  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.
