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.

翻译：卷积神经网络的成功很大程度上归功于对平移等变性的硬编码。量子卷积神经网络（QCNNs）被提出作为近期的量子类比，但相关的平移概念取决于数据编码方式。对于FRQI等地址/振幅编码，像素平移表现为索引寄存器上的模加法，而许多受MERA启发的QCNN仅对物理量子比特的循环排列具有等变性。我们形式化了这一不匹配问题，并构建了与编码所诱导的像素循环平移（PCS）对称性精确对易的QCNN层。我们的主要技术成果是对所有PCS等变酉算子的一种构造性表征：通过量子傅里叶变换（QFT）的对角化操作实现了平移的对角化，因此任何PCS等变层都是一个傅里叶模式复用器后接逆量子傅里叶变换（IQFT）。基于这一表征，我们提出了一种深度PCS-QCNN，采用了测量诱导池化、延迟条件操作和层间QFT抵消技术。我们还分析了随机初始化下的可训练性，并证明了期望平方梯度范数的下界在深度缩放机制下保持恒定，从而排除了该意义上深度诱导的贫瘠高原效应。