Abstract
We give algorithms for the optimization problem: \(\max_\rho \left\langle Q , \rho\right\rangle \), 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 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 ≥ 0, our algorithm runs in time exponential in ||Q|| F . While the existence of such an algorithm was first proved recently by Brandão, Christandl and Yard [Proceedings of the 43rd annual ACM Symposium on Theory of Computation , 343–352, 2011], our algorithm is conceptually simpler.
A full version of this paper is available at arXiv:1112.0808. This research was supported in part by National Basic Research Program of China Awards 2011CBA00300 and 2011CBA00301, and by NSF of United States Award 1017335.
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Shi, Y., Wu, X. (2012). Epsilon-Net Method for Optimizations over Separable States. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_67
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