quant-phApr 7, 2026
SavePixel-Translation-Equivariant Quantum Convolutional Neural Networks via Fourier Multiplexers
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.
Figure 2: Fixed benchmark controls used to expose translation-sensitive inductive bias. Training uses a balanced subset of images per class, and the standard MNIST test split membership is kept unchanged. Digits are resized from to , placed on a canvas, and randomly translated with maximum offset 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 th– th percentile range over 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 , applies the mode multiplexer , and maps back with before pooling measurements are performed on the least significant qubits of the active computational index registers. The resulting classical outcome is recorded and used as a classical control for the next multiplexer block. After repeated pooling steps (indicated by ), the final layer acts on a smaller active index register and is followed by full measurement of the remaining quantum output. Here and denote all remaining index qubits with labels .
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 -class logits.
Figure 5: Infinite-shot test-accuracy dynamics for the PCS-QCNN sweeps. (a) Translated-MNIST architecture sweep over quantum layers and feature-qubit counts with full readout on the translated benchmark with digits placed on a canvas; colors encode and line styles encode . (b) Full-MNIST size sweep for the canonical , PCS-QCNN across four preprocessing settings: direct , direct , centered in a canvas, and direct . In both panels, solid curves show mean test accuracy and shaded bands show the th– th percentile range over seeds.
| Model / baseline | Params | Accuracy (%) |
|---|---|---|
| Classical CNN | ||
| Classical MLP | ||
| PCS-QCNN | ||
| RBC-QCNN |
Table 1: Final mean test accuracies for the translated benchmark entries in Fig. 2 ( training examples per class, with digits resized to , placed on a canvas, and translated by at most pixels along each axis for the classical and quantum runs). The parameter column reports the total number of trainable parameters. Both quantum variants use , , and full readout. The reported value in each row is the mean over seeds.
Figure 6: Initialization-time quantum-gradient diagnostics in the depth-scaling family. Solid curves show the empirical-loss gradient norm and dashed curves show the per-sample RMS gradient norm , 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- full-MNIST reference PCS-QCNN ( , ). Each curve corresponds to an inference-time shot budget ( , , , , , 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.
Figure 8: Batch-mean test cross-entropy distributions for one representative direct- , model under finite-shot readout. (a) After training epochs. (b) After training epochs. Colored histograms correspond to , , , , and infinite-shot readout. Each finite-shot entry is the mean cross-entropy on one -sample test batch under one of independent multinomial resampling passes, giving entries per finite-shot histogram; the exact reference has 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 | 85–88 | |
| Huang et al. [ 17 ] | 10-class | 96 | |
| Li et al. [ 27 ] | Binary | (label pairs) | 99.80 (4 vs 5), 91.51 (3 vs 8) |
| Oh et al. [ 35 ] | 10-class | 95 | |
| Huggins et al. [ 18 ] | Binary | 95 | |
| Jing et al. [ 20 ] | 10-class | and | 94–95 |
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. (%) |
|---|---|---|---|---|---|
| , | 11 | 4 096 | 20 490 | 51.29 | |
| , | 12 | 16 384 | 40 970 | 62.50 | |
| , | 13 | 65 536 | 81 930 | 66.36 | |
| , | 11 | 8 192 | 5 130 | 59.02 | |
| , | 12 | 32 768 | 10 250 | 76.00 | |
| , | 13 | 131 072 | 20 490 | 76.85 |
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 | 9 | 4 096 | 5 130 | 93.21 | |
| direct | 11 | 16 384 | 20 490 | 97.96 | |
| on | 13 | 65 536 | 81 930 | 97.96 | |
| direct | 13 | 65 536 | 81 930 | 98.13 |
Table S3: Exact parameter accounting for the PCS-QCNN size sweep in Fig. 5 (b).
Figure S1: Brightness-range sensitivity sweep for the FRQI encoder scale. Panels show accuracy at epochs and as a function of the upper endpoint in the map with fixed lower endpoint . The article-wide benchmark convention was fixed independently of this sweep. Each curve corresponds to one architecture ; lines show the mean over seeds and shaded bands show the seedwise minimum–maximum range.
Figure S2: Classical reference architectures used in Fig. 2 (a). The same 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 trainable parameters. (b) Pure-dense MLP control with trainable parameters.
Figure S3: Classical CNN and MLP controls on the full standard MNIST split resized directly to 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 th– th percentile range over 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- reference run after epochs and finite-shot budgets , , , , and ; shaded bands show the interquartile range over test samples.
Figure S5: Readout-space loss landscape for the representative direct- reference PCS-QCNN after (a) and (b) 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 grid covering in sigma units for shots, and cells with valid fraction below are shown in white. Each panel has its own colorbar.
Figure S6: Qualitative error analysis for the representative final direct- full-MNIST PCS-QCNN after 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 into controlled operations on the feature qubits. The example uses a image (one qubit in each of the and registers) for clarity.
Figure S8: Reduction of the Fourier junction between two adjacent PCS-QCNN layers. Left: the original junction , where measures the spatial parity bit in the split . Right: the reduced implementation from Lemma S3 : in the Fourier alias split , apply and measure , then conditionally (for outcome ) apply the single-qubit -rotations on the -register.