Permutation-equivariant quantum convolutional neural networks

TL;DR

It is demonstrated that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of Sn and its subgroups, and the novel EQCNN architecture corresponding to the full permutation group Sn is built by applying all possible QCNNs with equal probability.

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. The subgroups of $S_{n}$ arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, e.g. reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this work 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.

Figures (10)

• Figure 1: In the center, a $4\times 4$ image with pixel values $\{a, b, ..., p\}$ indicated in the top left corner of each pixel and corresponding pixel positions in the bottom right corner. On the right, the image after vertical reflection. On the left, the image after rotation by angle $\pi/2$.
• Figure 2: (a) The structure of reflection-equivariant QCNN for 16 qubits. The cyan and the orange boxes represent respectively the parametrized convolutional and the pooling ansatze. For the control qubits in pooling layer, we use half-filled circles to indicate that the pooling ansatze depends on both the cotrol states $\{\vert 0\rangle, \vert 1\rangle\}$. (b) Reflection-equivariant QCNN using only nearest-neighbour connections in a 16-qubit quantum register with square-lattice architecture.
• Figure 3: EQCNN for reflection and $\pi/2$-rotation symmetry for 16 qubits. For non-nearest-neighbour applications of two-qubit convolutional ansatze in the first and third layer, we use same integers to clearly indicate qubits on which they apply within a sub-layer .
• Figure 4: (a) A schematic diagram showing all possible QCNNs and the qubit indices on which they act for an $S_{4}$-equivariant QCNN. Each arrow indicates a possible pooling operation. The number of different pooling operations in a layer is shown in the vertical column on the left. We highlight two examples of these QCNNs with red and blue arrows. (b) A circuit representation of the red and blue QCNNs and corresponding measurements. $C_{1}$ and $C_{2}$ are respectively the first and second convolutional layer.
• Figure 5: The circuit for realizing EQCNN for symmetric group $S_{4}$ on the input state $\rho$. In the controlled gates, the circles with no fillings and black fillings represent states $\vert 0\rangle$ and $\vert 1\rangle$ respectively.