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A promising approach to useful computational quantum advantage is to usevariational quantum algorithms for optimisation problems. Crucial for theperformance of these algorithms is to ensure that the algorithm converges withhigh probability to a near-optimal solution in a small time. In Barkoutsos etal (Quantum 2020) an alternative class of objective functions, calledConditional Value-at-Risk (CVaR), was introduced and it was shown that theyperform better than standard objective functions. Here we extend that work byintroducing an evolving objective function, which we call Ascending-CVaR andthat can be used for any optimisation problem. We test our proposed objectivefunction, in an emulation environment, using as case-studies three differentoptimisation problems: Max-Cut, Number Partitioning and Portfolio Optimisation.We examine multiple instances of different sizes and analyse the performanceusing the Variational Quantum Eigensolver (VQE) with hardware-efficient ansatzand the Quantum Approximate Optimization Algorithm (QAOA). We show thatAscending-CVaR in all cases performs better than standard objective functionsor the "constant" CVaR of Barkoutsos et al (Quantum 2020) and that it can beused as a heuristic for avoiding sub-optimal minima. Our proposal achieveshigher overlap with the ideal state in all problems, whether we consider easyor hard instances -- on average it gives up to ten times greater overlap atPortfolio Optimisation and Number Partitioning, while it gives an 80%improvement at Max-Cut. In the hard instances we consider, for the numberpartitioning problem, standard objective functions fail to find the correctsolution in almost all cases, CVaR finds the correct solution at 60% of thecases, while Ascending-CVaR finds the correct solution in 95% of the cases.

An evolving objective function for improved variational quantum optimisation