An End-to-End Survey of Quantum Computing Applications and Algorithmic Primitives

Thursday - November 23 2023, 10:22 UTC - 1 year ago

This survey presents an end-to-end overview of quantum computing applications, their underlying algorithmic primitives, and their end-to-end complexities. It focuses on quantum circuit models and considers the dominant cost of quantum algorithms (nonClifford cost) in the context of quantum error correction and fault-tolerance schemes.

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.They summarize the end-to-end complexities of a number of leading quantum application proposals (by which we mean quantum algorithms applied to a well-defined real-world problem). The complexities of these applications are deduced from the complexities of their underlying primitives, which they review in detail. The modular structure of the survey aids the highlevel understanding of the costs and tradeoffs coming from the various choices one makes when designing and compiling a quantum algorithm, as well as identifying the bottlenecks for a given application. On the technical front, this survey does not attempt to advance the state of the art; rather, it aims to collect, synthesize, and contextualize key results in the literature. They consider algorithms in the quantum circuit model, which is arguably the best-studied model for quantum computation, and renders the presented complexities hardware agnostic. In order to obtain concrete bounds we require oracles to be explicitly instantiated. They generally assume that quantum error correction of some form will be necessary, due to unavoidable imperfections inherent to all known quantum hardware modalities. As such, they typically consider the nonClifford cost of quantum algorithms as the dominant cost, in keeping with leading quantum fault-tolerance schemes. Due to the general lack of application-scale experimental data, they focus on evaluations of the complexity of key algorithmic primitives, as well as their end-to-end abstract complexities.