A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem

TL;DR

This paper demonstrates that the level-one Quantum Approximate Optimization Algorithm (QAOA) can efficiently produce solutions satisfying a greater fraction of equations in a bounded occurrence Max E3LIN2 problem than random guessing, outperforming some classical algorithms.

Contribution

The paper applies QAOA to a specific combinatorial problem and shows it can outperform classical algorithms in solution quality for bounded occurrence Max E3LIN2.

Findings

QAOA achieves a solution satisfying more equations than random chance.

QAOA outperforms recent classical algorithms in the same problem.

Quantum solutions are effective in typical cases for the problem.

Abstract

We apply our recent Quantum Approximate Optimization Algorithm to the combinatorial problem of bounded occurrence Max E3LIN2. The input is a set of linear equations each of which contains exactly three boolean variables and each equation says that the sum of the variables mod 2 is 0 or is 1. Every variable is in no more than D equations. A random string will satisfy 1/2 of the equations. We show that the level one QAOA will efficiently produce a string that satisfies $\left(\frac{1}{2} + \frac{1}{101 D^{1/2}\, l n\, D}\right)$ times the number of equations. A recent classical algorithm achieved $\left(\frac{1}{2} + \frac{constant}{D^{1/2}}\right)$. We also show that in the typical case the quantum computer will output a string that satisfies $\left(\frac{1}{2}+ \frac{1}{2\sqrt{3e}\, D^{1/2}}\right)$ times the number of equations.