NDLI: On the power of quantum proofs

Content Provider	IEEE Xplore Digital Library
Author	Raz, R. Shpilka, A.
Copyright Year	2004
Description	Author affiliation: Dept. of Comput. Sci. & Appl. Math., Weizmann Inst. of Sci., Rehovot, Israel (Raz, R.; Shpilka, A.)
Abstract	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 sub-spaces distance problem. The problem is of geometrical nature: each player gets a linear subspace of R/sup 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 /spl middot/ /spl radic/2) and 0 if the distance is very large (say, larger than 0.9 /spl middot/ /spl radic/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 /spl Omega/(m/sup /spl epsi//) (for some /spl epsi/ > 0). 3. The (standard) quantum communication complexity of the problem is /spl Omega/(/spl radic/m). 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 /spl Omega/(N) for the QMA black box complexity of a random function, and a tight lower bound of /spl Omega/(/spl radic/N) for the QMA black box query complexity of NOR(X/sub 1/,..., X/sub n/). In particular, this shows that any attempt to give short quantum proofs for the class of languages Co - NP have to go beyond black box arguments. We also observe that for any Boolean function G(X/sub 1/,..., X/sub n/), if for both G and 7minus;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/sup 6/) queries to the black box. In particular, this shows that in the black box model QMA /spl cap/ Co - QMA = P. On the positive side, we observe that any (total or partial) Boolean function G(X/sub 1/,..., X/sub n/) has a QMA black box protocol with proofs of length N that makes only 0(/spl radic/N) 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).
Sponsorship	IEEE Comput. Soc. Tech. Committee on Found. of Comput. (TCMF)
Starting Page	260
Ending Page	274
File Size	461774
Page Count	15
File Format	PDF
ISBN	0769521207
ISSN	10930159
DOI	10.1109/CCC.2004.1313849
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2004-06-24
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Complexity theory Quantum computing Protocols Mathematics Mathematical model Computational modeling Polynomials Boolean functions Radio access networks Computer science
Content Type	Text
Resource Type	Article

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