On the power of quantum proofs

We study the power of quantum proofs, or more precisely, the power of Quantum Merlin-Arthur (QMA) protocols, in two well studied models of quantum computation: the black box model and the communication complexity model. Our main results are obtained for the communication complexity model. For this model, we identify a complete promise problem for QMA protocols, the Linear Subspaces Distance problem. The problem is of geometrical nature: Each player gets a linear subspace of R(m) and considers the sphere of unit vectors in that subspace. Their goal is to output 1 if the distance between the two spheres is very small (say, smaller than 0.1 (.) root2) and 0 if the distance is very large (say, larger than 0.9 (.) root2)). We show that: 1. The QMA communication complexity of the problem is 0 (log m). 2. The (classical) MA communication complexity of the problem is Omega(m(epsilon)) (for some epsilon > 0). 3. The (standard) quantum communication complexity of the problem is Omega(rootm). In particular, this gives an exponential separation between QMA communication complexity and MA communication complexity. For the black box model we give several observations. First, we observe that the block sensitivity method, as well as the polynomial method for proving lower bounds for the number of queries, can both be extended to QMA protocols. We use these methods to obtain lower bounds for the QMA black box complexity of functions. In particular, we obtain a tight lower bound of Omega (N) for the QMA black box complexity of a random function, and a tight lower bound of Omega(rootN) for the QMA black box query complexity of NOR(X(1),..., X(N)) . In particular, this shows that any attempt to give short quantum proofs for the class of languages Co - NP will have to go beyond black box arguments. We also observe that for any boolean function G (X(1), -, X(N)), if for both G and -G there are QMA black box protocols that make at most T queries to the black box, then there is a classical deterministic black box protocol for G that makes 0 (T(6)) queries to the black box. In particular, this shows that in the black box model QMA boolean AND Co - QMA = P. On the positive side, we observe that any (total or partial) boolean function G(X(1), ---, XN) has a QMA black box protocol with proofs of length N that makes only 0(rootN) queries to the black box. Finally, we observe a very simple proof for the exponential separation (for promise problems) between QMA black box complexity and (classical) MA black box complexity (first obtained by Watrous).