# Tensor-product algorithms for quantum control problems

Lead Research Organisation:
University of Brighton

Department Name: Sch of Computing, Engineering & Maths

### Abstract

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.

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.

### Planned Impact

Quantum technologies are everywhere around us. As the recent Blackett review mentions, "there can be few homes in the UK without quantum devices. We wear them on our wrists, we talk into them, we watch them, and we travel in vehicles controlled by them." Quantum technologies are behind magnetic resonance imaging (MRI) in medicine, nuclear magnetic resonance (NMR) in chemistry, satellite navigators and mobile phones, secure communication and precise sensors. The new generation of quantum technologies is almost at our doorstep, including large-scale quantum computers, which are expected to appear in one-two decades.

To help quantum engineers and physicists to design more energy-efficient and reliable quantum devices, this project aims to equip them with faster and more accurate numerical algorithms, capable of simulating the behaviour of quantum systems with hundred(s) of particles. Existing algorithms have to rely on heuristic assumptions and unreliable approximations to cope with the exponentially growing complexity of the required calculations, known as the curse of dimensionality. This project aims to deliver algorithms which do not rely on heuristics (and thus are more reliable), solve problems faster (and thus reduce IT bills and save researchers' time to do more research), and solve them more accurately (and thus make experiments and devices more precise).

This project aims specifically at two applications: large-scale quantum systems that can be used in a quantum computer (if we can fully control their behaviour), and nuclear magnetic resonance (NMR) spectrography, in particular for experiments involving spatial dynamics (rotation experiments in NMR, spin-spin coupling in MRI, etc). This will mean better and faster drug development for pharmacy, improved patient diagnostics, a breakthrough in information security, machine learning and energy efficient computing. Potential beneficiaries in industry include groups working on the development of a quantum computer and related quantum technologies (e.g. Photonics research centre, Southampton), and researchers from quantum chemistry, pharmacy and medicine using NMR and MRI experiments.

To ensure that potential end users can benefit from the results of this project, the developed algorithms will be incorporated into free open-source software packages, such as Spinach, a popular tool among NMR and MRI researchers, and made publicly available. The code will be complete with illustrative examples and tutorials. The project team will promote the algorithms in academic and industrial forums (through reviews and mailing lists) and invite potential users to test them for their problems. Continuous support will be provided for the users by collecting their feedback, by responding to their queries, and where possible by implementing the requested features and functions of the code.

Tensor methods are still in a very early stage of their development and are not embedded in computational practice beyond academia. However, these methods have huge potential, reducing the time required for computations from weeks to hours and minutes. The important goal of this project is to promote them to potential users and to learn from their indispensable expertise and feedback. The project team will collaborate with academic partners, including Prof Dieter Hans Jaksch (U Oxford) and Dr Ilya Kuprov (U Southampton), and through this collaboration aims to establish closer links with end users of the developed methods. The main impact that we seek for the project team itself is establishing a network of collaboration with industrial partners that will secure further development of tensor product algorithms and enable their effective use for the development of quantum technologies and other challenging high-dimensional applications.

To help quantum engineers and physicists to design more energy-efficient and reliable quantum devices, this project aims to equip them with faster and more accurate numerical algorithms, capable of simulating the behaviour of quantum systems with hundred(s) of particles. Existing algorithms have to rely on heuristic assumptions and unreliable approximations to cope with the exponentially growing complexity of the required calculations, known as the curse of dimensionality. This project aims to deliver algorithms which do not rely on heuristics (and thus are more reliable), solve problems faster (and thus reduce IT bills and save researchers' time to do more research), and solve them more accurately (and thus make experiments and devices more precise).

This project aims specifically at two applications: large-scale quantum systems that can be used in a quantum computer (if we can fully control their behaviour), and nuclear magnetic resonance (NMR) spectrography, in particular for experiments involving spatial dynamics (rotation experiments in NMR, spin-spin coupling in MRI, etc). This will mean better and faster drug development for pharmacy, improved patient diagnostics, a breakthrough in information security, machine learning and energy efficient computing. Potential beneficiaries in industry include groups working on the development of a quantum computer and related quantum technologies (e.g. Photonics research centre, Southampton), and researchers from quantum chemistry, pharmacy and medicine using NMR and MRI experiments.

To ensure that potential end users can benefit from the results of this project, the developed algorithms will be incorporated into free open-source software packages, such as Spinach, a popular tool among NMR and MRI researchers, and made publicly available. The code will be complete with illustrative examples and tutorials. The project team will promote the algorithms in academic and industrial forums (through reviews and mailing lists) and invite potential users to test them for their problems. Continuous support will be provided for the users by collecting their feedback, by responding to their queries, and where possible by implementing the requested features and functions of the code.

Tensor methods are still in a very early stage of their development and are not embedded in computational practice beyond academia. However, these methods have huge potential, reducing the time required for computations from weeks to hours and minutes. The important goal of this project is to promote them to potential users and to learn from their indispensable expertise and feedback. The project team will collaborate with academic partners, including Prof Dieter Hans Jaksch (U Oxford) and Dr Ilya Kuprov (U Southampton), and through this collaboration aims to establish closer links with end users of the developed methods. The main impact that we seek for the project team itself is establishing a network of collaboration with industrial partners that will secure further development of tensor product algorithms and enable their effective use for the development of quantum technologies and other challenging high-dimensional applications.

### Publications

D. QuiĆ±ones Valles
(2019)

*Tensor product approach to quantum control*
Dolgov S
(2020)

*Parallel cross interpolation for high-precision calculation of high-dimensional integrals*in Computer Physics CommunicationsDescription | One of the key questions in quantum control tensor is whether a large system of quantum particles can be controlled as a whole, and how to control it to achieve the desired result. Typically this question is answered positively if/when some control protocol (such as a sequence of laser pulses) is offered which can take the quantum system from a given to a desired state. This protocol is typically constructed from a sequence of quantum gates and is usually not optimal in terms of required energy and/or required time. Calculation of the optimal control pulse for large spin systems is prohibitively expensive and not available for spin systems of about 50 spins, which is the current target for quantum computing. We used tensor product approximations to develop an algorithm of a new class that can describe the dynamics of large quantum systems using the compressed low-rank representations of their high-dimensional states. Using this algorithm, we were able to calculate the optimal control pulses for large linear systems, such Heisenberg spin wires, which are essential for emerging quantum computing technologies. At this stage we can control chains with up to 40 spins and the optimisation of the pulse can be performed on a single workstation. We also used tensor product interpolation to calculate with high precision high-dimensional integrals, describing the magnetic susceptibility of 2D Ising model. This problem was proposed as a computational challenge and benchmark by Prof David H. Bailey (Berkley) but remained unsolved for 15 years. Using tensor product methods, we constructed an adaptive numerical quadrature with radically improved convergence rate, and managed to calculate the required integrals to 30 accurate decimal digits, where the state of the art algorithms provided results accurate only to 6 digits. In the framework of experimental mathematics, proposed by Prof. Bailey, our results may help to uncover the physically relevant analytic expressions behind the integrals and hence further progress our understanding of superconductivity and other properties of ferromagnetic materials. |

Exploitation Route | Quantum control is one of the main areas of development of the whole computational science today, leading to potentially transforming applications in Informational Systems, Security, Electronics and Energy applications. Reducing computational time (and hence energy) required to calculate optimal pulses can have a major effect on environment. |

Sectors | Digital/Communication/Information Technologies (including Software),Electronics,Energy,Environment,Security and Diplomacy |