Commutative version of the k-local Hamiltonian problem and common eigenspace problem

Abstract

We study the complexity of a problem "Common Eigenspace" -- verifying consistency of eigenvalue equations for composite quantum systems. The input of the problem is a family of pairwise commuting Hermitian operators H_1,...,H_r on a Hilbert space (C^d)^{\otimes n} and a string of real numbers h_1,...,h_r. The problem is to determine whether a common eigenspace specified by equalities (H_a - h_a)|\psi>=0, a=1,...,r, has a positive dimension. We consider two cases: (i) all operators H_a are k-local; (ii) all operators H_a are factorized. It can be easily shown that both problems belong to the class QMA - the quantum analogue of NP, and that some NP-complete problems can be reduced to either (i) or (ii). A non-trivial question is whether the problems (i) or (ii) belong to NP? We show that the answer is positive for some special values of k and d. Also we prove that the problem (ii) can be reduced to its special case, such that all operators H_a are factorized projectors and all h_a=0.