Forrelation: A Problem That Optimally Separates Quantum from Classical Computing
Scott T. Aaronson, Andris Ambainis · SIAM Journal on Computing · 2018
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 $\Omega(\sqrt{N}/\log N)$ queries (improving an $\Omega(N^{1/4})$ lower bound of Aaronson). Conversely, we show that this 1 versus $\widetilde{\Omega}(\sqrt{N})$ 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. [ SIAM J. Comput., 37 (2008), pp. 1387--1400] 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 $\Omega(N^{1-1/2t})$ separation for all $t$. As a bonus, we show that this generalization is ${BQP}$-complete. This yields what is arguably the simplest ${BQP}$-complete problem yet known and gives a second sense in which Forrelation “captures the maximum power of quantum computation.”