Resource-efficient equivariant quantum convolutional neural networks

TL;DR

This paper tackles the resource bottlenecks of equivariant quantum neural networks by introducing an equivariant split-parallelizing QCNN (sp-QCNN) that encodes general symmetries through pooling-layer circuit splitting. It provides a group-theoretic subgroup-coset construction to build symmetry-preserving, parallelizable QCNN layers, achieving improved measurement and gradient efficiencies while maintaining trainability and avoiding barren plateaus. The authors prove general properties of the sp-QCNN, including potential $O(n)$ reductions in measurement resources and a barren-plateau-free landscape under modest assumptions, and demonstrate superior performance on a noisy quantum data classification task with $D_4$ symmetry. This work advances practical quantum machine learning on near-term devices and offers a scalable framework for incorporating broader symmetries into QCNNs, with future directions in classical simulability and higher-dimensional quantum systems.

Abstract

Equivariant quantum neural networks (QNNs) are promising quantum machine learning models that exploit symmetries to provide potential quantum advantages. 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 neural 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.

Figures (7)

• Figure 1: Basic structures of (a) conventional QCNN Cong2019-ov, (b) equivariant randomized QCNN Nguyen2022-go, and (c) equivariant sp-QCNN. (a) In the conventional QCNN, some qubits are discarded at each pooling layer, and the remaining qubits are measured at the end of the circuit. (b) The equivariant randomized QCNN randomly chooses which qubits to discard at each pooling layer to ensure the equivariance. The blue (red) bars indicate the pooling layer that discards even-(odd-)numbered qubits. (c) The equivariant sp-QCNN splits the circuit to impose the equivariance rather than discarding the qubits. This splitting structure with equivariance results in high trainability, generalization, and measurement efficiency.
• Figure 2: Correct and wrong examples of circuit splitting for translational symmetry $G=\{T^0,T^1,\cdots,T^{n-1}\}$, where $T$ is the translational operation acting as $T(q_i)=q_{i-1}$ ($q_i$ is the $i$th qubit). In the correct example, the $G$-invariance holds: $T(Q_1)=Q_2$ with $Q_1=\{q_1,q_3,q_5,q_7\}$ and $Q_2=\{q_2,q_4,q_6,q_8\}$. In contrast, the $G$-invariance does not hold in the wrong example: $T(Q_1)\neq Q_2$ with $Q_1=\{q_1,q_2,q_3,q_4\}$ and $Q_2=\{q_5,q_6,q_7,q_8\}$.
• Figure 3: An example of circuit splitting for $D_4=\{e, c_4, (c_4)^2, (c_4)^3, \sigma_1, \sigma_2, \sigma_3, \sigma_4 \}$ symmetry ($e$ is an identity operator, $c_4$ is a rotation by $\pi/2$, and $\sigma_i$'s are inversion operations around each axis). Each circle represents a qubit, and qubits with the same color are $G$-equivalent (see Definition \ref{['def: G-equivalence']}). We can construct the circuit splitting from a subgroup $H^{(\ell)}$ and a qubit subset $P^{(\ell)}$.
• Figure 4: (a) $2\times2\times2$ cubic lattice that consists of three types of bonds: $A$- (blue), $B$- (green), and $C$- (red) bonds. (b) This lattice is invariant under the action of $D_4=\{e,c_4,c_4^2, c_4^3,\sigma_1,\sigma_2,\sigma_3,\sigma_4 \}$. (c) The circuit structure of $D_4$-equivariant sp-QCNN used in this work. The quantum gates with the same color in each layer indicate that the rotation angles are shared. The details are provided in Appendix \ref{['ap: unitary ansatz']}.
• Figure 5: (a)--(c) Changes in training loss, test loss, and test accuracy during training. The solid, dotted, and dashed lines denote the results for equivariant sp-QCNN, non-equivariant sp-QCNN, and equivariant randomized QCNN, respectively. The shaded areas indicate the standard deviation for the twenty sets of initial parameters. The number of training data is $2N_t=20$. (d)--(f) Training loss, test loss, and test accuracy after sufficiently long training processes for various numbers of training data. The circles, triangles, and squares denote the results for equivariant sp-QCNN, non-equivariant sp-QCNN, and equivariant randomized QCNN, respectively. The shaded areas indicate the standard deviation for the twenty sets of initial parameters.

Theorems & Definitions (16)

• Definition 1: $G$-equivalence of qubits
• Definition 2: $G$-independence of qubits
• Definition 3: $G$-completeness of qubits
• Definition 1: $G$-equivalence of qubits
• Definition 2: $G$-independence of qubits
• Definition 3: $G$-completeness of qubits
• Theorem 1
• proof
• Corollary 1
• proof