A Quantum Approximate Optimization Algorithm

TL;DR

This paper presents a quantum algorithm for combinatorial optimization that improves approximation quality with a parameter p, offering a scalable approach with proven performance on specific graph problems.

Contribution

The paper introduces a new quantum approximate optimization algorithm with adjustable approximation quality and analyzes its performance on regular graphs.

Findings

For p=1, on 3-regular graphs, the algorithm finds cuts at least 69.24% of optimal.

The quantum circuit depth scales linearly with p and the number of constraints.

Efficient classical preprocessing is used when p is fixed, enabling practical implementation.

Abstract

We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that is, independent of the input size, the algorithm makes use of efficient classical preprocessing. If p grows with the input size a different strategy is proposed. We study the algorithm as applied to MaxCut on regular graphs and analyze its performance on 2-regular and 3-regular graphs for fixed p. For p = 1, on 3-regular graphs the quantum algorithm always finds a cut that is at least 0.6924 times the size of the optimal cut.