Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Abstract

Recently, Bravyi, Gosset, and König (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC0 circuits). In other words, they exhibited a search problem in QNC0 that is not in NC0. We strengthen their result by proving that the 2D HLF problem is not contained in AC0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem.