A Super-Grover Separation Between Randomized and Quantum Query Complexities

TL;DR

This paper constructs a total Boolean function demonstrating a super-Grover separation, showing that randomized query complexity can significantly exceed quantum query complexity, thus refuting a long-standing conjecture.

Contribution

It provides the first explicit total Boolean function with a super-Grover separation between randomized and quantum query complexities.

Findings

Established a total Boolean function with R(f)=Ω(Q(f)^{5/2})

Refuted the conjecture that R(f)=O(Q(f)^2) for all total functions

Improved the separation to R(f)=Ω(Q(f)^3) under a conjecture

Abstract

We construct a total Boolean function $f$ satisfying $R(f)=\tilde{\Omega}(Q(f)^{5/2})$, refuting the long-standing conjecture that $R(f)=O(Q(f)^2)$ for all total Boolean functions. Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions, we improve this to $R(f)=\tilde{\Omega}(Q(f)^3)$. Our construction is motivated by the G\"o\"os-Pitassi-Watson function but does not use it.