Convergence guarantee for linearly-constrained combinatorial optimization with a quantum alternating operator ansatz

TL;DR

This paper introduces a quantum algorithm, QAOA$^+$, with proven convergence guarantees for a class of linearly constrained combinatorial optimization problems, expanding the scope of quantum optimization methods.

Contribution

The authors develop QAOA$^+$ circuits with convergence guarantees for linearly constrained problems, including asymmetric mixing Hamiltonians and initialization methods, broadening previous quantum optimization results.

Findings

QAOA$^+$ converges to the optimal solution with increasing circuit layers.

The approach generalizes previous guarantees for unconstrained and symmetric problems.

Incorporates arbitrary feasible solutions as initial states.

Abstract

We present a quantum alternating operator ansatz (QAOA$^+$) that solves a class of linearly constrained optimization problems by evolving a quantum state within a Hilbert subspace of feasible problem solutions. Our main focus is on a class of problems with a linear constraint containing sequential integer coefficients. For problems in this class, we devise QAOA$^+$ circuits that provably converge to the optimal solution as the number of circuit layers increases, generalizing previous guarantees for solving unconstrained problems or problems with symmetric constraints. Our approach includes asymmetric ``mixing" Hamiltonians that drive transitions between feasible states, as well as a method to incorporate an arbitrary known feasible solution as the initial state, each of which can be applied beyond the specific linear constraints considered here. This analysis extends QAOA$^+$ performance guarantees to a more general set of linearly-constrained problems and provides tools for future generalizations.