Two-Stage Estimation and Variance Modeling for Latency-Constrained Variational Quantum Algorithms

The Quantum Approximate Optimization Algorithm (QAOA) has enjoyed increasing attention in noisy intermediate-scale quantum computing due to its application to combinatorial optimization problems. Because combinatorial optimization problems are NP-hard, QAOA could serve as a potential demonstration of quantum advantage in the future. As a hybrid quantum-classical algorithm, the classical component of QAOA resembles a simulation optimization problem, in which the simulation outcomes are attainable only through the quantum computer. The simulation that derives from QAOA exhibits two unique features that can have a substantial impact on the optimization process: (i) the variance of the stochastic objective values typically decreases in proportion to the optimality gap, and (ii) querying samples from a quantum computer introduces an additional latency overhead. In this paper, we introduce a novel stochastic trust-region method, derived from a derivative-free adaptive sampling trust-region optimization (ASTRO-DF) method, intended to efficiently solve the classical optimization problem in QAOA, by explicitly taking into account the two mentioned characteristics. The key idea behind the proposed algorithm involves constructing two separate local models in each iteration: a model of the objective function, and a model of the variance of the objective function. Exploiting the variance model allows us to both restrict the number of communications with the quantum computer, and also helps navigate the nonconvex objective landscapes typical in the QAOA optimization problems. We numerically demonstrate the superiority of our proposed algorithm using the SimOpt library and Qiskit, when we consider a metric of computational burden that explicitly accounts for communication costs.