On the power of lower bound methods for one-way quantum communication complexity

• Authors: Shengyu Zhang
• Venue: ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
• Year: 2011

One of the most fundamental questions in communication complexity is the largest gap between classical and quantum one-way communication complexities, and it is conjectured that they are polynomially related for all total Boolean functions f. One approach to proving the conjecture is to first show a quantum lower bound L(f), and then a classical upper bound U(f) = poly(L(f)). Note that for this approach to be possibly successful, the quantum lower bound L(f) has to be polynomially tight for all total Boolean functions f. This paper studies all the three known lower bound methods for one-way quantum communication complexity, namely the Partition Tree method by Nayak, the Trace Distance method by Aaronson, and the two-way quantum communication complexity. We deny the possibility of using the aforementioned approach by any of these known quantum lower bounds, by showing that each of them can be at least exponentially weak for some total Boolean functions. In particular, for a large class of functions generated from Erdös-Rényi random graphs G(N, p), with p in some range of 1/poly(N), though the two-way quantum communication complexity is linear in the size of input, the other two methods (particularly for the one-way model) give only constant lower bounds. En route of the exploration, we also discovered that though Nayak's original argument gives a lower bound by the VC-dimension, the power of its natural extension, the Partition Tree method, turns out to be exactly equal to another measure in learning theory called the extended equivalence query complexity.