Evolving objective function for improved variational quantum optimization

Abstract

A promising approach to useful computational quantum advantage is to use variational quantum algorithms for optimization problems. Crucial for the performance of these algorithms is to ensure that the algorithm converges with high probability to a near-optimal solution in a small time. In Barkoutsos et al. [Quantum 4, 256 (2020)2521-327X10.22331/q-2020-04-20-256], an alternative class of objective functions, called conditional value at risk (CVaR), was introduced and it was shown that they perform better than standard objective functions. Here we extend that work by introducing an evolving objective function, which we call ascending-CVaR and that can be used for any optimization problem. We test our proposed objective function in an emulation environment, using as case studies three different optimization problems: MaxCut, number partitioning, and portfolio optimization. We examine multiple instances of different sizes and analyze the performance using the variational quantum eigensolver with hardware-efficient ansatz and the quantum approximate optimization algorithm. We show that ascending-CVaR in all cases performs better than standard objective functions or the constant CVaR of Barkoutsos et al. [Quantum 4, 256 (2020)2521-327X10.22331/q-2020-04-20-256] and that it can be used as a heuristic for avoiding suboptimal minima. Our proposal achieves higher overlap with the ideal state in all problems, whether we consider easy or hard instances - on average, it gives up to ten times greater overlap at portfolio optimization and number partitioning, while it gives an 80% improvement at MaxCut. In the hard instances we consider, for the number partitioning problem, standard objective functions fail to find the correct solution in almost all cases, CVaR finds the correct solution at 60% of the cases, while ascending-CVaR finds the correct solution in 95% of the cases.