Quantum and classical communication-space tradeoffs from rectangle bounds

• Authors: Hartmut Klauck
• Venue: FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
• Year: 2004

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 problem f : X × Y → Z the multicolor discrepancy of the communication matrix of f is 1/2d, then any bounded error quantum protocol with space S, in which Alice receives some l inputs, Bob r inputs, and they compute f (xi ,yj ) for the l · r pairs of inputs (xi,yj ) needs communication C =Ω (lrd log | Z | /S). In particular, n × n-matrix multiplication over a finite field F requires C = Θ (n3 log2 | F |/S), matrix-vector multiplication C= Θ (n2 log2 | F |/S). We then turn to randomized bounded error protocols, and, utilizing a new direct product result for the one-sided rectangle lower bound on randomized communication complexity, derive the bounds C = Ω (n3 /S2) for Boolean matrix multiplication and C = Ω (n2/S2) for Boolean matrix-vector multiplication. These results imply a separation between quantum and randomized protocols when compared to quantum bounds in [KSW04] and partially answer a question by Beame et al.[BTY94].