Near-Optimal Separations of Certificate Complexity from Randomized and Quantum Query Complexity

RESUMO

We study how large the certificate complexity ${C}(f)$ of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function $f$ whose randomized query complexity with one-sided error satisfies ${R}_1(f) = \Theta(\sqrt{{C}(f)})$. This separation is optimal even when two-sided error is allowed. From this construction, we obtain another total Boolean function $F$ with bounded-error quantum query complexity ${Q}(F) = \widetilde O({C}(F)^{1/4})$, attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are $\widetilde O(\sqrt{{C}(F)})$. We also prove matching lower bounds for these two measures on $F$.