| EPSRC Reference: |
EP/P033954/1 |
| Title: |
Tensor-product algorithms for quantum control problems |
| Principal Investigator: |
Savostyanov, Dr D |
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| Department: |
Sch of Computing, Engineering & Maths |
| Organisation: |
University of Brighton |
| Scheme: |
First Grant - Revised 2009 |
| Starts: |
01 January 2018 |
Ends: |
30 June 2019 |
Value (£): |
100,878
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.bton.ac.uk |