Quantum algorithms for problems in number theory, algebraic geometry, and group theory

TL;DR

This paper reviews quantum algorithms that solve algebraic problems in number theory, algebraic geometry, and group theory, highlighting their potential for significant speedups over classical methods.

Contribution

It provides a comprehensive overview of quantum algorithms for algebraic problems, emphasizing their current development and potential for superpolynomial speedups.

Findings

Quantum algorithms can outperform classical algorithms on algebraic problems.

Shor's algorithm exemplifies exponential speedup in integer factoring.

Current research focuses on extending quantum advantages to broader algebraic problems.

Abstract

Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.