Separations in query complexity using cheat sheets

Scott Aaronson, Shalev Ben-David, Robin Kothari
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap
more » ... en quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al., to be shown in a unified manner.
doi:10.1145/2897518.2897644 dblp:conf/stoc/AaronsonBK16 fatcat:x77uykivsvhfle3icc2ocbqcai