Quantum and Classical CommunicationSpace Tradeoffs from Rectangle Bounds
Abstract
We derive lower bounds for tradeoffs between the communication C and space S for communicating circuits. The first such bound applies to quantum circuits. If for any function f with image Z the multicolor discrepancy of the communication matrix of f is 1/2^d, then any bounded error quantum protocol with space S, in which Alice receives some l inputs, Bob r inputs, and they compute f(x_i,y_j) for the lr pairs of inputs (x_i,y_j) needs communication C=\Omega(lrd \log Z/S). In particular, n\times nmatrix multiplication over a finite field F requires C=\Theta(n^3\log^2 F/S). We then turn to randomized bounded error protocols, and derive the bound C=\Omega(n^3/S^2) for Boolean matrix multiplication, utilizing a new direct product result for the onesided rectangle lower bound on randomized communication complexity. This implies a separation between quantum and randomized protocols.
 Publication:

arXiv eprints
 Pub Date:
 December 2004
 arXiv:
 arXiv:quantph/0412088
 Bibcode:
 2004quant.ph.12088K
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 17 pages, appears at FSTTCS '04