Equivariant quantum circuits for learning on weighted graphs

Variational quantum algorithms are the leading candidate for near-term advantage on noisy quantum hardware. When training a parametrized quantum circuit to solve a specific task, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. Problem-tailored ansatzes have become the standard for tasks in optimization or quantum chemistry, and yield more efficient algorithms with better performance than unstructured approaches. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. Considering that it is widely known that unstructured ansatzes can become untrainable with increasing system size and circuit depth, it is of key importance to also study problem-tailored circuit architectures in a QML context. 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 on weighted graphs, where a ML model is used to implement a heuristic for a combinatorial optimization problem. We analytically study the expressivity of our ansatz at depth one, and numerically compare the performance of our model on instances with up to 20 qubits to ansatzes where the equivariance property is gradually broken. We show that our ansatz outperforms all others even in the small-instance regime. Our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML and should be an active area of research in order to enable near-term advantages in this field.