[论文解读] 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 Architecture被引用 47

一句话总结

本文综述了量子算法技术，如 qubitization、block-encodings 和 phase estimation，讨论 Grover-like 运算符如何对 A 的函数进行编码并实现用于端到端算法复杂度的 Chebyshev 多项式表示。

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.

研究动机与目标

• 通过探索实际应用及其端到端复杂度来激发对量子算法的研究。
• 解释关键算子构造，如 qubitization 和 block-encoding，将给定算子转变为 Grover-like 形式。
• 说明量子 phase estimation 如何访问本征值并实现动力学的多项式近似。
• 强调通过对带结构的单位ary 的重复应用与 Chebyshev 多项式之间的联系，以进行复杂度分析。

提出的方法

• 描述从单位ary UA 使用 Z|0⟩ 反射构造 Grover-like 运算符 W 以获得 2x2 块形式。
• 显示 W 在不变子空间上的限制产生一个角度 θλ = arccos(λ) 的旋转，特征值为 e ± i θλ。
• 证明 W 在跨子空间作为一个块对角组合 Wd = ⨁λ(Td(λ)1 − λ^2Ud−1(λ) − 1 − λ^2Ud−1(λ)Td(λ))，将其与第一类和第二类 Chebyshev 多项式 (Td, Ud) 相联系。
• 将重复应用与 Chebyshev 多项式表示联系起来，使得量子算法的端到端复杂度分析成为可能。

实验结果

研究问题

• RQ1 如何利用像 qubitization 和 block-encoding 这样的算子编码实现端到端的量子算法复杂度？
• RQ2 Grover-like 反射和通过量子 phase estimation 访问本征值在实际算法设计中起到什么作用？
• RQ3 Chebyshev 多项式表示（Td, Ud）如何描述重复单位ary 在不变子空间的作用？
• RQ4 本征值结构 e^{±iθλ} 对量子过程的性能与误差分析有何影响？

主要发现

• 从 base UA 和一个反射导出的 Grover-like 运算符 W 在不变子空间上产生 θλ = arccos(λ) 的旋转。
• 本征值在 e^{±i arccos(λ)}，对应于由 |0^m⟩|λ⟩ 及其正交伴随构建的本征向量。
• 在子空间内对 W 的重复作用映射到第一类和第二类 Chebyshev 多项式，支持基于多项式的复杂度推理。
• block-encoding 框架允许将 A 表达在一个支持 phase estimation 和本征值相关技术的单位ary 形式中。
• 这些构造通过将算子表示与经典多项式近似联系起来，支撑端到端的算法策略。