Elsevier

Theoretical Computer Science

Volume 598, 20 September 2015, Pages 51-63
Theoretical Computer Science

Epsilon-net method for optimizations over separable states

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Abstract

We give algorithms for the optimization problem: maxρQ,ρ, where Q is a Hermitian matrix, and the variable ρ is a bipartite 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 a logarithmic number of elementary gates on both proofs, as well as the promise problem of deciding if a bipartite local Hamiltonian has a large or small ground energy. For Q0, our algorithm runs in time exponential in QF. While the existence of such an algorithm was first proved recently by Brandão, Christandl and Yard (2011) [8], our algorithm is conceptually simpler.

Keywords

Epsilon-net
Separable states
QMA(2)
PSPACE
Local Hamiltonian problem

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1

This work was completed when the author was a student at Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48105, USA.