The Quantum Alternating Operator Ansatz on Maximum k-Vertex Cover

TL;DR

This paper explores the Quantum Alternating Operator Ansatz for maximum k-vertex cover, comparing initial states and mixers, analyzing solution distributions, and proposing efficient angle strategies, advancing quantum optimization methods.

Contribution

It provides a comprehensive analysis of the Quantum Alternating Operator Ansatz applied to maximum k-vertex cover, including performance comparisons and new insights into state preparation and mixer choices.

Findings

Dicke states outperform classical initial states

Complete graph mixer outperforms ring mixer

Solution distribution's standard deviation decreases exponentially with rounds

Abstract

The Quantum Alternating Operator Ansatz is a generalization of the Quantum Approximate Optimization Algorithm (QAOA) designed for finding approximate solutions to combinatorial optimization problems with hard constraints. In this paper, we study Maximum $k$-Vertex Cover under this ansatz due to its modest complexity, while still being more complex than the well studied problems of Max-Cut and Max E3-LIN2. Our approach includes (i) a performance comparison between easy-to-prepare classical states and Dicke states as starting states, (ii) a performance comparison between two $XY$-Hamiltonian mixing operators: the ring mixer and the complete graph mixer, (iii) an analysis of the distribution of solutions via Monte Carlo sampling, and (iv) the exploration of efficient angle selection strategies. Our results are: (i) Dicke states improve performance compared to easy-to-prepare classical states, (ii) an upper bound on the simulation of the complete graph mixer, (iii) the complete graph mixer improves performance relative to the ring mixer, (iv) numerical results indicating the standard deviation of the distribution of solutions decreases exponentially in $p$ (the number of rounds in the algorithm), requiring an exponential number of random samples to find a better solution in the next round, and (v) a correlation of angle parameters which exhibit high quality solutions that behave similarly to a discretized version of the Quantum Adiabatic Algorithm.