A Filter Function Perspective on Faulty Quantum Approximate Optimization Algorithms

In combinatorial optimization, approximation algorithms aim to find approximate solutions with provable guarantees on the distance between the returned solution and the global optimum. The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical algorithm that seeks to achieve approximate solutions by iteratively alternating between intervals of controlled quantum evolution. Here, we examine the effect of analog precision errors on QAOA performance both from the perspective of algorithmic training and canonical distance metrics between quantum states. Leveraging cumulant expansions and the filter function formalism (FFF), we recast the faulty QAOA as a control problem in which precision errors are synonymous with multiplicative control noise. We show that the FFF proves to be a useful tool for understanding QAOA evolution subject to precision errors. Furthermore, we show that the FFF approach to QAOA lends itself to more general noise scenarios and the calculation of error bounds on QAOA performance and broader classes of variational quantum algorithms.