A Strong XOR Lemma for Randomized Query Complexity

TL;DR

This paper establishes a strong direct sum theorem for the randomized query complexity of XOR functions, confirming a conjecture and providing new insights into the complexity of composed functions.

Contribution

It proves a tight bound for the randomized query complexity of XOR of multiple instances, confirming a conjecture and answering an open question in the field.

Findings

The randomized query complexity of xor of k instances is proportional to k times the complexity of a single instance.

The paper confirms a conjecture by Blais and Brody regarding the direct sum theorem.

It provides a total function with complexity scaling as k log(k) times the original complexity.

Abstract

We give a strong direct sum theorem for computing $xor \circ g$. Specifically, we show that for every function g and every $k\geq 2$, the randomized query complexity of computing the xor of k instances of g satisfies $\overline{R}_\eps(xor\circ g) = \Theta(k \overline{R}_{\eps/k}(g))$. This matches the naive success amplification upper bound and answers a conjecture of Blais and Brody (CCC19). As a consequence of our strong direct sum theorem, we give a total function g for which $R(xor \circ g) = \Theta(k \log(k)\cdot R(g))$, answering an open question from Ben-David et al.(arxiv:2006.10957v1).