The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs

TL;DR

This paper investigates the effectiveness of the quantum adiabatic algorithm on random instances of 3-regular 3-XORSAT and Max-Cut problems, finding it fails to solve them efficiently due to different underlying reasons.

Contribution

It provides a comparative analysis of the quantum adiabatic algorithm's performance on two related hypergraph-based optimization problems using advanced simulation techniques.

Findings

Quantum adiabatic algorithm fails to solve both problems efficiently.

Different failure mechanisms are identified for each problem.

Results are obtained using sign-problem free quantum Monte Carlo and cavity methods.

Abstract

In this paper we study the performance of the quantum adiabatic algorithm on random instances of two combinatorial optimization problems, 3-regular 3-XORSAT and 3-regular Max-Cut. The cost functions associated with these two clause-based optimization problems are similar as they are both defined on 3-regular hypergraphs. For 3-regular 3-XORSAT the clauses contain three variables and for 3-regular Max-Cut the clauses contain two variables. The quantum adiabatic algorithms we study for these two problems use interpolating Hamiltonians which are stoquastic and therefore amenable to sign-problem free quantum Monte Carlo and quantum cavity methods. Using these techniques we find that the quantum adiabatic algorithm fails to solve either of these problems efficiently, although for different reasons.