权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Tensor-product algorithms for quantum control problems

量子控制问题的张量积算法

基本信息

批准号：
EP/P033954/1
负责人：
Dmitry Savostyanov
金额：
$ 12.85万
依托单位：
University of Brighton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP033954%2F1
关键词：
Tensor product algorithms quantum control

项目摘要

We know the laws of quantum physics, by which tiny particles (like atoms, electrons and photons) live. But can we use this knowledge to control their behaviour and make them really useful?It is control that turns knowledge into technology. Even with full understanding of the physics behind counter-intuitive quantum phenomena even with advanced instruments capable of acting on a quantum scale (such as lasers, magnets or single photons), we rely on numerical algorithms to solve equations and tell us how to drive a quantum system the way we want it to go. Mathematical quantum control paves the way from the first principles of quantum physics to high-end engineering applications, demanded by modern technology, science and society. The quantum technologies quickly grow in size --- in a few decades we expect quantum computers to appear, where hundred(s) of quantum particles are working together as a single system. The complexity of such systems grows exponentially with their size --- just like a football game depends on every player on the field, the state of a quantum system depends on all states of individual particles. This problem, known as the curse of dimensionality, is probably the biggest computational challenge of the 21st century. Traditional algorithms now used to control the quantum devices are not fit for the challenge, even assuming that computational power will increase in line with optimistic estimates of Moore's law.My project aims to beat the curse of dimensionality and prepare to solve the problems which the future poses not by the brute force of supercomputers, but by developing smarter numerical algorithms, which exploit the internal structure of the problem.At the heart of this project are tensor product formats. They are based on the general idea of the separation of variables, which is described mathematically by a low-rank decomposition of matrices and high-dimensional arrays (tensors, wavefunctions). It is crucial to keep the data in a compressed representation throughout the whole calculation, which requires us to rewrite all the algorithms we use, starting with elementary operations like +, - and *.Not every quantum state can be compressed. Some states have low entanglement, which means that quantum particles barely depend on each other. Some states are fully entangled, and the change which happens with one particle immediately affects the state of the others. Only states with low and moderate entanglement can be compressed and thus are computationally accessible. When algorithms are restricted to the manifold of computationally accessible states, we have new mathematical questions to be answered, new computational strategies to be proposed, implemented, tested and promoted to applications. This project aims to achieve it.I will develop fast and accurate tensor product algorithms for quantum control problems using recently proposed alternating minimal energy algorithm (AMEn, successor to DMRG and MPS methods) and optimisation on Riemaniann manifolds, which mathematically describe the set of computationally achievable states.Algorithms are flexible, and the tensor product algorithms can be used in any high-dimensional problem. In this project I will describe the algorithms and ideas in general language of numerical linear algebra, which researchers from other disciplines can understand. All algorithms created in this project will be made publicly available. The algorithms I developed are already used by researchers aiming to understand complex gene reaction networks, to solve stochastic and parametric problems faster, and to design more accurate nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) experiments. I am excited by the possibility that the methods I will develop in this project to control a quantum computer could to be useful in a variety of applications, which I can and which I can not yet predict.

我们知道量子物理的定律，微小的粒子(如原子、电子和光子)都是靠它生存的。但我们能用这些知识来控制他们的行为，让他们真正有用吗？正是控制将知识转化为技术。即使对反直觉量子现象背后的物理现象有了充分的理解，即使使用了能够在量子尺度上发挥作用的先进仪器(如激光、磁铁或单光子)，我们仍然依赖数值算法来求解方程，并告诉我们如何驱动量子系统按我们希望的方式运行。数学量子控制为从量子物理的基本原理到现代技术、科学和社会所要求的高端工程应用铺平了道路。量子技术在规模上迅速增长-我们预计几十年后将出现量子计算机，数百个(S)量子粒子将作为一个单一系统一起工作。这类系统的复杂性随着它们的大小呈指数增长-就像足球比赛取决于球场上的每个球员一样，量子系统的状态取决于单个粒子的所有状态。这个被称为维度诅咒的问题可能是21世纪最大的计算挑战。现在用来控制量子设备的传统算法不适合这一挑战，即使假设计算能力将按照摩尔定律的乐观估计增加。我的项目旨在克服维度诅咒，并准备解决未来提出的问题，而不是超级计算机的蛮力，而是通过开发更智能的数值算法，利用问题的内部结构。该项目的核心是张量积格式。它们基于分离变量的一般思想，这在数学上由矩阵和高维数组(张量、波函数)的低阶分解来描述。在整个计算过程中保持数据的压缩表示是至关重要的，这需要我们重写我们使用的所有算法，从+、-和*等基本操作开始。一些态的纠缠度很低，这意味着量子粒子几乎不相互依赖。有些状态是完全纠缠的，一个粒子发生的变化会立即影响其他粒子的状态。只有低纠缠和中等纠缠的态才能被压缩，因此可以通过计算获得。当算法被限制在计算可访问的流形上时，我们有新的数学问题需要回答，新的计算策略需要提出、实施、测试和推广到应用中。这个项目旨在实现这一目标。我将使用最近提出的交替最小能量算法(AMEN，DMRG和MPS方法的后续算法)和Riemaniann流形上的优化来开发快速而准确的张量积算法来解决量子控制问题，该流形从数学上描述了可计算可实现的状态集。在这个项目中，我将用数值线性代数的通用语言描述算法和思想，其他学科的研究人员可以理解。在这个项目中创建的所有算法都将公开可用。我开发的算法已经被研究人员使用，目的是了解复杂的基因反应网络，更快地解决随机和参数问题，并设计更准确的核磁共振(NMR)和磁共振成像(MRI)实验。令我兴奋的是，我将在这个项目中开发的控制量子计算机的方法可能会在各种应用中有用，我可以预测，但我还不能预测。