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Variational quantum algorithms (VQAs) have demonstrated considerablepotential in solving NP-hard combinatorial problems in the contemporary nearintermediate-scale quantum (NISQ) era. The quantum approximate optimisationalgorithm (QAOA) is one such algorithm, used in solving the maximum cut(Max-Cut) problem for a given graph by successive implementation of $L$ quantumcircuit layers within a corresponding Trotterised ansatz. The challenge ofexploring the cost function of VQAs arising from an exponential proliferationof local minima with increasing circuit depth has been well-documented.However, fewer studies have investigated the impact of circuit depth on QAOAperformance in finding the correct Max-Cut solution. Here, we employbasin-hopping global optimisation methods to navigate the energy landscapes forQAOA ans\"atze for various graphs, and analyse QAOA performance in finding thecorrect Max-Cut solution. The structure of the solution space is alsoinvestigated using discrete path sampling to build databases of local minimaand the transition states that connect them, providing insightfulvisualisations using disconnectivity graphs. We find that the correspondinglandscapes generally have a single funnel organisation, which makes itrelatively straightforward to locate low-lying minima with good Max-Cutsolution probabilities. In some cases below the adiabatic limit the secondlowest local minimum may even yield a higher solution probability than theglobal minimum. This important observation has motivated us to develop broadermetrics in evaluating QAOA performance, based on collections of minima obtainedfrom basin-hopping global optimisation. Hence we establish expectationthresholds in elucidating useful solution probabilities from local minima, anapproach that may provide significant gains in elucidating reasonable solutionprobabilities from local minima.

Energy Landscapes for the Quantum Approximate Optimisation Algorithm