A Quantum Approximate Optimization Algorithm

TLDR

The paper introduces a quantum algorithm that generates approximate solutions for combinatorial optimization problems. The algorithm, parameterized by an integer p, constructs a quantum circuit of depth linear in p and the number of constraints, with gate locality bounded by that of the objective function, and employs efficient classical preprocessing when p is fixed or a different strategy when p scales with input size. On 3‑regular graphs with p = 1, the algorithm guarantees a cut at least 0.6924 of optimal, and its performance on 2‑ and 3‑regular graphs is analyzed for fixed p.

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.