Improving Variational Quantum Optimization using CVaR. (arXiv:1907.04769v1 [quant-ph])

Hybrid quantum/classical variational algorithms can be implemented on noisy
intermediate-scale quantum computers and can be used to find solutions for
combinatorial optimization problems. Approaches discussed in the literature
minimize the expectation of the problem Hamiltonian for a parameterized trial
quantum state. The expectation is estimated as the sample mean of a set of
measurement outcomes, while the parameters of the trial state are optimized
classically. This procedure is fully justified for quantum mechanical
observables such as molecular energies. In the case of classical optimization
problems, which yield diagonal Hamiltonians, we argue that aggregating the
samples in a different way than the expected value is more natural. In this
paper we propose the Conditional Value-at-Risk as an aggregation function. We
empirically show - using classical simulation as well as real quantum hardware
- that this leads to faster convergence to better solutions for all
combinatorial optimization problems tested in our study. We also provide
analytical results to explain the observed difference in performance between
different variational algorithms.

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