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，量子物理学的定律（例如原子，电子和光子）生存。但是，我们能否利用这些知识来控制其行为并使其真正有用？这是控制知识变成技术的控制。即使有能够以量子尺度起作用（例如激光器，磁铁或单个光子）的高级仪器，我们也依靠数值算法来求解方程，并告诉我们如何以我们希望的方式驱动量子系统。数学量子控制铺平了从现代技术，科学和社会要求的量子物理学的第一原理到高端工程应用的道路。量子技术很快就会大小生长 - 在几十年中，我们预计量子计算机将出现，其中数百（s）的量子颗粒作为单个系统都一起工作。这种系统的复杂性随着它们的大小而成倍增长 - 就像足球比赛取决于场上的每个玩家一样，量子系统的状态取决于各个粒子的所有状态。这个问题被称为维度的诅咒，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挑战，即使假设计算能力将与摩尔定律的乐观估计相符。格式。它们基于变量分离的一般思想，该矩阵的低排列分解和高维阵列（张量，波形）在数学上描述了。在整个计算过程中，将数据保持在压缩表示中至关重要，这要求我们重写我们使用的所有算法，从 + +， - - - 和 *的基本操作开始。一些状态的纠缠较低，这意味着量子颗粒几乎不依赖于彼此。有些状态完全纠缠，一个粒子发生的变化立即影响另一种粒子的状态。只有低和中等纠缠的状态才能被压缩，因此可以在计算上访问。当算法仅限于可访问的计算态的多种状态时，我们需要回答新的数学问题，要提出，实施，测试和推广到应用程序的新计算策略。该项目旨在实现这一目标。我将使用最近提议的交替交替的最小能量算法（Amen，DMRG和MPS方法的继任者和MPS方法）开发快速，准确的张量产品算法，以用于量子控制问题，并对Riemaniann流形进行了优化，并在数学上描述了可以实现的稳定性。高维问题。在这个项目中，我将以数值线性代数的一般语言描述算法和思想，来自其他学科的研究人员可以理解。该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旨在了解复杂基因反应网络的研究人员使用，以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。我为控制量子计算机可以在这个项目中开发的方法可能在各种应用程序中有用的可能性感到兴奋，而我可以和我无法预测的应用程序。