[Paper Review] Quantum algorithms: A survey of applications and end-to-end complexities

Alexander M. Dalzell, Sam McArdle+11 more|arXiv (Cornell University)|Oct 4, 2023

Quantum Computing Algorithms and Architecture47 citations

TL;DR

The paper surveys quantum algorithm techniques such as qubitization, block-encodings, and phase estimation, discussing how Grover-like operators encode functions of A and enable Chebyshev polynomial representations for end-to-end algorithmic complexity.

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.

Motivation & Objective

• Motivate the study of quantum algorithms by exploring practical applications and their end-to-end complexity.
• Explain key operator constructions like qubitization and block-encoding that transform a given operator into a Grover-like form.
• Illustrate how quantum phase estimation accesses eigenvalues and enables polynomial approximations of dynamics.
• Highlight the connection between repeated applications of structured unitaries and Chebyshev polynomials for complexity analysis.

Proposed method

• Describe the construction of a Grover-like operator W from a unitary UA using a Z|0⟩ reflection to obtain a 2x2 block form.
• Show that W restricted to invariant subspaces yields a rotation with angle θλ = arccos(λ) and eigenvalues e ± i θλ.
• Demonstrate that W acts as a block-diagonal combination Wd = ⨁λ(Td(λ)1 − λ^2Ud−1(λ) − 1 − λ^2Ud−1(λ)Td(λ)) across subspaces, linking to Chebyshev polynomials of the first and second kind (Td, Ud).
• Relate repeated applications to Chebyshev polynomial representations, enabling end-to-end complexity analysis of quantum algorithms.

Experimental results

Research questions

• RQ1 How can we realize end-to-end quantum algorithmic complexity using operator encodings like qubitization and block-encoding?
• RQ2 What is the role of Grover-like reflections and eigenvalue access via quantum phase estimation in practical algorithm design?
• RQ3 How do Chebyshev polynomial representations (Td, Ud) characterize the action of repeated unitaries on invariant subspaces?
• RQ4 What are the implications of eigenvalue structure e^{±iθλ} for performance and error analysis of quantum procedures?

Key findings

• A Grover-like operator W derived from a base UA and a reflection yields a rotation on invariant subspaces with angle θλ = arccos(λ).
• Eigenvalues of W occur at e^{±i arccos(λ)} corresponding to eigenvectors built from |0^m⟩|λ⟩ and its orthogonal companion.
• Repeated applications of W within subspaces map to Chebyshev polynomials of the first and second kinds, enabling polynomial-based complexity reasoning.
• The block-encoding framework allows expressing A within a unitary form that supports phase estimation and eigenvalue-based techniques.
• These constructions underpin end-to-end algorithmic strategies by linking operator representations to classical polynomial approximations.