Forrelation: A Problem that Optimally Separates Quantum from Classical Computing

TL;DR

This paper introduces the Forrelation problem, demonstrating the largest known separation between quantum and classical query complexities, and proves its optimality, establishing fundamental limits of quantum advantage in query complexity.

Contribution

The authors define Forrelation, prove its optimal quantum-classical query complexity separation, and show it is BQP-complete, resolving a longstanding open problem and illustrating the maximum quantum computational power in this context.

Findings

Quantum algorithm solves Forrelation with 1 query.

Classical randomized algorithms require ~sqrt(N)/log(N) queries.

Forrelation is BQP-complete and captures maximum quantum computational power.

Abstract

We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs ~sqrt(N)/log(N) queries (improving an ~N^{1/4} lower bound of Aaronson). Conversely, we show that this 1 versus ~sqrt(N) separation is optimal: indeed, any t-query quantum algorithm whatsoever can be simulated by an O(N^{1-1/2t})-query randomized algorithm. Thus, resolving an open question of Buhrman et al. from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal t versus ~N^{1-1/2t} separation for all t. As a bonus, we show that this generalization is BQP-complete. This yields what's arguably the simplest BQP-complete problem yet known, and gives a second sense in which Forrelation "captures the maximum power of quantum computation."