Optimally Band-Limited Parameter Training for the Quantum Approximate Optimization Algorithm

The quantum approximate optimization algorithm (QAOA) forms the cornerstone for solving combinatorial optimization problems on modern quantum processors. Known as a hybrid quantum-classical algorithm, QAOA leverages a variational ansatz composed of alternating parametrized layers that are tuned via a classical optimization algorithm. Parameter training can be particularly challenging for deep circuits, where the high dimensional search space can result in vanishing gradients—the so-called barren plateau problem. Here, we show that parameter training can be significantly improved through functional basis expansions. Specifically, we utilize the discrete prolate spheroidal sequences (DPSS) to define the variational parameter arrays. The DPSS afford an inherent bandwidth parameter that controls both the number of basis functions and the dynamic range of the variational parameters. We show QAOA performance is dependent on bandwidth, revealing an optimal minimum bandwidth required to achieve high approximation ratios. Through this analysis, we find that it is possible to obtain optimized solutions with significantly fewer, highly constrained variational parameters. This work showcases a new approach to variational parameter training that provides insight into the interplay between characteristics of the QAOA parameter search space and algorithmic performance.