Easy and hard functions for the Boolean hidden shift problem

TL;DR

This paper analyzes the quantum query complexity of the Boolean hidden shift problem, revealing that the difficulty varies with the function type, with some instances solvable in one query and others requiring more, using Fourier-based quantum algorithms.

Contribution

It characterizes the complexity for different classes of functions, introduces an improved quantum algorithm for random functions, and connects the problem to Fourier analysis and quantum measurement techniques.

Findings

Exact one-query solution for bent functions

Two queries suffice for random functions

Hard instances include delta functions

Abstract

We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f. We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice. Our algorithm for random functions is based on performing the pretty good measurement on several copies of a certain state; its analysis relies on the Fourier transform. We also use this approach to improve the quantum rejection sampling approach to the Boolean hidden shift problem.