Tensor-product algorithms for quantum control problems

我们知道量子物理学的定律，微小的粒子（如原子、电子和光子）是据此存在的。但我们能利用这些知识来控制它们的行为，让它们真正有用吗？正是控制将知识转化为技术。即使完全理解了反直觉量子现象背后的物理原理，即使有了能够在量子尺度上起作用的先进仪器（如激光、磁铁或单光子），我们仍然依靠数值算法来解方程，并告诉我们如何驱动量子系统按照我们想要的方式运行。数学量子控制为现代技术、科学和社会所需要的从量子物理的第一原理到高端工程应用铺平了道路。量子技术的规模迅速扩大——在几十年内，我们预计量子计算机将出现，在那里，数百个量子粒子作为一个单一的系统一起工作。这种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。现在用于控制量子器件的传统算法并不适合这一挑战，即使假设计算能力将按照摩尔定律的乐观估计增长。我的项目旨在打破维度的诅咒，并准备解决未来提出的问题，而不是通过超级计算机的蛮力，而是通过开发更智能的数值算法，利用问题的内部结构。这个项目的核心是张量积格式。它们基于变量分离的一般思想，这是由矩阵和高维数组（张量，波函数）的低秩分解在数学上描述的。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。不是每个量子态都可以被压缩。一些状态有低纠缠，这意味着量子粒子几乎不依赖于彼此。有些状态是完全纠缠的，一个粒子发生的变化会立即影响到其他粒子的状态。只有具有低和中等纠缠的状态才能被压缩，因此在计算上是可访问的。当算法被限制在计算可访问状态的流形时，我们有新的数学问题需要回答，新的计算策略需要提出、实现、测试和推广到应用中。本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方法的继任者）和黎曼流形的优化，为量子控制问题开发快速准确的张量积算法，黎曼流形在数学上描述了计算上可实现的状态集。算法是灵活的，张量积算法可用于任何高维问题。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，这是其他学科的研究人员可以理解的。在这个项目中创建的所有算法都将公开提供。我开发的算法已经被研究人员用于理解复杂的基因反应网络，更快地解决随机和参数问题，以及设计更精确的核磁共振（NMR）和磁共振成像（MRI）实验。我很兴奋，我将在这个项目中开发的控制量子计算机的方法可能在各种应用中有用，这是我可以预测的，也是我还不能预测的。

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.