Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation

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.

Automated summary

The quantum approximate optimisation algorithm combines quantum processing and classical outer-loop optimisation to approximately minimise the problem's generator. The optimal algorithm parameters reduce to one free variable in certain conditions, and a linear relation between circuit parameters is found due to the vanishing gradients of the overlap function.