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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.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| _version_ | 1862730154963369984 |
|---|---|
| author | Critch, A. Morton, J. |
| author_facet | Critch, A. Morton, J. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T19:18:55Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146599 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:18:55Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Critch, A. Morton, J. 2019-02-10T09:46:25Z 2019-02-10T09:46:25Z 2014 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 https://nasplib.isofts.kiev.ua/handle/123456789/146599 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. AC and JM were supported in part by DARPA under awards FA8650-10-C-7020 and N66001-10-
 1-4040 respectively. We would like to thank J. Biamonte, J. Eisert, B. Sturmfels, F. Vaccarino,
 F. Verstraete, and G. Vidal for helpful discussions. We are also grateful to anonymous referees
 who provided helpful comments and corrections. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Algebraic Geometry of Matrix Product States Article published earlier |
| spellingShingle | Algebraic Geometry of Matrix Product States Critch, A. Morton, J. |
| title | 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_short | Algebraic Geometry of Matrix Product States |
| title_sort | algebraic geometry of matrix product states |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146599 |
| work_keys_str_mv | AT critcha algebraicgeometryofmatrixproductstates AT mortonj algebraicgeometryofmatrixproductstates |