Quantum algorithms for algebraic problems

TL;DR

This paper reviews the current state of quantum algorithms, especially those with algebraic problems, highlighting their potential for superpolynomial speedups over classical methods and their importance in advancing quantum computing.

Contribution

It provides a comprehensive overview of quantum algorithms for algebraic problems, emphasizing their significance and recent developments in the field.

Findings

Quantum algorithms can solve algebraic problems faster than classical algorithms.

Shor's algorithm exemplifies exponential speedup for factoring integers.

The review identifies key open challenges in quantum algorithm development.

Abstract

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for 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 reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.