Quantum Algorithms

TL;DR

This survey reviews the current landscape of quantum algorithms, covering foundational, advanced, and specialized algorithms, highlighting their development, techniques, and potential for quantum advantage.

Contribution

It provides a comprehensive overview of quantum algorithms, including recent advances and generalizations, serving as a valuable resource for understanding the field's progress.

Findings

Summarizes key quantum algorithms like Shor's factoring and Grover's search.

Highlights the development of non-Abelian Hidden Subgroup algorithms.

Discusses the potential of quantum algorithms for tasks beyond classical capabilities.

Abstract

This article surveys the state of the art in quantum computer algorithms, including both black-box and non-black-box results. It is infeasible to detail all the known quantum algorithms, so a representative sample is given. This includes a summary of the early quantum algorithms, a description of the Abelian Hidden Subgroup algorithms (including Shor's factoring and discrete logarithm algorithms), quantum searching and amplitude amplification, quantum algorithms for simulating quantum mechanical systems, several non-trivial generalizations of the Abelian Hidden Subgroup Problem (and related techniques), the quantum walk paradigm for quantum algorithms, the paradigm of adiabatic algorithms, a family of ``topological'' algorithms, and algorithms for quantum tasks which cannot be done by a classical computer, followed by a discussion.