Efficient quantum protocols for XOR functions

We show that for any Boolean function f : {0, 1} → {0, 1}, the bounded-error quantum communication complexity Q (f ◦ ⊕) of XOR functions f(x ⊕ y) satisfies that Q (f ◦ ⊕) = O ( 2 ( log ‖f̂‖1, + log n ) log(1/ ) ) , where d = deg2(f) is the F2-degree of f , and ‖f̂‖1, = ming:‖f−g‖∞≤ ‖ĝ‖1. This implies that the previous lower bound Q (f ◦ ⊕) = Ω(log ‖f̂‖1, ) by Lee and Shraibman [LS09] is tight for f with low F2-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 QE(f) = O(2 d log ‖f̂‖0), where ‖f̂‖0 is the number of nonzero Fourier coefficients of f . This matches the previous lower bound QE(f(x, y)) = Ω(log rank(Mf )) by Buhrman and de Wolf [BdW01] for low-degree XOR functions.