A Strong XOR Lemma for Randomized Query Complexity
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We give a strong direct sum theorem for computing xor ∘ g. Specifically, we show that for every function g and every k≥ 2, the randomized query complexity of computing the xor of k instances of g satisfies R_(xor∘ g) = Θ(k R_/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 ∘ g) = Θ(k log(k)· R(g)), answering an open question from Ben-David et al.(arxiv:2006.10957v1).
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