Quantum algorithms for hidden nonlinear structures

TL;DR

This paper introduces quantum algorithms for discovering hidden nonlinear structures over finite fields, demonstrating cases where quantum methods outperform classical approaches and analyzing their query complexity.

Contribution

It proposes a new class of problems involving hidden nonlinear structures and provides quantum algorithms that solve them efficiently, expanding the scope beyond hidden subgroup problems.

Findings

Quantum algorithms solve certain hidden nonlinear structure problems efficiently.

Classical algorithms cannot efficiently solve these hidden nonlinear problems.

Quantum query complexity for these problems is favorably bounded.

Abstract

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.