Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation

Abstract

The quantum approximate optimisation algorithm is a p layer, time variable split operator method executed on a quantum processor and driven to convergence by classical outer-loop optimisation. The classical co-processor varies individual application times of a problem/driver propagator sequence to prepare a state which approximately minimises the problem’s generator. Analytical solutions to choose optimal application times (called parameters or angles) have proven difficult to find, whereas outer-loop optimisation is resource intensive. Here we prove that the optimal quantum approximate optimisation algorithm parameters for p=1 layer reduce to one free variable and in the thermodynamic limit, we recover optimal angles. We moreover demonstrate that conditions for vanishing gradients of the overlap function share a similar form which leads to a linear relation between circuit parameters, independent of the number of qubits. Finally, we present a list of numerical effects, observed for particular system size and circuit depth, which are yet to be explained analytically.