Twisted hybrid algorithms for combinatorial optimization

TL;DR

This paper introduces a twisted hybrid quantum-classical algorithm for MaxCut on 3-regular graphs, improving efficiency by reducing circuit depth and variational parameters while maintaining approximation quality.

Contribution

It proposes a novel twisted hybrid approach combining classical greedy post-processing with variational quantum algorithms, enhancing performance and reducing quantum resource requirements.

Findings

Reduced quantum circuit depth by 4 levels for certain approximation ratios.

Maintained expected approximation ratios with fewer variational parameters.

Demonstrated the effectiveness of the twisted approach on MaxCut problems.

Abstract

Proposed hybrid algorithms encode a combinatorial cost function into a problem Hamiltonian and optimize its energy by varying over a set of states with low circuit complexity. Classical processing is typically only used for the choice of variational parameters following gradient descent. As a consequence, these approaches are limited by the descriptive power of the associated states. We argue that for certain combinatorial optimization problems, such algorithms can be hybridized further, thus harnessing the power of efficient non-local classical processing. Specifically, we consider combining a quantum variational ansatz with a greedy classical post-processing procedure for the MaxCut-problem on $3$-regular graphs. We show that the average cut-size produced by this method can be quantified in terms of the energy of a modified problem Hamiltonian. This motivates the consideration of an improved algorithm which variationally optimizes the energy of the modified Hamiltonian. We call this a twisted hybrid algorithm since the additional classical processing step is combined with a different choice of variational parameters. We exemplify the viability of this method using the quantum approximate optimization algorithm (QAOA), giving analytic lower bounds on the expected approximation ratios achieved by twisted QAOA. These show that the necessary non-locality of the quantum ansatz can be reduced compared to the original QAOA: We find that for levels $p=2,\ldots, 6$, the level $p$ can be reduced by one while roughly maintaining the expected approximation ratio. This reduces the circuit depth by $4$ and the number of variational parameters by $2$.