Algebraic Geometry of Matrix Product States
We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with...
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| Дата: | 2014 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2014
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146599 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Algebraic Geometry of Matrix Product States / A. Critch, J. Morton // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1465992019-02-11T01:24:02Z Algebraic Geometry of Matrix Product States Critch, A. Morton, J. We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters. 2014 Article Algebraic Geometry of Matrix Product States / A. Critch, J. Morton // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14J81; 81Q80; 14Q15 DOI:10.3842/SIGMA.2014.095 http://dspace.nbuv.gov.ua/handle/123456789/146599 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters. |
| format |
Article |
| author |
Critch, A. Morton, J. |
| spellingShingle |
Critch, A. Morton, J. Algebraic Geometry of Matrix Product States Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Critch, A. Morton, J. |
| author_sort |
Critch, A. |
| title |
Algebraic Geometry of Matrix Product States |
| title_short |
Algebraic Geometry of Matrix Product States |
| title_full |
Algebraic Geometry of Matrix Product States |
| title_fullStr |
Algebraic Geometry of Matrix Product States |
| title_full_unstemmed |
Algebraic Geometry of Matrix Product States |
| title_sort |
algebraic geometry of matrix product states |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146599 |
| citation_txt |
Algebraic Geometry of Matrix Product States / A. Critch, J. Morton // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT critcha algebraicgeometryofmatrixproductstates AT mortonj algebraicgeometryofmatrixproductstates |
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2023-05-20T17:25:11Z |
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2023-05-20T17:25:11Z |
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