Quantum algorithms: A survey of applications and end-to-end complexities

TL;DR

This survey reviews quantum algorithms across various fields, analyzing their complexities, challenges, and potential advantages over classical methods, with a focus on end-to-end evaluation and practical implementation considerations.

Contribution

It provides a modular, detailed overview of quantum algorithmic primitives and applications, emphasizing technical subtleties and comprehensive complexity analysis.

Findings

Quantum algorithms can offer speedups over classical methods in specific applications.

End-to-end complexity analysis reveals hidden costs and implementation challenges.

Comparison with classical approaches highlights potential quantum advantages and limitations.

Abstract

The anticipated applications of quantum computers span across science and industry, ranging from quantum chemistry and many-body physics to optimization, finance, and machine learning. Proposed quantum solutions in these areas typically combine multiple quantum algorithmic primitives into an overall quantum algorithm, which must then incorporate the methods of quantum error correction and fault tolerance to be implemented correctly on quantum hardware. As such, it can be difficult to assess how much a particular application benefits from quantum computing, as the various approaches are often sensitive to intricate technical details about the underlying primitives and their complexities. Here we present a survey of several potential application areas of quantum algorithms and their underlying algorithmic primitives, carefully considering technical caveats and subtleties. We outline the challenges and opportunities in each area in an "end-to-end" fashion by clearly defining the problem being solved alongside the input-output model, instantiating all "oracles," and spelling out all hidden costs. We also compare quantum solutions against state-of-the-art classical methods and complexity-theoretic limitations to evaluate possible quantum speedups. The survey is written in a modular, wiki-like fashion to facilitate navigation of the content. Each primitive and application area is discussed in a standalone section, with its own bibliography of references and embedded hyperlinks that direct to other relevant sections. This structure mirrors that of complex quantum algorithms that involve several layers of abstraction, and it enables rapid evaluation of how end-to-end complexities are impacted when subroutines are altered.