SIAM Review, Volume 62, Issue 4, Page 899-900, January 2020.
This issue of SIAM Review presents two papers in the Education section. The first paper, “Time Correlation Functions of Equilibrium and Nonequilibrium Langevin Dynamics: Derivations and Numerics Using Random Numbers,” is coauthored by Xiaocheng Shang and Martin Krog̈er. Langevin dynamics is a term describing a mathematical model of the dynamics of molecular systems which uses stochastic differential equations. This model extends the molecular dynamics to allow for describing effects due to friction caused by solvent or air molecules or other phenomena that perturb the system. In this paper, the authors present the model equation for the behavior of a harmonic oscillator in two or three dimensions in the presence of friction, additive noise, and an external field with both rotational and deformational components. They study the time correlation function of the dynamics analytically and numerically. The paper provides a good illustration of the tools of stochastic calculus on a nontrivial application and helps evaluate the numerical solvers for stochastic differential equations (SDEs). The authors argue that the fair assessment of the efficiency and accuracy of numerical solvers for SDEs requires access to analytical reference solutions. However, deriving relevant analytical formulae is possible only for very simple cases. Here a two-dimensional nontrivial benchmark problem is analyzed and an exact solution is derived. The analysis in the paper includes also a discussion on inertial effects. The model equation formulated in section 2 is interesting because it arises in multiple applications. Furthermore, the absence of the restoring force or the external field in the model leads to popular special cases. Among those are the random walk, diffusion models, the motion of atoms in the presence of gravitational or centrifugal potential, nanomagnets subjected to magnetic fields, Brownian oscillators, rotational relaxation of molecules trapped in a three-dimensional crystal, and many others. Relevant literature describing those models is provided. The authors describe a direct approach to the derivation of the time correlation functions for general Langevin dynamics, as well as an approach that uses Fourier transforms and the Wiener--Khinchin theorem, and a complementary approach via the associated Fokker--Planck equation. The last portion of the paper contains the description of two numerical methods and numerical experience. The second paper presents “An Introduction to Quantum Computing, without the Physics,” written by Giacomo Nannicini. This paper introduces discrete mathematicians and computer scientists to quantum computing. While quantum computing uses certain quantum phenomena to perform computation, many scientists may learn about its principles without acquiring deeper knowledge of quantum physics. The author starts with a set of assumptions that are satisfied by a quantum computing device. From those assumptions further properties are derived in a rigorous way. The necessary concepts are formally introduced and all statements are supplied with mathematical proofs. The author adopts a model of computation, which is known as the quantum circuit model. Quoting from the paper, it works as follows: - The quantum computer has a state that is contained in a quantum register and is initialized in a predefined way. - The state evolves by applying operations specified in advance in the form of an algorithm. - At the end of the computation, some information on the state of the quantum register is obtained by means of a special operation, called a measurement. At the beginning of the paper, some properties of the tensor product are reviewed, which are used to express states and operations in a quantum device. For example, qubits are the quantum counterparts of the bits in our computers and the state of a $q$-qubit quantum register is identified with a unit vector in the space $(\mathbb{C}^2)^{\otimes q}$, where $\mathbb{C}^2$ is the complex Euclidean space. The exposition proceeds with the introduction and analysis of the types of states and their properties, the quantum phenomena of superposition and enlargement, measurement gates, and the input-output model for quantum computations. The second portion of the paper discusses several quantum algorithms including a numerical implementation of Grover's algorithm. It may be of interest that, although quantum computing is able to solve some problems much faster than the classical computer, it is still not known to solve NP-complete problems efficiently. The paper concludes with a section containing notes for further reading. According to the author's experience, the paper provides material for a graduate-level module in quantum computing for mathematicians and computer scientists. It is recommended to split the material into five lectures/meetings of 90--120 minutes each. The last meeting would aim at providing a hands-on numerical experience based on section 6 of the paper. The audience should be familiar with linear algebra and with basic concepts in computing such as Turing machines and algorithmic complexity.

中文翻译：

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教育

SIAM评论，第62卷，第4期，第899-900页，2020年1月。
本期《 SIAM评论》在“教育”部分提供了两篇论文。尚小成和马丁·克罗格尔（MartinKrog̈er）合着了第一篇论文，“平衡和非平衡兰格文动力学的时间相关函数：使用随机数的导数和数值”。Langevin动力学是一个描述使用随机微分方程的分子系统动力学数学模型的术语。该模型扩展了分子动力学，可以描述由于溶剂或空气分子或其他干扰系统的现象引起的摩擦而产生的影响。在本文中，作者提出了在存在摩擦，附加噪声以及具有旋转和变形分量的外场的情况下，二维或三次谐波振荡器行为的模型方程。他们通过分析和数值研究动力学的时间相关函数。这篇文章很好地说明了在非平凡应用中的随机演算工具，并有助于评估随机微分方程（SDE）的数值求解器。作者认为，对SDE数值求解器的效率和准确性进行公正的评估需要获得分析参考解决方案。但是，只有在非常简单的情况下，才能得出相关的分析公式。在此分析二维非平凡基准问题，并得出精确解。本文的分析还包括对惯性效应的讨论。第2节中公式化的模型方程式很有趣，因为它出现在多种应用中。此外，模型中缺少恢复力或没有外部场会导致流行的特殊情况。其中包括随机游走，扩散模型，在存在重力或离心势的情况下原子的运动，受到磁场作用的纳米磁铁，布朗振荡器，在三维晶体中捕获的分子的旋转弛豫等。提供了描述那些模型的相关文献。作者介绍了用于导出一般Langevin动力学时间相关函数的直接方法，以及使用傅立叶变换和Wiener-Khinchin定理的方法，以及通过相关的Fokker-Planck方程的补充方法。本文的最后一部分包含两种数值方法和数值经验的描述。第二篇论文由Giacomo Nannicini撰写，题为“没有物理的量子计算入门”。本文向离散数学家和计算机科学家介绍了量子计算。尽管量子计算使用某些量子现象来执行计算，但许多科学家可能无需了解更深入的量子物理学知识就能了解其原理。作者从一组量子计算设备可以满足的假设开始。从这些假设中，可以进一步得出严格的属性。正式介绍了必要的概念，并且所有陈述都提供了数学证明。作者采用了一种计算模型，称为量子电路模型。引用本文，其工作方式如下：-量子计算机具有包含在量子寄存器中并以预定义方式初始化的状态。-通过应用以算法形式预先指定的操作来演变状态。-在计算结束时，有关量子寄存器状态的某些信息是通过称为测量的特殊操作获得的。在本文的开头，对张量积的一些属性进行了回顾，这些属性用于表达量子设备中的状态和操作。例如，量子位是我们计算机中位的量子对应物，并且$ q $-量子位量子寄存器的状态用空间$（\ mathbb {C} ^ 2）^ {\ otimes q } $，其中$ \ mathbb {C} ^ 2 $是复杂的欧几里得空间。博览会继续介绍和分析状态的类型及其性质，叠加和扩大的量子现象，测量门以及用于量子计算的输入输出模型。本文的第二部分讨论了几种量子算法，包括格罗弗算法的数值实现。有趣的是，尽管量子计算能够比传统计算机更快地解决某些问题，但仍然未知如何有效地解决NP完全问题。本文的结尾部分包含一些注释，供您进一步阅读。根据作者的经验，本文为数学家和计算机科学家提供了量子计算的研究生级模块的材料。建议将材料分成五节每次90--120分钟的讲座/会议。上次会议的目的是根据本文第6节提供动手的数值体验。听众应该熟悉线性代数和计算的基本概念，例如图灵机和算法复杂性。