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Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data

Koki Chinzei, Quoc Hoan Tran, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato

TL;DR

This work introduces the split-parallelizing QCNN (sp-QCNN), a symmetry-aware architecture that exploits translational invariance to parallelize quantum convolutional neural networks without increasing qubit count. By enforcing translational symmetry and performing circuit splitting at pooling layers, sp-QCNN achieves an $O(n)$ improvement in measurement efficiency for local observables and gradient estimates, addressing a key bottleneck in quantum machine learning on NISQ devices. Applied to quantum phase recognition in a translationally symmetric 1D model, sp-QCNN delivers comparable classification accuracy to conventional QCNN while requiring far fewer measurements, enabling faster and more robust training under limited resources. The approach connects to geometric quantum machine learning and opens avenues for symmetry-based hardware-efficient QNN designs with potential practical quantum advantages.

Abstract

The quantum convolutional neural network (QCNN) is a promising quantum machine learning (QML) model that is expected to achieve quantum advantages in classically intractable problems. However, the QCNN requires a large number of measurements for data learning, limiting its practical applications in large-scale problems. To alleviate this requirement, we propose a novel architecture called split-parallelizing QCNN (sp-QCNN), which exploits the prior knowledge of quantum data to design an efficient model. This architecture draws inspiration from geometric quantum machine learning and targets translationally symmetric quantum data commonly encountered in physics and quantum computing science. By splitting the quantum circuit based on translational symmetry, the sp-QCNN can substantially parallelize the conventional QCNN without increasing the number of qubits and improve the measurement efficiency by an order of the number of qubits. To demonstrate its effectiveness, we apply the sp-QCNN to a quantum phase recognition task and show that it can achieve comparable classification accuracy to the conventional QCNN while considerably reducing the measurement resources required. Due to its high measurement efficiency, the sp-QCNN can mitigate statistical errors in estimating the gradient of the loss function, thereby accelerating the learning process. These results open up new possibilities for incorporating the prior data knowledge into the efficient design of QML models, leading to practical quantum advantages.

Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data

TL;DR

This work introduces the split-parallelizing QCNN (sp-QCNN), a symmetry-aware architecture that exploits translational invariance to parallelize quantum convolutional neural networks without increasing qubit count. By enforcing translational symmetry and performing circuit splitting at pooling layers, sp-QCNN achieves an improvement in measurement efficiency for local observables and gradient estimates, addressing a key bottleneck in quantum machine learning on NISQ devices. Applied to quantum phase recognition in a translationally symmetric 1D model, sp-QCNN delivers comparable classification accuracy to conventional QCNN while requiring far fewer measurements, enabling faster and more robust training under limited resources. The approach connects to geometric quantum machine learning and opens avenues for symmetry-based hardware-efficient QNN designs with potential practical quantum advantages.

Abstract

The quantum convolutional neural network (QCNN) is a promising quantum machine learning (QML) model that is expected to achieve quantum advantages in classically intractable problems. However, the QCNN requires a large number of measurements for data learning, limiting its practical applications in large-scale problems. To alleviate this requirement, we propose a novel architecture called split-parallelizing QCNN (sp-QCNN), which exploits the prior knowledge of quantum data to design an efficient model. This architecture draws inspiration from geometric quantum machine learning and targets translationally symmetric quantum data commonly encountered in physics and quantum computing science. By splitting the quantum circuit based on translational symmetry, the sp-QCNN can substantially parallelize the conventional QCNN without increasing the number of qubits and improve the measurement efficiency by an order of the number of qubits. To demonstrate its effectiveness, we apply the sp-QCNN to a quantum phase recognition task and show that it can achieve comparable classification accuracy to the conventional QCNN while considerably reducing the measurement resources required. Due to its high measurement efficiency, the sp-QCNN can mitigate statistical errors in estimating the gradient of the loss function, thereby accelerating the learning process. These results open up new possibilities for incorporating the prior data knowledge into the efficient design of QML models, leading to practical quantum advantages.
Paper Structure (17 sections, 1 theorem, 35 equations, 10 figures)

This paper contains 17 sections, 1 theorem, 35 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Review of QCNN
  3. Split-parallelizing QCNN
  4. Translational symmetry
  5. Circuit splitting
  6. Relation with geometric quantum machine learning
  7. Measurement efficiency in sp-QCNN
  8. Measurement efficiency of local observable
  9. Measurement efficiency of gradient
  10. Measurement efficiency for random state
  11. Application to quantum phase recognition
  12. Formulation of problem
  13. Performance with limited measurement resources
  14. Quantification of measurement efficiency
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 1

For $z\neq \pm n$, the following relation holds in the limit of $n\to\infty$: with $\ell_z=(n+z)/2$. Here $\binom{\cdot}{\cdot}$ denotes the binomial coefficient. This theorem states that the asymptotic form of $\text{dim}V_z$ is $\binom{n}{\ell_z}/n$.

Figures (10)

  • Figure 1: Basic structures of (a) conventional and (b) sp-QCNNs. In the figure, C, P, and FC represent convolutional, pooling, and fully-connected layers, respectively. (a) In the conventional QCNN, some qubits are discarded at each pooling layer, and only one of the remaining qubits is measured in the end to classify the quantum data. (b) In the sp-QCNN, the translational symmetry of data is used as prior knowledge to design an efficient QML model. The circuit of the sp-QCNN (the left circuit) consists of translationally symmetric layers and splitting structures, allowing us to substantially parallelize the nonsplitting QCNN (the right circuit) to improve the measurement efficiency.
  • Figure 2: Example of translationally symmetric unitary layer. Single qubit rotations are applied in parallel, followed by ZZ rotations on the nearest neighboring qubits. These procedures are repeated $d$ times. The rotation angles are translationally symmetric, and thus the number of independent parameters is $4d$.
  • Figure 3: (a) An illustration of translationally symmetric circuit splitting. In this circuit, the entire circuit structure is invariant under the translation operation.
  • Figure 4: Mechanism of parallelization in the sp-QCNN. (a) In the sp-QCNN, the expectation value of a local observable is equivalent for all the qubits. This can be proved by virtually translating the entire circuit. The translation does not change the input state and quantum circuit due to their translational symmetry but shifts the position of the measured qubit, showing the equivalence of expectation values at different qubits. (b) The gradient measurement can be parallelized in the sp-QCNN. In accordance with the chain rule, the gradient is the sum of several derivatives, $\partial \braket{Z_1}/\partial \theta = \sum_j \partial \braket{Z_1}/\partial \theta_j$. For example, we suppose that the parameter $\theta$ is in the first convolutional layer as shown in the figure (the red boxes denote $\partial/\partial \theta_2$ and $\partial/\partial \theta_1$). Then translating the circuit proves $\partial \braket{Z_1}/\partial \theta_j = \partial \braket{Z_{j-2}}/\partial \theta_1$ and thus $\partial \braket{Z_1}/\partial \theta = \sum_j \partial \braket{Z_j}/\partial \theta_1$, which can be computed with only two circuits by measuring all the qubits.
  • Figure 5: (a) Quantification of measurement efficiency. In actual experiments, statistical errors arise in estimating the expectation value of an observable. This figure shows the probability distribution of the estimated expectation value. Here, we define the relative measurement efficiency $r$ as the ratio of the variances in the sp-QCNN and the nonsplitting QCNN. (b) Number of eigenstates of $Z_1$ and $Z_\text{avg}$ with an eigenvalue $s$. While the possible measurement outcome is $\pm1$ in the nonsplitting QCNN (left panel), it is widely distributed in the range of $-1$ to 1 with a width of $\mathcal{O}(1/\sqrt{n})$ in the sp-QCNN (right panel).
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof