Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm

TL;DR

This paper introduces quantum algorithms that efficiently solve the hidden shift problem for quadratic Boolean functions and functions with large Gowers U_3 norm, outperforming classical methods in query complexity.

Contribution

The paper presents the first quantum algorithms for hidden shift problems of quadratic functions and functions close to quadratics, with improved query efficiency and robustness.

Findings

Quantum algorithms identify hidden shifts in quadratic Boolean functions efficiently.

Quantum approach reduces query complexity from Theta(n^2) to linear in n.

Algorithms are robust to functions close to quadratics with large Gowers U_3 norm.

Abstract

Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor's algorithm, sampling from the quantum Fourier spectrum is used to discover periodicity of the modular exponentiation function. In a generalization of this idea, quantum Fourier sampling can be used to discover hidden subgroup structures of some functions much more efficiently than it is possible classically. Another problem for which the Fourier transform has been recruited successfully on a quantum computer is the hidden shift problem. Quantum algorithms for hidden shift problems usually have a slightly different flavor from hidden subgroup algorithms, as they use the Fourier transform to perform a correlation with a given reference function, instead of sampling from the Fourier spectrum directly. In this paper we show that hidden shifts can be extracted efficiently from Boolean functions that are quadratic forms. We also show how to identify an unknown quadratic form on n variables using a linear number of queries, in contrast to the classical case were this takes Theta(n^2) many queries to a black box. What is more, we show that our quantum algorithm is robust in the sense that it can also infer the shift if the function is close to a quadratic, where we consider a Boolean function to be close to a quadratic if it has a large Gowers U_3 norm.