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

TL;DR

This paper introduces a new class of quantum convolutional neural networks that are exactly equivariant under pixel cyclic shifts by leveraging Fourier multiplexers and the quantum Fourier transform.

Contribution

It formalizes the notion of translation symmetry in quantum CNNs and constructs layers that commute with pixel cyclic shifts, enabling more effective quantum image processing.

Findings

Constructive characterization of all PCS-equivariant unitaries.

Deep PCS-QCNN with measurement-induced pooling and inter-layer QFT cancellation.

Proved a lower bound on gradient norm, indicating no depth-induced barren plateau.

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.