On the Power of Quantum Proofs

• Authors: Ran Raz;Amir Shpilka
• Venue: CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
• Year: 2004

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 complexitymodel.Our main results are obtained for the communicationcomplexity model. For this model, we identify a completepromise problem for QMA protocols, the Linear SubspacesDistance problem. The problem is of geometricalnature: Each player gets a linear subspace of R^m and considersthe sphere of unit vectors in that subspace. Theirgoal is to output 1 if the distance between the two spheresis very small (say, smaller than 0.1 \cdot \sqrt 2 ) and 0 if the distanceis very large (say, larger than 0.9 \cdot \sqrt 2 ). We show that:1. The QMA communication complexity of the problem is O(logm).2. The (classical) MA communication complexity of the problem is \Omega (m^驴) (for some 驴 0).3. The (standard) quantum communication complexity of the problem is \Omega (\sqrt m).In particular, this gives an exponential separation betweenQMA communication complexity and MA communicationcomplexity.For the black box model we give several observations.First, we observe that the block sensitivity-method, as wellas the polynomialmethod for proving lower bounds for thenumber of queries, can both be extended to QMA protocols.We use thesemethods to obtain lower bounds for theQMA black box complexity of functions. In particular,weobtain a tight lower bound of \Omega(N) for the QMA black boxcomplexity of a random function, and a tight lower boundof \Omega (\sqrt N) for the QMA black box query complexity ofNOR (X_1, ...,X_N). In particular, this shows that any attemptto give short quantum proofs for the class of languagesCo - NP will have to go beyond black box arguments.We also observe that for any boolean functionG(X_1, ...,X_N), if for both G and -G there are QMAblack box protocols that make at most T queries tothe black box, then there is a classical deterministicblack box protocol for G that makes O(T^6) queries tothe black box. In particular, this shows that in the blackbox model QMA 驴 Co - QMA = P.On the positive side, we observe that any (total or partial)boolean function G(X_1, ...,X_N) has a QMA blackbox protocol with proofs of length N that makes only0(\sqrt N) queries to the black box.Finally, we observe a very simple proof for the exponentialseparation (for promise problems) betweenQMAblack box complexity and (classical) MA black box complexity(first obtained by Watrous).