Energy landscapes for the quantum approximate optimization algorithm

Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of $L$ quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima.