Application of the Level-$2$ Quantum Lasserre Hierarchy in Quantum Approximation Algorithms

TL;DR

This paper introduces the first use of the level-2 quantum Lasserre hierarchy in an approximation algorithm for a QMA-complete problem, Quantum Max Cut, showing modest improvements and highlighting the hierarchy's potential in quantum optimization.

Contribution

It demonstrates the application of the level-2 quantum Lasserre hierarchy in a quantum approximation algorithm, providing new insights and potential for higher-level hierarchies in quantum problem solving.

Findings

Achieved modest improvements in Quantum Max Cut approximation factors.

Level-2 hierarchy satisfies more physically-motivated constraints than level-1.

Higher levels of the quantum Lasserre hierarchy may enhance quantum approximation algorithms.

Abstract

The Lasserre Hierarchy is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more accurate relaxation. High level programs have proven to be invaluable components of approximation algorithms for many NP-hard optimization problems. There is a natural analogous quantum hierarchy, which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy. Here we provide the first ever use of the level-$2$ hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut. We obtain modest improvements on state-of-the-art approximation factors for this problem, as well as demonstrate that the level-$2$ hierarchy satisfies many physically-motivated constraints that the level-$1$ does not satisfy. Indeed, this observation is at the heart of our analysis and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for QMA-complete problems.