PDE and mathematical physics · Oct 6, 2026

The optimal quartic separation between randomized and quantum queries

Shows that the universal bound R(f)=O((1+Q(f))4)R(f)=O((1+Q(f))^4) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.

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