Tensor product algorithms for quantum control problems

The aim of this research is to provide faster and more reliable numerical tools for fellow researchers who work with highdimensional problems in quantum physics, chemistry and other areas.

At the heart of the project are tensor product algorithms. Based on the idea of separation of variables, they substitute the exponentially large state vector with a structured representation in tensor product format, e.g. tensor train (TT) format, also known as MPS/MPO/DMRG.

By combining the optimisation framework of DMRG/MPS methods from quantum physics with classical ideas of iterative methods, I have recently developed an alternating minimal energy (AMEn) method, a faster and more reliable algorithm for solving high-dimensional linear systems. The method has been adopted for solving highdimensional problems in quantum physics, plasma dynamics, fluid dynamics, mathematical biology, machine learning, uncertainty quantification. The project aims to extend the success of tensor product algorithms to quantum control problems.

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.