Iteration Complexity of Variational Quantum Algorithms

Abstract

There has been much recent interest in near-term applications of quantum computers. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. In this paper, we analyze the iteration complexity of VQA, that is, the number of steps VQA required until the iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, the form of the noise makes the evaluations of the objective function via circuits biased, necessitating the perspective of convergence analysis of variants of these classical optimization procedures, wherein the evaluations exhibit systematic bias. We apply our reasoning to the most often used procedures, including SPSA the parameter shift rule, which can be seen as zeroth-order, or derivative-free, optimization algorithms with biased function evaluations. We show that the asymptotic rate of convergence is unaffected by the bias, but the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity.Comment: 39 pages, 11 figure