Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments

Abstract

We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis and the spectral method of Barnum, Saks, and Szegedy. As an immediate consequence of our main theorem, it can be shown that adversary methods can only prove lower bounds for Boolean functions f in $O(\min(\sqrt{n C_0(f)},\sqrt{n C_1(f)}))$, where $C_0, C_1$ is the certificate complexity and n is the size of the input.