Pixel-Translation-Equivariant Quantum Convolutional Neural Networks via Fourier Multiplexers
This work addresses a foundational problem in quantum machine learning by enabling exact translation equivariance in quantum neural networks, which is incremental but crucial for improving their performance and scalability.
The paper tackles the mismatch between translation equivariance in classical convolutional neural networks and its quantum analogues by constructing quantum convolutional neural network layers that exactly commute with pixel cyclic shift symmetry induced by data encoding, resulting in a deep architecture with proven trainability that avoids depth-induced barren plateaus.
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.