Epsilon-net method for optimizations over separable states

We give algorithms for the optimization problem: $\max_ρ\ip{Q}ρ$, where $Q$ is a Hermitian matrix, and the variable $ρ$ 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{ã}o, Christandl and Yard [{\em Proceedings of the 43rd annual ACM Symposium on Theory of Computation}, 343--352, 2011], our algorithm is conceptually simpler.

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