Pixel-Translation-Equivariant Quantum Convolutional Neural Networks via Fourier Multiplexers

Dmitry Chirkov; Igor Lobanov

quant-phApr 7, 2026

Pixel-Translation-Equivariant Quantum Convolutional Neural Networks via Fourier Multiplexers

Authors: Dmitry Chirkov, Igor Lobanov

Organizations: School of Physics and Engineering, ITMO University, St. Petersburg, Russia

Abstract

Convolutional neural networks owe much of their success to hard-coding translation equivariance. Quantum convolutional neural networks (QCNNs) have been proposed as near-term quantum analogues, but the relevant notion of translation depends on the data encoding. For address/amplitude encodings such as FRQI, a pixel shift acts as modular addition on an index register, whereas many MERA-inspired QCNNs are equivariant only under cyclic permutations of physical qubits. We formalize this mismatch and construct QCNN layers that commute exactly with the pixel cyclic shift (PCS) symmetry induced by the encoding. Our main technical result is a constructive characterization of all PCS-equivariant unitaries: conjugation by the quantum Fourier transform (QFT) diagonalizes translations, so any PCS-equivariant layer is a Fourier-mode multiplexer followed by an inverse QFT (IQFT). Building on this characterization, we introduce a deep PCS-QCNN with measurement-induced pooling, deferred conditioning, and inter-layer QFT cancellation. We also analyze trainability at random initialization and prove a lower bound on the expected squared gradient norm that remains constant in a depth-scaling regime, ruling out a depth-induced barren plateau in that sense.

Figures & tables

Figure 1: Schematic comparison of qubit- vs pixel-translation symmetry for address/amplitude encodings. (a) A typical QCNN convolution pattern based on repeating the same local unitary on different qubit pairs (adapted from Ref. [ 13 ] ). (b) Under address encoding, translating the input image corresponds to modular addition on the index register, so reusing the same local unitary across qubit positions does not, in general, enforce pixel-translation equivariance. (c) The same pattern can still be equivariant under cyclic permutations of qubits (QCS). (d) Required symmetry: equivariance under cyclic shifts of pixel indices (PCS), i.e., commutation with the translation action induced by the encoding.

Panel of Figure 2 — Figure 2: Fixed benchmark controls used to expose translation-sensitive inductive bias. Training uses a balanced subset of $1000$ images per class, and the standard MNIST test split membership is kept unchanged. Digits are resized from $28\times 28$ to $16\times 16$ , placed on a $32\times 32$ canvas, and randomly translated with maximum offset $8$ pixels per axis. (a) Classical baselines on this translated benchmark: a convolutional CNN reference and a pure-dense MLP control. (b) Quantum models: PCS-QCNN and a matched non-PCS random-basis control (RBC) QCNN on the same translated benchmark. Solid lines show mean test accuracy, train curves show mean train accuracy, and shaded bands show the $25$ th– $75$ th percentile range over $3$ seeds. The corresponding full-MNIST classical control without translations is shown separately in Supplemental Fig. .

Figure 3: Schematic of a multilayer PCS-QCNN. Each non-final layer maps the active index registers to the Fourier basis with

\mathcal{F}^{(\ell)}

, applies the mode multiplexer

\mathcal{B}^{(\ell)}

, and maps back with

\mathcal{F}^{(\ell)\dagger}

before pooling measurements are performed on the least significant qubits of the active computational index registers. The resulting classical outcome

m_{\ell}

is recorded and used as a classical control for the next multiplexer block. After repeated pooling steps (indicated by

\cdots

), the final layer acts on a smaller active index register and is followed by full measurement of the remaining quantum output. Here

x_{Q-1\ldots}

and

y_{Q-1\ldots}

denote all remaining index qubits with labels

Q-1,Q,\ldots

Figure 4: Overview of the hybrid model used in our experiments. The Encoding block prepares an FRQI-like image state on the spatial index registers and the feature register. The PCS-QCNN block applies one or more PCS-equivariant quantum layers with measurement-induced pooling implemented through deferred conditioning. The final measured probability tensor (or its finite-shot histogram estimate at evaluation time) is fed into a linear-softmax head that produces $10$ -class logits.

Panel of Figure 5 — Figure 5: Infinite-shot test-accuracy dynamics for the PCS-QCNN sweeps. (a) Translated-MNIST architecture sweep over quantum layers $Q\in\{1,\dots,5\}$ and feature-qubit counts $n_{f}\in\{1,2,3\}$ with full readout on the translated benchmark with $16\times 16$ digits placed on a $32\times 32$ canvas; colors encode $Q$ and line styles encode $n_{f}$ . (b) Full-MNIST size sweep for the canonical $Q=1$ , $n_{f}=3$ PCS-QCNN across four preprocessing settings: direct $8\times 8$ , direct $16\times 16$ , centered $28\times 28$ in a $32\times 32$ canvas, and direct $32\times 32$ . In both panels, solid curves show mean test accuracy and shaded bands show the $25$ th– $75$ th percentile range over $3$ seeds.

Model / baseline	Params	Accuracy (%)
Classical CNN	$47{,}034$	$97.68$
Classical MLP	$47{,}947$	$48.93$
PCS-QCNN	$39{,}434$	$75.89$
RBC-QCNN	$39{,}434$	$40.82$

Table 1: Final mean test accuracies for the translated benchmark entries in Fig. 2 (

1000

training examples per class, with digits resized to

16\times 16

, placed on a

32\times 32

canvas, and translated by at most

8

pixels along each axis for the classical and quantum runs). The parameter column reports the total number of trainable parameters. Both quantum variants use

Q=3

n_{f}=2

, and full readout. The reported value in each row is the mean over

3

seeds.

Figure 6: Initialization-time quantum-gradient diagnostics in the depth-scaling family. Solid curves show the empirical-loss gradient norm

\|G_{\mathcal{D}}\|_{2}

and dashed curves show the per-sample RMS gradient norm

R_{\mathcal{D}}

, defined in Eqs. ( 17 ) and ( 18 ). Colors indicate whether the selected coordinates are all quantum parameters, the first quantum layer, or the last quantum layer.

Figure 7: Finite-shot reevaluation of one representative direct-

16\times 16

full-MNIST reference PCS-QCNN (

Q=1

n_{f}=3

). Each curve corresponds to an inference-time shot budget (

128

256

512

1024

2048

, or infinite-shot exact readout). Each finite-shot point is one stochastic full-test-set pass with one independently sampled multinomial readout histogram per test image; the exact curve is deterministic. Training always uses exact probabilities; only inference is resampled.

Panel of Figure 8 — Figure 8: Batch-mean test cross-entropy distributions for one representative direct- $16\times 16$ $Q=1$ , $n_{f}=3$ model under finite-shot readout. (a) After $100$ training epochs. (b) After $800$ training epochs. Colored histograms correspond to $128$ , $256$ , $512$ , $1024$ , and infinite-shot readout. Each finite-shot entry is the mean cross-entropy on one $250$ -sample test batch under one of $100$ independent multinomial resampling passes, giving $4000$ entries per finite-shot histogram; the exact reference has $40$ entries. Vertical lines mark sample-weighted mean losses.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Work	Task	Input / preprocessing	Reported accuracy (%)
Mattern et al. [ 31 ]	10-class	$14\times 14$	85–88
Huang et al. [ 17 ]	10-class	$14\times 14$	96
Li et al. [ 27 ]	Binary	$4\times 4$ (label pairs)	99.80 (4 vs 5), 91.51 (3 vs 8)
Oh et al. [ 35 ]	10-class	$10\times 10$	95
Huggins et al. [ 18 ]	Binary	$8\times 8$	95
Jing et al. [ 20 ]	10-class	$10\times 10$ and $20\times 20$	94–95

Appendix

Table S1: Overview of representative QCNN-like MNIST results in the literature. Entries are shown in the form reported by the corresponding references. Our main paper uses MNIST with a fixed preprocessing and evaluation protocol and includes within-family matched controls (Sec. VI.3 ).

Configuration	Total qubits	Quantum params	Classifier params	Readout shape	Final mean acc. (%)
$Q=1$ , $n_{f}=1$	11	4 096	20 490	$32\times 32\times 2$	51.29
$Q=1$ , $n_{f}=2$	12	16 384	40 970	$32\times 32\times 4$	62.50
$Q=1$ , $n_{f}=3$	13	65 536	81 930	$32\times 32\times 8$	66.36
$Q=2$ , $n_{f}=1$	11	8 192	5 130	$16\times 16\times 2$	59.02
$Q=2$ , $n_{f}=2$	12	32 768	10 250	$16\times 16\times 4$	76.00
$Q=2$ , $n_{f}=3$	13	131 072	20 490	$16\times 16\times 8$	76.85

Appendix

Table S2: Exact parameter accounting for the PCS-QCNN architecture sweep in Fig. 5 (a).

Preprocessing	Total qubits	Quantum params	Classifier params	Readout shape	Final mean acc. (%)
direct $8\times 8$	9	4 096	5 130	$8\times 8\times 8$	93.21
direct $16\times 16$	11	16 384	20 490	$16\times 16\times 8$	97.96
$28\times 28$ on $32\times 32$	13	65 536	81 930	$32\times 32\times 8$	97.96
direct $32\times 32$	13	65 536	81 930	$32\times 32\times 8$	98.13

Appendix

Table S3: Exact parameter accounting for the PCS-QCNN size sweep in Fig. 5 (b).

Panel of Figure 13 — Figure S1: Brightness-range sensitivity sweep for the FRQI encoder scale. Panels show accuracy at epochs $150$ and $300$ as a function of the upper endpoint $b/\pi$ in the map $p=a+(b-a)x$ with fixed lower endpoint $a=0$ . The article-wide benchmark convention $(a,b)=(0,\pi)$ was fixed independently of this sweep. Each curve corresponds to one architecture $(Q,n_{f})\in\{1,2\}\times\{1,2\}$ ; lines show the mean over $3$ seeds and shaded bands show the seedwise minimum–maximum range.

Panel of Figure 14 — Figure S2: Classical reference architectures used in Fig. 2 (a). The same $32\times 32$ input models are used for the translated benchmark in the main text and for the full-MNIST control in Sec. B.4 . (a) Convolutional CNN baseline with $47{,}034$ trainable parameters. (b) Pure-dense MLP control with $47{,}947$ trainable parameters.

Figure S3: Classical CNN and MLP controls on the full standard MNIST split resized directly to

32\times 32

without translations. The architectures are exactly those specified in Fig. S2 ; only the data regime changes. Solid lines show mean test accuracy, train curves show mean train accuracy, and shaded bands show the

25

th–

75

th percentile range over

3

seeds. The gap is much smaller than on the translated benchmark, which illustrates why non-translated MNIST alone is a less stringent benchmark for testing convolutional inductive bias.

Figure S4: Mean Shannon entropy of the full readout distribution versus training epoch for the representative direct-

16\times 16

reference run after epochs

100,200,\dots,2000

and finite-shot budgets

128

256

512

1024

, and

2048

; shaded bands show the interquartile range over test samples.

Panel of Figure 17 — Figure S5: Readout-space loss landscape for the representative direct- $16\times 16$ reference PCS-QCNN after (a) $100$ and (b) $800$ training epochs. Each test sample is perturbed in its own two-dimensional local shot-noise PCA basis, the resulting cross-entropy is averaged over valid test samples on an $81\times 81$ grid covering $[-3,3]^{2}$ in sigma units for $N=128$ shots, and cells with valid fraction below $0.10$ are shown in white. Each panel has its own colorbar.

Panel of Figure 18 — Figure S6: Qualitative error analysis for the representative final direct- $16\times 16$ full-MNIST PCS-QCNN after $2000$ training epochs. (a) Confusion matrix (rows: true labels, columns: predicted labels); darker cells indicate higher counts. (b) Examples of misclassified MNIST digits with predicted labels shown in the lower-left corner and true labels in the lower-right corner of each image.

Figure S7: Decomposition of a block-diagonal multiplexer

\mathcal{B}

into controlled operations

U_{i}

on the feature qubits. The example uses a

2\times 2

image (one qubit in each of the

x

and

y

registers) for clarity.

Figure S8: Reduction of the Fourier junction between two adjacent PCS-QCNN layers. Left: the original junction

F_{N/2}\,\mathcal{M}_{s}\,F_{N}^{\dagger}

, where

\mathcal{M}_{s}

measures the spatial parity bit in the split

r=2q+s

. Right: the reduced implementation from Lemma S3 : in the Fourier alias split

k=q+\sigma N/2

, apply

H

and measure

\sigma

, then conditionally (for outcome

b=1

) apply the single-qubit

Z

-rotations on the

q

-register.

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