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

<title>Abstract</title> <p>Symmetry is a central source of inductive bias in learning, but in quantum models the symmetry available to a circuit is shaped by the data encoding.We show that this changes what it means for a quantum circuit to be convolutional.For address/amplitude image encodings such as FRQI, pixel translations on the finite periodic lattice are represented by pixel cyclic shifts (PCS), which act as modular addition on the spatial index register, whereas many MERA-inspired QCNN templates are equivariant under cyclic permutations of physical qubits.We formalize this mismatch between PCS and qubit cyclic shifts (QCS), and construct QCNN layers that commute exactly with the PCS symmetry induced by the encoding.We characterize the resulting layer family as the unitary commutant of the encoded translation operator, yielding a Fourier-space constructive form for PCS-equivariant quantum convolution.Building on this characterization, we construct a multiscale PCS-QCNN with measurement-conditioned pooling and analyze its trainability with gradient diagnostics that separate parameter-count effects from suppression of the optimization-relevant gradient norm.Exact PCS equivariance is enforced for the unitary quantum convolutional blocks on their active registers; pooling yields a multiscale/coarse equivariance structure, and in the reported hybrid classifier an unconstrained linear-softmax head maps readout probabilities to labels, so label invariance is learned rather than imposed.To make the empirical comparison diagnostic of convolutional inductive bias, we use a translated-MNIST benchmark with matched classical CNN/MLP controls that exhibit a separation between translation-aware and dense architectures.On this benchmark, the PCS-QCNN achieves higher accuracy than a matched non-PCS random-basis quantum control.Full-MNIST experiments provide a non-translated reference, and finite-shot diagnostics identify a possible train--deploy mismatch at fixed shot budget, making sampling cost a deployment-relevant part of model selection.</p>