Epsilon-Net Method for Optimizations over Separable States

Abstract

We give algorithms for the optimization problem: max ρ Q, 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 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.Besides the connection with the weak membership problem for separability, Problem 1 can also be understood from many other aspects. Firstly, as the objective function is the inner-product of a Hermitian matrix and a quantum state, which represents the average value of some physical observable, the optimal value of Problem 1 inherently possesses certain physical meaning. Secondly, in the study of the tensor product space [DF92], the value OptSep(Q) is precisely the injective norm of Q in L(A 1 ) ⊗ L(A 2 ), where L(A) denote the Banach space of operators on A with the operator norm. Finally, one may be equally motivated from the study in operations research. The definition of Problem 1 appeared in an equivalent form in [LQNY09] with the new name of "Bi-Quadratic Optimization over Unit Spheres". Subsequent works [HLZ10, So11] demonstrate that Problem 1 is just a special case of a more general class of optimization problems called homogenous polynomial optimization with quadratic constraints, which is currently an active research topic in that field.Another motivation to study Problem 1 is the recent interest about the complexity class QMA(2). Originally the class QMA (also known as quantum proofs) was defined [KSV02] as the quantum counterpart of the classical complexity class NP. While the extension of NP to allow multiple provers trivially reduces to NP itself, the power of QMA(2), the extension for QMA with multiple unentangled provers, remains far from being well understood. The study of the multipleprover model was initiated in [KMY01, KMY03], where QMA(k) denotes the complexity class for the k-prover case. Much attention was attracted to this model because of the discovery that NP admits logarithmic-size unentangled quantum proofs [BT09]. This result was surprising because single prover quantum logarithm-size proofs only characterize BQP [MW05]. It seems adding one unentangled prover increases the power of the model substantially. There are several subsequent works on refining the initial protocol either with improved completeness and soundness bounds [Bei10, ABD+09...

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