Efficient quantum protocols for XOR functions
Shengyu Zhang · 2013
We show that for any Boolean function f : {0,1}n → {0,1}, the bounded-error quantum communication complexity Q∊(f ○ ⊕) of XOR functions f(x ⊕ y) satisfies that where d = deg2(f) is the ℱ2-degree of f, and ‖ ‖1,∊ = ming:‖f–g‖∞≤∊ ‖ĝ‖1. This implies that the previous lower bound Q∊(f ○ ⊕) = Ω(log ‖ ‖1,∊) by Lee and Shraibman [LS09] is tight for f with low 2-degree. The result also confirms the quantum version of the Log-rank Conjecture for low-degree XOR functions. In addition, we show that the exact quantum communication complexity satisfies where ‖ ‖0 is the number of nonzero Fourier coefficients of f. This matches the previous lower bound QE(f(x, y)) = Ω(logrank(Mf)) by Buhrman and de Wolf [BdW01] for low-degree XOR functions.