Epsilon-Net Method for Optimizations over Separable States

  • Yaoyun Shi
  • Xiaodi Wu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


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.


Density Operator Separable State Additive Error Polynomial Space CNOT Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yaoyun Shi
    • 1
  • Xiaodi Wu
    • 1
  1. 1.Department of EECSUniversity of MichiganAnn ArborUSA

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