Epsilon-net method for optimizations over separable states

TL;DR

This paper introduces a simpler algorithm for optimizing over bipartite separable quantum states, providing better bounds for certain quantum complexity problems despite the NP-hard nature of the task.

Contribution

It presents a new, conceptually simpler epsilon-net based algorithm for separable state optimization, improving bounds for quantum complexity problems.

Findings

Provides PSPACE upper bounds for QMA(2) promise problems.

Runs in exponential time in Frobenius norm for positive Q.

Offers a simpler alternative to recent algorithms by Brandão, Christandl, and Yard.

Abstract

We give algorithms for the optimization problem: $\max_\rho \ip{Q}{\rho}$, where $Q$ is a Hermitian matrix, and the variable $\rho$ is a bipartite {\em separable} quantum state. This problem lies at the heart of several problems in quantum computation and information, such as the complexity of QMA(2). While the problem is NP-hard, our algorithms are better than brute force for several instances of interest. In particular, they give PSPACE upper bounds on promise problems admitting a QMA(2) protocol in which the verifier performs only logarithmic number of elementary gate on both proofs, as well as the promise problem of deciding if a bipartite local Hamiltonian has large or small ground energy. For $Q\ge0$, our algorithm runs in time exponential in $\|Q\|_F$. While the existence of such an algorithm was first proved recently by Brand{\~a}o, Christandl and Yard [{\em Proceedings of the 43rd annual ACM Symposium on Theory of Computation}, 343--352, 2011], our algorithm is conceptually simpler.