condition (a) in both directions. The second term in Eq. (B7) vanishes if K̂(m) = h⋆ (m) at every m ∈ S, i.e., if h⋆ can be realized by a symmetric kernel transformation, which is condition (b) in Prop. 1. By Bochner’s theorem on ZN [29], the realizable envelopes K̂ of symmetric probability kernels are the symmetric positive-definite sequences normalized to K̂(0) = 1. A vanishing second term therefore forces h⋆ to coincide with one such envelope on the regular lags, which is condition (b). Three properties now immediately follow. Since p̂(0) = p̂g (0) = 1, Eq. (B2) gives h⋆ (0) = 1, so the Bochner normalization is not an extra assumption. Since p and pg are real, p̂(−m) = p̂(m)∗ and p̂g (−m) = p̂g (m)∗ , so h⋆ is automatically real and even, and the restriction to symmetric kernels in condition (b) costs nothing. Finally, every P probability kernel obeys K̂(m) ≤ x K(x) = 1, so |h⋆ (m)| ≤ 1 is necessary, which is the amplification channel of Sec. III C, appearing already here. Collecting the two directions: if Φg = 0, both terms of Eq. (B7) vanish at the attaining K, which is condition (a) at every lag and condition (b) on S. Conversely, (a) and (b) provide a K ⋆ ∈ K with K̂ ⋆ = h⋆ on S; both terms vanish, and ∥pg − p ⋆ K ⋆ ∥2 = 0, hence Φg = 0. Appendix C: Spectral Compactness

Proof of Lemma 1. The modulus of Wm bounds the Fourier support of p̂g , so the magnitude profile that sets how many modes are needed to describe the filtered distribution. For any filter g, we bound Eq. (5) 1 sup |Wm (k)| |p̂g | ≤ psucc k

X

|ρm (k)| ≤

k

P

1

sup |Wm (k)| =

psucc

k

1 psucc

sup |g(k)| |g(k − m)| .

(C1)

k

2 k |â(k)||â(k − m)| ≤ ∥â∥ = 1 by Cauchy-Schwarz.

P

Hence,

P P 1/2 1/2
1 2 2 ∥1 ≤ 12 ≤ 2 p1succ . TV pg , p≤M = 2 ∥pg − p≤M g g |m|>M |p̂g (m)| |m|>M supk |Wm (k)|

(C2)

In the second inequality, we have used keeping all frequencies with |m| ≤ M ,

k |ρm (k)| =

Here we have used Parseval’s theorem Eq. (B3) to bound the total variation by the sum of absolutes of the Fourier √ P transformed probabilities ∥f ∥1 ≤ N ∥f ∥2 = ( m |fˆ(m)|2 )1/2 for the first inequality and the second inequality follows from Eq. (C1).
Substituting the hypothesis supk |gσ (k)| |gσ (k − m)| ≤ s(|m|/σ) from Lemma 1 into Eq. (C2), we use that s is non-increasing and replace the sum with an integral, we obtain p σ T (M/σ) √ TV(pg , p≤M ) ≤ . (C3) g 2 psucc R∞ Here we have defined T (a) := a s(u)2 du. We are interested in the number of modes M ⋆ Eq. (14) required to approximate the distribution pg by the truncated distribution p≤M Eq. (15) to error ϵ, i.e., TV(pg , p≤M ) ≤ ϵ. g g Substituting Eq. (C3) then yields  2 (p⋆ ϵ)2  M ∗ (σ; ϵ) ≤ σ T −1 , σ

(C4)

where we have used psucc ≥ p⋆ such that the bound depends on the state only through psucc ≥ p⋆ . The decay of p̂g is controlled purely by the filter’s overlap width (≈ 2σ for families gσ as defined in the main text).

15 Appendix D: Proofs of Prop. 2 and Theorem 1

Throughout this appendix g is a magnitude filter, so that Wm (k) = |g(k)||g(k − m)| ∈ R≥0 , and {gσ } is a family with passband scale σ in the sense of Sec. III B. Furthermore, we assume the stretched form |gσ (k)| = G(|k|/σ) for a fixed profile G : [0, ∞) → [0, 1] that is continuous, non-increasing, and normalized to G(0) = 1; in particular, |gσ (k)| is of order unity for |k| ≲ σ and decays for |k| ≳ σ. Then, psucc (σ) is non-decreasing in σ, so Eq. (13) is well posed. a. Derivation of the phase certificate Eq. (19). Fix K ∈ K and set f = pg − p ⋆ K. Linearity and the convolution theorem give fˆ(m) = p̂g (m) − K̂(m)p̂(m), and |ω −mx | = 1 gives fˆ(m) ≤ ∥f ∥ = 2 TV(pg , p ⋆ K). At a regular lag 1

the orthogonal decomposition Eq. (B6) of App. B bounds the same quantity from below, fˆ(m)

2

2

2

= |Rg (m)| + h⋆ (m) − K̂(m) |p̂(m)|

2

2

≥ |Rg (m)| ,

so TV(pg , p ⋆ K) ≥ 12 |Rg (m)| for every m ∈ S and every K ∈ K. Maximizing over m and taking the infimum over K yields Φg ≥ Cg . Proof of Prop. 2. (i) The success probability Eq. (6) sees the input only through the spectral magnitudes |â(k)|, hence so does σ ⋆ (n) defined by Eq. (13). Evaluating the bound Eq. (C4) at σ = σ ⋆ (n), where psucc ≥ p⋆ holds by construction, gives M ⋆ (pg ; ϵ) ≤ σ ⋆ T −1 (2(p⋆ ϵ)2 /σ ⋆ ) in which the filter appears only through the overlap profile s of its modulus, the state only through psucc ≥ p⋆ , and the grid size N = 2n not at all. Since T is non-increasing, so is T −1 , and the right-hand side is non-decreasing in σ ⋆ . At fixed ϵ and p⋆ a bound σ ⋆ (n) ≤ σ0 therefore gives
⋆ −1 ⋆ 2 M ≤ σ0 T 2(p ϵ) /σ0 uniformly in n and for arbitrary input phases. (ii) At a regular lag p̂(m) ̸= 0 and psucc > 0, so Rg (m) ̸= 0 ⇐⇒ P Im⟨Wm ⟩ρm ̸= 0 by Eq. (11). For a magnitude filter θ ≡ 0 in Eq. (12) and Wm (k) ∈ R, so that with ⟨Wm ⟩ρm = k Wm (k)ρm (k)/p̂(m), Im⟨Wm ⟩ρm =

1

X 2

|p̂(m)|

  Wm (k) Im ρm (k) p̂∗ (m) .

(D1)

k

The real weights rotateP no term, so the imaginary part is sourced exclusively by the arguments of ρm (k) = â(k)â∗ (k − m) relative to p̂(m) = k ρm (k), i.e. by the spectral phase of the input. On a phase-trivial family, inserting â(k) = eiα ω tk rk makes the collinearity explicit, ρm (k) = â(k)â∗ (k − m) = eiλm rk rk−m ,

λ≡

2πt , N

(D2)

so the global phase α cancels and every term P carries the same k-independent phase eiλm , which is exactly the iλm hypothesis at work. Consequently p̂(m) = e k rk rk−m and the terms ρm (k) are collinear with their own sum, P ρm (k) p̂∗ (m) = rk rk−m k′ rk′ rk′ −m ∈ R for all k at every regular lag, so every summand of Eq. (D1) vanishes, ⟨Wm ⟩ρm ∈ R, and hence Rg ≡ 0 and Cg = 0 for every n. (iii) The lower bound is the certificate bound above. For the upper bound, condition (b) provides K ⋆ ∈ K with ⋆ K̂ = h⋆ on S. In the exact decomposition Eq. (B7) the envelope term then vanishes on S, and the √ degenerate P 2 2 2 lags contribute |p̂g (m)| = 0 by hypothesis, so ∥pg − p ⋆ K ⋆ ∥2 = N1 m∈S |Rg (m)| . With ∥f ∥1 ≤ N ∥f ∥2 and Φg ≤ TV(p ⋆ K ⋆ , pg ) this is the right-hand side of Eq. (20). Finally, Cg = 0 forces Rg (m) = 0 at every m ∈ S and hence Φg = 0 by that upper bound, while Φg = 0 forces Cg = 0 by the lower bound. Proof of Theorem 1. Exclusive and exhaustive. Split on whether σ ⋆ (n) is bounded, giving cell (I) or cell (II); within cell (II), split on whether Cg → 0, giving the certified branch (a) or the uncertified branch (b). Cell (I). Part (i) of Prop 2 at σ ⋆ (n) = O(1) gives M ⋆ (pg ; ϵ) = O(1) uniformly in n and for arbitrary input phases. ⋆ The truncation p≤M is real and normalized but need not be pointwise nonnegative. g Samplability: We expand p̂g (m) into real and imaginary parts, so the truncation Eq. (15) becomes

p≤M (x) = g

M X   1 2πmx . 1+2 Rep̂g (m) cos 2πmx − Imp̂ (m) sin g N N N m=1

(D3)

Here we have used that p̂g (−m) = p̂g (m)∗ for real p̂g and the truncation window |m| < M is symmetric in m. From this, it follows that p̂≤M is a real trigonometric polynomial, normalized by its m = 0 mode. Note that it can become g

16 negative below zero wherever pg is small, so it is only an approximation of distribution, not a distribution. This can be resolved by clipping the negative part and renormalizing,
⋆ X
max pg≤M , 0 ⋆ , Z= max p≤M (x), 0 , q= g Z x Clipping sends each negative point to 0, which is nearer the nonnegative pg than where it was before, so it cannot ⋆ ⋆ ⋆ , 0) − pg 1 ≤ p≤M − pg 1 ≤ 2ϵ. Since p≤M already sums to one, Z − 1 is exactly increase the error, max(p≤M g g g the negative mass the clip removes, bounded by the same discrepancy, Z − 1 ≤ 2ϵ. Renormalizing displaces the vector by ∥q − p̃∥1 = |1 − Z|, so ∥q − pg ∥1 ≤ 4ϵ and TV(q, pg ) ≤ 2ϵ. The distribution q is efficiently samplable: partial sums of the O(1) retained modes are geometric series in closed form, a real trigonometric polynomial of degree M ⋆ changes sign at most 2M ⋆ times, so the clipped region is a union of O(1) arcs computable to precision δ in poly(n, log(1/δ)) time, and inverse-transform sampling from q follows. The conclusion uses no property of Cg or Φg . Cell (II) If σ ⋆ (n) is unbounded then, by the stretching form of the family, |gσ⋆ (k)| → 1 at every fixed k along the subsequence realizing the divergence, hence Wm (k) → 1 on every fixed set of frequencies and lags. Branch (a), certified. By part (ii), Cg > 0 requires Im⟨Wm ⟩ρm ̸= 0 at some regular lag. The window Wm is real and by Eq. (D1) cannot supply it, so it is carried by arg ρm , that is by the spectral phase of the input. By Eq. (19), Φg ≥ Cg , so Cg ̸→ 0 forces Φg ̸→ 0: the output stays bounded away from every classical smoothing, and the separation is carried by the spectral phase of the input. Branch (b), uncertified. Cg → 0 means maxm∈S |Rg (m)| → 0: no regular lag separates pg from the smoothing class. If Φg → 0, then by Eq. (9) and the attainment of the infimum (App. B) there are symmetric probability kernels Kn with TV(p ⋆ Kn , pg ) → 0. The unfiltered samples reproduce pg in the limit of large n at no post-selection cost. This is consistent with the absence of attenuation since a filter that tends to the identity on every fixed band is matched by the trivial kernel K = δ0 , and a kernel that smooths at a fixed frequency would contradict Eq. (18). If instead Φg ̸→ 0, then along any subsequence with Φg bounded away from zero either a hypothesis of part (iii) of Prop. 2 fails, i.e. p̂g (m) ̸= 0 at a degenerate lag or h⋆ is not a realizable envelope, or both hypotheses hold and P 2 Eq. (20) forces m∈S |Rg (m)| ≥ 4Φ2g ̸→ 0, which is compatible with Cg → 0 since residuals of size N −1/2 at the |S| = Θ(N ) regular lags suffice. The two channels need no Im⟨Wm ⟩ρm ̸= 0 to open, and the third mechanism requires it at no lag above the vanishing certificate, so none certifies coherence through a single-lag observable. What all three certify is that pg has left the smoothing class, i.e. that the filter is not doing what it advertises. In neither cell is the separation attributable to the filter. The mode count of part (i) depends on g only through |g| and on the state only through psucc ≥ p⋆ , while the residual that carries a certified gap is sourced entirely by the phases of ρm . The filter supplies the real window Wm and the post-selection cost psucc , and nothing else. Appendix E: Spectral filtering on the hypercube

Character orthogonality, the convolution theorem, Parseval’s theorem and Bochner’s characterization exist on any finite abelian group [29], so Eqs. (5)–(8), Prop. 1 and the washout bound Eq. (23) hold there verbatim, with the lag difference k − m read as the group difference, on Zn2 the bitwise XOR k ⊕ m.
The exception is Lemma 1, which counts n lags along a line: on the Boolean cube the lags at distance w number w , so the tail sum of Eq. (C2) is no longer controlled by the decay of supk |Wm (k)| alone. Its role is played by the degree bound below, which, unlike Lemma 1, is a hypothesis on the input rather than on the filter. The characters χk (x) = (−1)k·x are real and every element is its own inverse, so kernel symmetry is automatic and K = K+ , which makes the reflection caveat of Def. 2 vacuous, p̂(m) and p̂g (m) are real, and every envelope is real with K̂(m) ≤ 1. Filter and kernel. Write the low-pass of Sec. III D as gη (k) = λ|k| with flip rate η ∈ [0, 21 ] and λ = 1 − 2η ∈ [0, 1]. Using |k| + |k ⊕ m| = |m| + 2|k \ m| , where |k \ m| =

Pn

j=1 kj (1 − mj ) counts the 1-bits of k at positions where m has a 0, the pair weight factorizes as

Wm (k) = gη (k)gη∗ (k ⊕ m) = λ|m| λ2 For the independent bit-flip kernel Kη (z) = η |z| (1 − η)n−|z| ,

|k\m|

.

17 the Walsh transform (the Fourier transform on the Boolean cube) is n Y   K̂η (m) = (1 − η) + η(−1)mj = λ|m| . j=1

Therefore Wm (k) = K̂η (m) λ2

|k\m|

,

which is Eq. (22). Thus the pair weight consists of the classical bit-flip transfer function and an additional k-dependent window. P Lemma 2. On Zn2 , for every diagonal filter g and every input state, Sm = k Wm (k)ρm (k) is real. Consequently ⟨Wm ⟩ρm ∈ R and Rg (m) = 0 at every regular lag. 2

2

Indeed W0 (k) = |g(k)| and ρ0 (k) = |â(k)| are real, and for m ̸= 0 the map k 7→ k ⊕ m is a fixed-point-free involution with Wm (k ⊕ m) = Wm (k)∗ and ρm (k ⊕ m) = ρm (k)∗ , so the sum splits into real pairs 2 Re[Wm (k)ρm (k)]; since p̂(m) is real so is ⟨Wm ⟩ρm , and Eq. (11) gives Rg (m) = 0. No reality assumption on g enters, so on the cube the certificate Eq. (19) is empty for phase filters as well as for magnitude ones, and part (ii) of Prop. 2 carries no information there. P P 2 Matched cost. With Aw = |k|=w |â(k)| , Eq. (6) reads psucc (η) = w λ2w Aw , continuous and non-increasing in η with psucc (0) = 1, so the affordable flip rate η ⋆ (n) = max{η ∈ [0, 21 ] : psucc (η) ≥ p⋆ } is attained. It is the counterpart of Eq. (13), with small η playing the role of large σ. For a low-degree input, Aw = 0 for w > w0 , one has psucc ≥ λ2w0 , so any η with λ2w0 ≥ p⋆ is affordable and η ⋆ = Ω(1) whenever w0 = O(1). Since ρm (k) ̸= 0 requires |k|, |k ⊕ m| ≤ w0 while |m| ≤ |k| + |k ⊕ m|, Eqs. (3) and (5) confine both p̂ and p̂g to |m| ≤ 2w0 for every η. The confinement is exact P rather than a truncation, so the clipping used in the proof of Theorem 1 is vacuous: prefix marginals of pg (x) = 2−n m p̂g (m)(−1)m·x are again Walsh sums of degree at most 2w0 , so its O(n2w0 ) coefficients fix every conditional bit probability and pg is sampled bit by bit in poly(n) time. P P For a broadband input, w≤L Aw → 0 at every fixed L, and splitting psucc at degree L gives psucc (η) ≤ w≤L Aw + λ2L , the analogue of Eq. (17). Affordability then forces (λ⋆ )2L ≥ p⋆ − o(1) at every fixed L, hence λ⋆ → 1, i.e. η ⋆ (n) → 0, and gη⋆ (k) → 1 at every fixed degree: no fixed degree is attenuated, which is Eq. (18) read on the cube. Between the two classes the argument is silent, since a bounded η ⋆ does not by itself bound the degree of pg . Appendix F: Washout for spectrally incoherent input

p Consider a magnitude filter g (Wm real, Wm (k) = |g(k)||g(k − m)| ≥ 0) and ax = p(x) eiφx with φx independent and uniform. In the following, we show that under these conditions, the spectrum effectively washes out such that Rg = 0 in expectation. The coherent residual Eq. (10) is carried by Im ⟨Wm ⟩ρm , so it suffices to show Im ⟨Wm ⟩ρm → 0 2 in mean square as the input spreads. Note that p(x) = |ax | is deterministic and the only randomnessP is in the phases. To isolate the action of the random phases on the spectrum, we write ⟨Wm ⟩ρm = Sm /p̂(m) with Sm ≡ k Wm (k)ρm (k) random and p̂(m) a constant. With this, the expectation of ⟨Wm ⟩ρm is E[⟨Wm ⟩ρm ] =

E[Sm ] , p̂(m)

Var(⟨Wm ⟩ρm ) =

Var(Sm ) |p̂(m)|

2

.

(F1)

The case m = 0 is trivial, since p̂g (0) = p̂(0) = 1 forces Rg (0) = 0. In the following, we thus fix a regular lag m ̸= 0. With the particular form of the amplitudes fixed as above and using Eq. (2), we have ρm (k) =

1 X√ px py ei(ϕx −ϕy ) ω −kx+(k−m)y , N x,y

(F2)

Taking the expectation value over the phases yields E[ei(φx −φy ) ] = δxy , such that E[Sm ] =

p̂(m) X Wm (k) . N k

(F3)

18 P Substituting this into Eq. (F1) yields E[⟨Wm ⟩ρm ] = k Wm (k)/N , and since Wm (k) is real by assumption. Consequently, E[Rg ] = 0 such that a magnitude filter thus produces no residual in expectation and any residual is a fluctuation about the mean whose size we bound next. 2 Expanding E[|Sm | ] yields   X 2 Wm (k)Wm (k ′ )ρm (k)ρ∗m (k ′ ) (F4) E[|Sm | ] = E  k,k′

=

X X

′

′

′

Wm (k)Wm (k ′ )ω −kx+(k−m)y+k x −(k −m)y

′

√

px py px′ py′ E [exp(i(ϕx − ϕy − ϕx′ + ϕy′ ))] . (F5)

x,y,x′ ,y ′ k,k′

Note that the expectation value is nonzero only for the pairings x = y, x′ = y ′ and x = x′ , y = y ′ . The first pairing 2 reconstructs |E[Sm ]| and cancels in the variance. Subtracting the coincidence x = y = x′ = y ′ once thus leads to ′ 1 XX Wm (k)Wm (k ′ )ω −(k−k )(x−y) px py (F6) Var(Sm ) = 2 N ′ x̸=y k,k

=

2 1 X cm (x − y) . p(x) p(y) W 2 N

(F7)

x̸=y

P Where we have introduced the Fourier transform of the pair-weights Ŵm (l) = k Wm (k)ω −kl . Dropping the constraint x ̸= y (the omitted terms are nonnegative), bounding one factor p(x) ≤ pmax ≡ maxx px , completing the sum over x then yields 2 pmax X Var(Sm ) ≤ (F8) Ŵm (k) . 2 N k

Using Parseval’s theorem Eq. (B3) finally gives pmax X Wm (x)2 ≤ pmax . (F9) N x P 2 2 where we have used Wm (k)2 = |g(k)| |g(k − m)| ≤ 1, so k Wm (k)2 ≤ N in the last step. We now return to the residual which in expectation is bounded as Var(Sm ) ≤

   |p̂(m)|2 Var(Sm ) pmax 2 ≤ ≤ ⋆ 2, (F10) E |Rg (m)| = E (Im ⟨Wm ⟩ρm )2 2 2 psucc psucc (p )   2 which is Eq. (23). Here we have used E (Im ⟨Wm ⟩ρm )2 ≤ Var(⟨Wm ⟩ρm ) = Var(Sm )/|p̂(m)| , which is bounded by Eq. (F9). The bound is uniform in m, so by Markov’s inequality Rg (m) → 0 in mean-square at every lag as pmax → 0 with psucc ≥ p⋆ fixed. Note that for a phase filter we have Wm (k) ∈ C. In this case, the residual does not vanish in general so the same incoherent input does not wash out. Scope of the bound. Equation (23) constrains one lag at a time and does not transfer to Φg . Summed over the |S| = Θ(N ) regular lags of a broadband output it returns at least 1/(p⋆ )2 > 1, since pmax ≥ 1/N always, so it never feeds the upper bound of Eq. (20). A residual of size N −1/2 spread over every lag, which is what the washout √ produces, saturates the conversion ∥f ∥1 ≤ N ∥f ∥2 behind that bound. What Eq. (23) establishes is that the certificate Eq. (19) P vanishes, not that the gap closes. The uniform member of Sec. IV A separates the two: there √ (p⋆K)(x) = N1 y K(y) = p(x) for every K ∈ K, so Φg = TV(pg , p) exactly, while matched cost fixes σ ⋆ /N → p⋆ / π, ⋆ giving the intensity N pg an n-independent nondegenerate law. Numerically, at p = 0.1 over 32 phase seeds per size, the exact gap plateaus at Φg = 0.345, 0.355, 0.352, 0.354, 0.354 for n = 8, 10, 12, 14, 16, while the mode count grows from 35 to 8684 and the modulus bound Eq. (21) falls from 0.115 to 0.013. Appendix G: Gaussian example 2

2

2

2

For g(k) = e−k /2σ , completing the square in Wm (k) = g(k)g(k − m) gives supk |Wm (k)| = e−m /4σ (attained at 2 2 k = m/2), so the overlap profile is s(u) = e−u /4 and s(u)2 = e−u /2 . Hence Z ∞ 2 2 2 e−u /2 du ≤ e−M /2σ , T (M/σ) = M/σ

19 TABLE I. Cost of dropping reflection symmetry, by input family, over all instances at 8 ≤ n ≤ 12. A dagger marks a deterministic family, with one instance per size. Φ+ g /Φg input instances median min phase-trivial 5† 1.000 1.000 spectrally incoherent 160 0.997 0.947 coherent chirp 5† 0.966 0.966 Gaussian spectrum 5† Φg = Φ+ g = 0 random-parameter circuits 320 0.931 0.201 trained circuits 320 0.998 0.833

a Chernoff tail bound (valid for M ≥ σ). Substituting into Eq. (C3), √ −M 2 /4σ2 σe ≤M √ TV(pg , pg ) ≤ . 2 psucc Imposing TV ≤ ϵ and using psucc ≥ p⋆ gives, at equality, e

−M 2 /2σ 2

2(p⋆ ϵ)2 = σ

=⇒

r M (σ; ϵ) = σ 2 ln ⋆

p
σ = O σ ln(1/ϵ) , 2(p⋆ ϵ)2

consistent with the general form M ⋆ ≤ σ T −1 2(p⋆ ϵ)2 /σ . Appendix H: Exact gaps for the numerical experiments

The gap Eq. (9) is an infimum of a piecewise-linear function of K over a polytope, hence a linear program: minimize

1 2 ∥CK − pg ∥1

subject to

K(x) ≥ 0 ,

K∈RN

X

K(x) = 1 ,

(H1)

x

K(−x) = K(x) , where Cxu = p(x − u) is the circulant of the input, so CK = p ⋆ K and the objective is TV(p ⋆ K, pg ). The last constraint selects K and returns Φg ; dropping it selects K+ and returns Φ+ g . We solve both programs exactly for every input reported in Fig. 3, at 8 ≤ n ≤ 12 where the O(N ) program is affordable: 1647 instances in total, comprising the four constructed families and the trained and random-parameter circuits together with their phase controls. This replaces the bracket of Eq. (19) by Φg itself, and returns Φ+ g alongside it. The instance counts differ by family only because the families differ in what there is to sample: the phase-trivial, Gaussian-spectrum and chirp inputs are deterministic constructions with one state per size, so their five instances are the five sizes and a median or minimum over them reports the n-dependence rather than an ensemble spread; the incoherent family draws 32 phase seeds per size, and each circuit family runs 32 seeds at d = 2 and d = 4. The phase-erased control is the one variant whose program is not affordable at every size, since its near-uniform filtered output is matched only by a kernel spread over a finite fraction of the torus; it is solved at n ≤ 10, where its gap is already flat (0.0380, 0.0380, 0.0383 at n = 8, 9, 10 for d = 4), and its instance count is correspondingly smaller. Two facts follow. First, dropping reflection symmetry from the kernel family does not result in meaningful advantages for the classical competitor, which justifies the restriction to smoothing kernels in the main text (Table I). + The phase-trivial input has Φg = Φ+ g = 0.039. The Gaussian-spectrum input attains Φg = Φg = 0 to within the solver residual, so its ratio carries no information and it is listed separately. No instance changes cell membership. Every trained circuit keeps Φ+ g ≥ 0.115, and the optimizer over K+ is strongly asymmetric throughout (median P 1 |K(x) − K(−x)| = 0.78), so the shift freedom is exercised but does not significantly improve the performance. x 2 There are two exceptions: The chirp loses a fixed 3.4% at every n, which is the deterministic shift of K+ acting on the displaced output of a quadratic phase. The random-parameter circuits are the loosest family, with isolated seeds retaining only 20% of their gap. They remain bounded away from zero (min Φ+ g = 0.036), but the trained circuits, which carry the claim, do not rely on this.

20 Second, the bracket is loose downwards and tight upwards: on the trained circuits, pooled over both depths, the certificate Cg understates Φg by a median factor 3.0 (range 1.2–34), whereas the Gaussian-kernel bound exceeds it by a median 0.06% and at most 3.3%. The true gap therefore sits at the top of the reported bracket for the circuits and, to 0.1%, for the spectrally incoherent family. The chirp is an exception, falling near mid-bracket (Φg = 0.515 against 0.449 and 0.579). Both programs are solved by column generation over the kernel lags with a dual-feasibility certificate, cross-checked −10 against the full design. Across all instances Φ+ , the returned kernel reproduces its own objective g − Φg ≤ 2.6 × 10 1 −8 to 3.8 × 10 , and 2 maxm |Rg (m)| ≤ Φg ≤ TV(p ⋆ Kσ , pg ) holds at every instance.

[1] Y. Wang and J. Liu, Reports on Progress in Physics 87, 116402 (2024). [2] Y. Ji, Z.-Y. Chen, M. Roth, D. A. Kreplin, C. Schiffer, M. King, O. Anton, M. S. Alam, M. Krutzik, D. Willsch, L. Mathey, F. K. Wilhelm, and G.-P. Guo, Quantum deep learning: A comprehensive review (2026), arXiv:2603.06644 [quant-ph]. [3] M. Cerezo, G. Verdon, H.-Y. Huang, L. Cincio, and P. J. Coles, Nature Computational Science 2, 567 (2022). [4] Y. Gujju, A. Matsuo, and R. Raymond, Phys. Rev. Appl. 21, 067001 (2024). [5] M. Schuld, R. Sweke, and J. J. Meyer, Phys. Rev. A 103, 032430 (2021). [6] M. Schuld and N. Killoran, Phys. Rev. Lett. 122, 040504 (2019). [7] M. C. Caro, H.-Y. Huang, M. Cerezo, K. Sharma, A. Sornborger, L. Cincio, and P. J. Coles, Nature Communications 13, 4919 (2022). [8] R. Sweke, J.-P. Seifert, D. Hangleiter, and J. Eisert, Quantum 5, 417 (2021). [9] H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Science 376, 1182 (2022), https://www.science.org/doi/pdf/10.1126/science.abn7293. [10] Y. Liu, S. Arunachalam, and K. Temme, Nature Physics 17, 1013 (2021). [11] H. Zhao, A. Zlokapa, H. Neven, R. Babbush, J. Preskill, J. R. McClean, and H.-Y. Huang, Exponential quantum advantage in processing massive classical data (2026), arXiv:2604.07639 [quant-ph]. [12] S. Thanasilp, S. Wang, M. Cerezo, and Z. Holmes, Exponential concentration in quantum kernel methods (2024), arXiv:2208.11060 [quant-ph]. [13] M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, Nature Communications 15, 7172 (2024). [14] M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Nature Reviews Physics 7, 174 (2025). [15] D. Wakeham and M. Schuld, Inference, interference and invariance: How the quantum fourier transform can help to learn from data (2024), arXiv:2409.00172 [quant-ph]. [16] P. Simidzija, E. Koskin, E. Y. Zhu, M. Dascal, and M. Schuld, Solving approximate hidden subgroup problems: quantum heuristics to detect weak entanglement (2026), arXiv:2603.15733 [quant-ph]. [17] L. Coffman, N. L. Diaz, M. Larocca, M. Schuld, and M. Cerezo, Group fourier filtering of quantum resources in quantum phase space (2026), arXiv:2601.14225 [quant-ph]. [18] V. Belis, J. Bowles, R. Gupta, E. Peters, and M. Schuld, Spectral methods: crucial for machine learning, natural for quantum computers? (2026), v2, arXiv:2603.24654 [quant-ph]. [19] J.-G. Liu and L. Wang, Phys. Rev. A 98, 062324 (2018). [20] M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Quantum Science and Technology 4, 043001 (2019). [21] B. Coyle, D. Mills, V. Danos, and E. Kashefi, npj Quantum Information 6, 60 (2020). [22] X. Gao, E. R. Anschuetz, S.-T. Wang, J. I. Cirac, and M. D. Lukin, Phys. Rev. X 12, 021037 (2022). [23] M. S. Rudolph, S. Lerch, S. Thanasilp, O. Kiss, O. Shaya, S. Vallecorsa, M. Grossi, and Z. Holmes, npj Quantum Information 10, 116 (2024). [24] T. Hastie, R. Tibshirani, and J. Friedman, Model assessment and selection, in The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer New York, New York, NY, 2009) pp. 219–259. [25] J. M. Kübler, S. Buchholz, and B. Schölkopf, in Advances in Neural Information Processing Systems, Vol. 34 (2021) pp. 12661–12673. [26] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Prentice Hall, Upper Saddle River, NJ, 2009). [27] (American Mathematical Society, 2002) pp. 53–74. [28] R. O’Donnell, Analysis of Boolean Functions (Cambridge University Press, Cambridge, 2014). [29] W. Rudin, Fourier Analysis on Groups (Interscience Publishers, New York, 1962).

Related documents

• Control incompatibility in multiparameter quantum metrology#1000479
• Hybrid Quantum-Classical Generative Adversarial Networks with Transfer Learning#139208
• Fixed, Driven, and Autonomous Coherent Generators in Quantum Wasserstein Geometry Trajectory compatibility, driven lifting, and model-level invariance in open quantum dynamics#650036
• Generative models of 21cm Epoch of Reionisation lightcones with 3D scattering transforms#126765
• Manifold-Aware Perturbations for Constrained Generative Modeling#1144756