Progress towards analytically optimal angles in quantum approximate optimisation

TL;DR

This paper advances understanding of the Quantum Approximate Optimisation Algorithm by deriving analytical solutions for optimal parameters at one layer and exploring gradient conditions, reducing resource needs for quantum optimization.

Contribution

It proves that for a single-layer QAOA, optimal parameters depend on only one variable and identifies conditions for vanishing gradients, simplifying parameter selection.

Findings

Optimal angles for p=1 reduce to one free variable.

In the thermodynamic limit, optimal angles are recovered.

Gradient vanishing conditions lead to a linear relation between parameters.

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 minimizes the problem's generator. Analytical solutions to choose optimal application times (called angles) have proven difficult to find, whereas outer loop optimisation is resource intensive. Here we prove that 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 on 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.