A Strong XOR Lemma for Randomized Query Complexity

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).