Learning Cut Distributions with Quantum Optimization

TL;DR

This paper introduces a quantum approach to optimize distributions in combinatorial problems, demonstrating theoretical universality and empirical advantages over classical methods.

Contribution

It shows that a finite-layer QAOA circuit can represent any distribution and outperforms classical algorithms on certain graph structures.

Findings

QAOA can capture any distribution over bitstrings with finite layers.

The quantum algorithm effectively solves the Fair Cut Cover problem.

Empirical results show quantum advantage on tested instances.

Abstract

Many combinatorial optimization problems admit a maximin fairness variant, where the aim is to find a distribution over possible solutions which maximizes an expected worst-case outcome. However, the support for an optimal distribution may be exponential, which can be intractable to represent in the worst case. To this end, we propose a quantum based approach to solving distribution optimization problems. Expanding on work analyzing the Dynamical Lie Algebras of the Quantum Approximate Optimization Algorithm (QAOA), we show that with a finite number of layers, a QAOA ansatz can be constructed to capture any distribution over bitstrings. We show that the resulting circuit is able to effectively solve the Fair Cut Cover, a fair interpretation of the classical Fractional Cut Cover Problem. In addition, we show that our algorithm is provably better than classical approximations on certain graph structures and empirically outperforms these classical algorithms on tested instances.