Efficient quantum protocols for XOR functions

TL;DR

This paper establishes tight bounds on the quantum communication complexity of XOR functions based on their Fourier spectrum and degree, confirming conjectures for low-degree cases and providing new efficient protocols.

Contribution

It introduces new quantum protocols with tight bounds for XOR functions, linking Fourier analysis to quantum communication complexity and confirming the quantum Log-rank Conjecture for low-degree functions.

Findings

Quantum protocols are efficient for low-degree XOR functions.

Bounds match previous lower bounds, confirming tightness.

Supports the quantum Log-rank Conjecture for low-degree XOR functions.

Abstract

We show that for any Boolean function f on {0,1}^n, the bounded-error quantum communication complexity of XOR functions $f\circ \oplus$ satisfies that $Q_\epsilon(f\circ \oplus) = O(2^d (\log\|\hat f\|_{1,\epsilon} + \log \frac{n}{\epsilon}) \log(1/\epsilon))$, where d is the F2-degree of f, and $\|\hat f\|_{1,\epsilon} = \min_{g:\|f-g\|_\infty \leq \epsilon} \|\hat f\|_1$. This implies that the previous lower bound $Q_\epsilon(f\circ \oplus) = \Omega(\log\|\hat f\|_{1,\epsilon})$ by Lee and Shraibman \cite{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 $Q_E(f) = O(2^d \log \|\hat f\|_0)$, where $\|\hat f\|_0$ is the number of nonzero Fourier coefficients of f. This matches the previous lower bound $Q_E(f(x,y)) = \Omega(\log rank(M_f))$ by Buhrman and de Wolf \cite{BdW01} for low-degree XOR functions.