Abstract 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.
Abstract 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.
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.
Abstract The Symmetric group S n manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. Subgroups of S n arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to S n and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of S n . Our novel EQCNN architecture corresponding to the full permutation group S n is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of S n , our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The S n -equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the S n -equivariant QCNN shows better average performance compared to S n -equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.
This work presents an implementation-grounded quantum convolutional neural network (QCNN) pipeline in which classical images are amplitude-encoded with the quantum signal-induced heap transform (QsiHT), transformed by a 2-D quantum Fourier transform (2-D QFT), and classified by a two-stage QCNN. Each convolution stage applies shared single-qubit rotations and ring CNOT entanglement, pooling is unitary-only, and prediction is obtained from the final readout qubit probability via thresholding. Training uses parameter-shift gradients with Adam-style updates and includes practical regularization through random-affine data augmentation, ℓ2 weight decay, and class-balanced subset sampling. Experiments in IBM Qiskit on a binary MNIST task (classes (0, 1)) evaluate 4-qubit (4 × 4) and 6-qubit (8 × 8) models under finite-shot simulation (256 shots), yielding final validation accuracies of 91.41% and 99.80%, respectively. These results indicate that the QsiHT+QFT front-end integrates effectively with shallow QCNN layers and benefits from increased qubit resolution in this setting.
As image datasets grow in scale and complexity, classical convolutional neural networks (CNNs) increasingly face limitations in computational efficiency, scalability, and generalisation in low-data regimes. Quantum machine learning (QML) offers a promising alternative, with quantum convolutional neural networks (QCNNs) exploiting quantum parallelism and entanglement for feature extraction. This paper introduces the Novel Quantum Convolutional Neural Network (No-QCNN), which differs from recent hybrid QCNN/QCQ-CNN architectures by employing an end-to-end quantum convolution-pooling pipeline without inserting an intermediate classical CNN block. No-QCNN is a hybrid quantum-classical model that employs a variational quantum classifier (VQC) optimised via the COBYLA algorithm and is designed for binary and multiclass classification of low-resolution colour images on near-term quantum devices. A key innovation is a problem-specific quantum feature map that pre-processes each image into a structured three-dimensional block-matrix representation, jointly encoding pixel colour (R, G, B) and spatial position before mapping this information into a hierarchical ZZFeatureMap. This encoding captures spatial-chromatic correlations at shallow circuit depth, making it compatible with noisy intermediate-scale quantum (NISQ) constraints. The model is implemented using IBM-Qiskit on a local quantum simulator and benchmarked against a classical CNN. For a six-class classification task on a 50-image dataset, No-QCNN achieves a validation accuracy of 82.05%, substantially outperforming the classical CNN's 40.00%, indicating improved generalisation and reduced overfitting in low-data, multiclass settings. Conversely, for a simpler binary classification task, the classical CNN achieves perfect validation accuracy of 100%, surpassing the No-QCNN's 89.7%. We further observe that No-QCNN performance declines with increasing dataset size and training time due to limited circuit expressibility, data-encoding overhead, and finite-sampling noise inherent to NISQ models. Overall, these results position No-QCNN as a complementary framework best suited to low-data, correlation-rich classification tasks, defining a realistic niche for quantum-enhanced artificial vision and quantum perception in the NISQ era.
Abstract Equivariant quantum neural networks (QNNs) are promising variational models that exploit symmetries to improve machine learning capabilities. Despite theoretical developments in equivariant QNNs, their implementation on near-term quantum devices remains challenging due to limited computational resources. This study proposes a resource-efficient model of equivariant quantum convolutional networks (QCNNs) called equivariant split-parallelizing QCNN (sp-QCNN). Using a group-theoretical approach, we encode general symmetries into our model beyond the translational symmetry addressed by previous sp-QCNNs. We achieve this by splitting the circuit at the pooling layer while preserving symmetry. This splitting structure effectively parallelizes QCNNs to improve measurement efficiency in estimating the expectation value of an observable and its gradient by order of the number of qubits. Our model also exhibits high trainability and generalization performance, including the absence of barren plateaus. Numerical experiments demonstrate that the equivariant sp-QCNN can be trained and generalized with fewer measurement resources than a conventional equivariant QCNN in a noisy quantum data classification task. Our results contribute to the advancement of practical quantum machine learning algorithms.
Quantum convolutional neural networks (QCNNs) offer a promising architecture for near-term quantum machine learning by combining hierarchical feature extraction with modest parameter growth. However, any QCNN operating on classical data must rely on an encoding scheme to embed inputs into quantum states, and this choice can dominate both performance and resource requirements. This work presents an implementation-level comparison of three representative encodings -- Angle, Amplitude, and a Hybrid phase/angle scheme -- for QCNNs under depolarizing noise. We develop a fully differentiable PyTorch--Qiskit pipeline with a custom autograd bridge, batched parameter-shift gradients, and shot scheduling, and use it to train QCNNs on downsampled binary variants of MNIST and Fashion-MNIST at 4×4 and 8×8 resolutions.
Our experiments reveal regime-dependent trade-offs. On aggressively downsampled 4×4 inputs, Angle encoding attains higher accuracy and remains comparatively robust as noise increases, while the Hybrid encoder trails and exhibits non-monotonic trends. At 8×8, the Hybrid scheme can overtake Angle under moderate noise, suggesting that mixed phase/angle encoders benefit from additional feature bandwidth. Amplitude-encoded QCNNs are sparsely represented in the downsampled grids but achieve strong performance in lightweight and full-resolution configurations, where training dynamics closely resemble classical convergence. Taken together, these results provide practical guidance for choosing QCNN encoders under joint constraints of resolution, noise strength, and simulation budget.