An exponential time upper bound for Quantum Merlin-Arthur games with unentangled provers

TL;DR

This paper establishes a deterministic exponential time upper bound for Quantum Merlin-Arthur games with unentangled provers, significantly improving previous bounds and impacting quantum complexity and chemistry problems.

Contribution

It provides the first non-trivial exponential time upper bound for QMA(k) with unentangled provers, using perturbation theory and epsilon-net methods.

Findings

Exponential time upper bound for QMA(k) with unentangled provers

Implications for quantum chemistry problems like N-Representability

Advances understanding of separable Hamiltonian problems

Abstract

We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian problem to a variant of the Separable Local Hamiltonian problem with an exponentially small promise gap, and then to decide this instance using epsilon-net methods. Our results imply an exponential time algorithm for the Pure State N-Representability problem in quantum chemistry, which is in QMA(2), but is not known to be in QMA. We also discuss the implications of our results on the Best Separable State problem.