Ising Hamiltonians for Constrained Combinatorial Optimization Problems and the Metropolis-Hastings Warm-Starting Algorithm

TL;DR

This paper introduces a general method to derive Ising Hamiltonians for constrained combinatorial optimization problems and proposes a Metropolis-Hastings warm-starting algorithm for QAOA that converges to global optima, demonstrated on several NP-hard problems.

Contribution

A novel method for obtaining Ising Hamiltonians for CCOPs and a warm-starting algorithm for QAOA with proven convergence to global solutions.

Findings

First-time derivation of Ising Hamiltonian for MWVC.

Metropolis-Hastings warm-start improves QAOA performance.

Numerical validation on multiple NP-hard problems.

Abstract

Quantum approximate optimization algorithm (QAOA) is a promising variational quantum algorithm for combinatorial optimization problems. However, the implementation of QAOA is limited due to the requirement that the problems be mapped to Ising Hamiltonians and the nonconvex optimization landscapes. Although the Ising Hamiltonians for many NP hard problems have been obtained, a general method to obtain the Ising Hamiltonians for constrained combinatorial optimization problems (CCOPs) has not yet been investigated. In this paper, a general method is introduced to obtain the Ising Hamiltonians for CCOPs and the Metropolis-Hastings warm-starting algorithm for QAOA is presented which can provably converge to the global optimal solutions. The effectiveness of this method is demonstrated by tackling the minimum weight vertex cover (MWVC) problem, the minimum vertex cover (MVC) problem, and the maximal independent set problem as examples. The Ising Hamiltonian for the MWVC problem is obtained first time by using this method. The advantages of the Metropolis-Hastings warm-starting algorithm presented here is numerically analyzed through solving 30 randomly generated MVC cases with 1-depth QAOA.