Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies
Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on the manifolds of...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2021-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/ac0b02 |
| _version_ | 1827872857096454144 |
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| author | Ilia A Luchnikov Mikhail E Krechetov Sergey N Filippov |
| author_facet | Ilia A Luchnikov Mikhail E Krechetov Sergey N Filippov |
| author_sort | Ilia A Luchnikov |
| collection | DOAJ |
| description | Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on the manifolds of unitary and isometric matrices as well as the cone of positive-definite matrices. Combining this approach with the up-to-date computational methods of automatic differentiation, we demonstrate the efficacy of the Riemannian optimization in the study of the low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of a tensor network in the form of the multiscale entanglement-renormalization ansatz, preparation of arbitrary states (including highly entangled ones) in the circuit implementation of quantum computation, decomposition of quantum gates, and tomography of quantum states. Universality of the developed approach together with the provided open source software enable one to apply the Riemannian optimization to complex quantum architectures well beyond the listed problems, for instance, to the optimal control of noisy quantum systems. |
| first_indexed | 2024-03-12T16:29:30Z |
| format | Article |
| id | doaj.art-7407deca460546c1844efffe5f590c7b |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:29:30Z |
| publishDate | 2021-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-7407deca460546c1844efffe5f590c7b2023-08-08T15:34:10ZengIOP PublishingNew Journal of Physics1367-26302021-01-0123707300610.1088/1367-2630/ac0b02Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologiesIlia A Luchnikov0https://orcid.org/0000-0003-3010-740XMikhail E Krechetov1Sergey N Filippov2https://orcid.org/0000-0001-6414-2137Moscow Institute of Physics and Technology , Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia; Skolkovo Institute of Science and Technology , Skolkovo, Moscow Region 121205, Russia; Russian Quantum Center , Skolkovo, Moscow 143025, RussiaSkolkovo Institute of Science and Technology , Skolkovo, Moscow Region 121205, RussiaMoscow Institute of Physics and Technology , Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia; Steklov Mathematical Institute of Russian Academy of Sciences , Gubkina Street 8, Moscow 119991, Russia; Valiev Institute of Physics and Technology of Russian Academy of Sciences , Nakhimovskii Prospect 34, Moscow 117218, RussiaOptimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on the manifolds of unitary and isometric matrices as well as the cone of positive-definite matrices. Combining this approach with the up-to-date computational methods of automatic differentiation, we demonstrate the efficacy of the Riemannian optimization in the study of the low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of a tensor network in the form of the multiscale entanglement-renormalization ansatz, preparation of arbitrary states (including highly entangled ones) in the circuit implementation of quantum computation, decomposition of quantum gates, and tomography of quantum states. Universality of the developed approach together with the provided open source software enable one to apply the Riemannian optimization to complex quantum architectures well beyond the listed problems, for instance, to the optimal control of noisy quantum systems.https://doi.org/10.1088/1367-2630/ac0b02Riemannian optimizationStiefel manifoldquantum state engineeringmultiscale entanglement-renormalization ansatzquantum tomography |
| spellingShingle | Ilia A Luchnikov Mikhail E Krechetov Sergey N Filippov Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies New Journal of Physics Riemannian optimization Stiefel manifold quantum state engineering multiscale entanglement-renormalization ansatz quantum tomography |
| title | Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies |
| title_full | Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies |
| title_fullStr | Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies |
| title_full_unstemmed | Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies |
| title_short | Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies |
| title_sort | riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies |
| topic | Riemannian optimization Stiefel manifold quantum state engineering multiscale entanglement-renormalization ansatz quantum tomography |
| url | https://doi.org/10.1088/1367-2630/ac0b02 |
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