# Improving Variational Quantum Optimization using CVaR

## TL;DR

This paper introduces the use of Conditional Value-at-Risk (CVaR) as a new aggregation method in variational quantum algorithms for combinatorial optimization, demonstrating faster convergence and better solutions.

## Contribution

The paper proposes CVaR as an alternative to expectation in variational quantum algorithms, improving optimization performance for classical combinatorial problems.

## Key findings

- CVaR leads to faster convergence in quantum optimization.
- Empirical results show improved solutions using CVaR.
- Analytical insights explain performance differences.

## Abstract

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 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.