Approximately Equivariant Quantum Neural Network for $p4m$ Group Symmetries in Images

TL;DR

This work proposes equivariant Quantum Convolutional Neural Networks (EquivQCNNs) for image classification under planar p4m symmetry, including reflectional and 90° rotational symmetry, and presents the results tested in different use cases, proving that the equivariance fosters better generalization of the model.

Abstract

Quantum Neural Networks (QNNs) are suggested as one of the quantum algorithms which can be efficiently simulated with a low depth on near-term quantum hardware in the presence of noises. However, their performance highly relies on choosing the most suitable architecture of Variational Quantum Algorithms (VQAs), and the problem-agnostic models often suffer issues regarding trainability and generalization power. As a solution, the most recent works explore Geometric Quantum Machine Learning (GQML) using QNNs equivariant with respect to the underlying symmetry of the dataset. GQML adds an inductive bias to the model by incorporating the prior knowledge on the given dataset and leads to enhancing the optimization performance while constraining the search space. This work proposes equivariant Quantum Convolutional Neural Networks (EquivQCNNs) for image classification under planar $p4m$ symmetry, including reflectional and $90^\circ$ rotational symmetry. We present the results tested in different use cases, such as phase detection of the 2D Ising model and classification of the extended MNIST dataset, and compare them with those obtained with the non-equivariant model, proving that the equivariance fosters better generalization of the model.

Figures (6)

• Figure 1: Schematic diagram of the action of $p4m$ symmetry on 2D images of size $4 \times 4$ encoded using CAA embedding method with $2$ qubits. The pixel at position $(i,j)$ is associated with a computational basis $\ket{i}\ket{j}$.
• Figure 2: A Schematic diagram of (a) EquivQCNN and (b) Appr-EquivQCNN for an example of 8 qubits to classify image of size $16\times 16$. They consist of $U_2$ (yellow rectangle) and $U_4$ (blue rectangle) convolutional filters (c.f. Fig. \ref{['fig:circuit']}), followed by pooling layers (green circle). Both models contain a preliminary scanning phase, where $U_2$ acts on $q_{1:n}$ and $q_{n+1:2n}$ separately. EquivQCNN then consists of $U_4$ ansatz, while Appr-EquivQCNN is subject to a small noise by connecting $q_{1:n}$ and $q_{n+1:2n}$ with $U_2$ gate.
• Figure 3: Parameterized quantum circuits ansatz, $U_2$ (yellow rectangle) and $U_4$ (blue rectangle), used the convolutional filters equivariant with respect to $p4m$ symmetry group.
• Figure 4: Approximately Invariant Measurement process. We apply $R_z(\phi)$ and $H$ gate on two qubits, $q_{i_m} \in q_{1:n}$ and $q_{i_m+n} \in q_{n+1:2n}$, and measure the probability distribution on each qubits separately. The final label is computed by summing up the two distributions and taking its half.
• Figure 5: Examples of the Ising model and the extended MNIST image samples with size $16\times16$ used for binary classification. Each row corresponds to each class.