Equivariant quantum circuits for learning on weighted graphs

Abstract

Variational quantum algorithms are the leading candidate for advantage on near-term quantum hardware. When training a parametrized quantum circuit in this setting to solve a specific problem, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. In this work, we introduce an ansatz for learning tasks on weighted graphs that respects an important graph symmetry, namely equivariance under node permutations. We evaluate the performance of this ansatz on a complex learning task, namely neural combinatorial optimization, where a machine learning model is used to learn a heuristic for a combinatorial optimization problem. We analytically and numerically study the performance of our model, and our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML.

On the universality of Sn-equivariant k-body gates

“…i.e. there are as many elements in the center as irreps in equation (10). Finally, let us note that su…”

“…where Q is a representation defined by the right-hand-side of equation (10). We then define the maximal special subalgebra su…”

“…Numerous endeavors have been undertaken to create learning models that are tailored specifically to a given task. Among these, geometric QML (GQML) has emerged as one of the most promising approaches [8][9][10][11][12][13][14][15]. The fundamental idea behind GQML is to leverage the symmetries present in the task to develop sharp inductive biases for the learning models.…”

A Comparison between Invariant and Equivariant Classical and Quantum Graph Neural Networks

“…The EQGNN takes the same form as the QGNN; however, we aggregate the final elements of the product state 1 2 n ∑ 2 n k=1 |ψ P k ⟩ via mean pooling before sending this complex value to a fully connected NN [31,40,41]. See Appendix C for a proof of the quantum product state permutation equivariance over the sum of its elements.…”

Building spatial symmetries into parameterized quantum circuits for faster training

“…Following conceptual advances in the field of classical machine learning [16][17][18], emphasis has recently been put forward to develop the framework of geometric quantum machine learning (QML), where one structures quantum learning models based on symmetries of the task to be solved [19][20][21]. In the context of quantum circuit design, respecting (all or some of) the symmetries of a problem can be achieved by an appropriate choice of the type of gates constituting the circuit employed, and also, imposing correlation patterns in the parameters of such gates.…”