Simulación cuántica del Modelo de Jaynes-Cummings
Este trabajo implementa la simulación cuántica del modelo de Jaynes-Cummings y del modelo de Rabi en la plataforma Qiskit de IBM, utilizando los mapeos de Holstein-Primakoff y binario. Mientras que la simulación del modelo de Jaynes-Cummings reproduce con precisión las oscilaciones de Rabi, la del m...
- Autores:
-
Vélez Peña, Esneider
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2025
- Institución:
- Universidad de Antioquia
- Repositorio:
- Repositorio UdeA
- Idioma:
- spa
- OAI Identifier:
- oai:bibliotecadigital.udea.edu.co:10495/46047
- Acceso en línea:
- https://hdl.handle.net/10495/46047
- Palabra clave:
- Sistemas cuánticos
Quantum systems
Luz - Dispersión
Light - Scattering
Óptica cuántica
Quantum optics
Computación cuántica
Quantum computing
Electrodinámica cuántica
Quantum electrodynamics
Teoría de la aproximación
Approximation theory
Sistemas hamiltonianos
Hamiltonian systems
Simulación por computador
Computer simulation
Modelo de Jaynes–Cummings
http://id.loc.gov/authorities/subjects/sh2013002642
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http://id.loc.gov/authorities/subjects/sh85029533
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by-nc-sa/4.0/
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Simulación cuántica del Modelo de Jaynes-Cummings |
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Simulación cuántica del Modelo de Jaynes-Cummings |
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Simulación cuántica del Modelo de Jaynes-Cummings Sistemas cuánticos Quantum systems Luz - Dispersión Light - Scattering Óptica cuántica Quantum optics Computación cuántica Quantum computing Electrodinámica cuántica Quantum electrodynamics Teoría de la aproximación Approximation theory Sistemas hamiltonianos Hamiltonian systems Simulación por computador Computer simulation Modelo de Jaynes–Cummings http://id.loc.gov/authorities/subjects/sh2013002642 http://id.loc.gov/authorities/subjects/sh85076877 http://id.loc.gov/authorities/subjects/sh85109465 http://id.loc.gov/authorities/subjects/sh2014002839 http://id.loc.gov/authorities/subjects/sh85109459 http://id.loc.gov/authorities/subjects/sh85006190 http://id.loc.gov/authorities/subjects/sh85058563 http://id.loc.gov/authorities/subjects/sh85029533 |
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Simulación cuántica del Modelo de Jaynes-Cummings |
| title_full |
Simulación cuántica del Modelo de Jaynes-Cummings |
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Simulación cuántica del Modelo de Jaynes-Cummings |
| title_full_unstemmed |
Simulación cuántica del Modelo de Jaynes-Cummings |
| title_sort |
Simulación cuántica del Modelo de Jaynes-Cummings |
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Vélez Peña, Esneider |
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Sanz Vicaro, José Luis |
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Vélez Peña, Esneider |
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Grupo de Física Atómica y Molecular |
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Sistemas cuánticos Quantum systems Luz - Dispersión Light - Scattering Óptica cuántica Quantum optics Computación cuántica Quantum computing Electrodinámica cuántica Quantum electrodynamics Teoría de la aproximación Approximation theory Sistemas hamiltonianos Hamiltonian systems Simulación por computador Computer simulation |
| topic |
Sistemas cuánticos Quantum systems Luz - Dispersión Light - Scattering Óptica cuántica Quantum optics Computación cuántica Quantum computing Electrodinámica cuántica Quantum electrodynamics Teoría de la aproximación Approximation theory Sistemas hamiltonianos Hamiltonian systems Simulación por computador Computer simulation Modelo de Jaynes–Cummings http://id.loc.gov/authorities/subjects/sh2013002642 http://id.loc.gov/authorities/subjects/sh85076877 http://id.loc.gov/authorities/subjects/sh85109465 http://id.loc.gov/authorities/subjects/sh2014002839 http://id.loc.gov/authorities/subjects/sh85109459 http://id.loc.gov/authorities/subjects/sh85006190 http://id.loc.gov/authorities/subjects/sh85058563 http://id.loc.gov/authorities/subjects/sh85029533 |
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Modelo de Jaynes–Cummings |
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http://id.loc.gov/authorities/subjects/sh2013002642 http://id.loc.gov/authorities/subjects/sh85076877 http://id.loc.gov/authorities/subjects/sh85109465 http://id.loc.gov/authorities/subjects/sh2014002839 http://id.loc.gov/authorities/subjects/sh85109459 http://id.loc.gov/authorities/subjects/sh85006190 http://id.loc.gov/authorities/subjects/sh85058563 http://id.loc.gov/authorities/subjects/sh85029533 |
| description |
Este trabajo implementa la simulación cuántica del modelo de Jaynes-Cummings y del modelo de Rabi en la plataforma Qiskit de IBM, utilizando los mapeos de Holstein-Primakoff y binario. Mientras que la simulación del modelo de Jaynes-Cummings reproduce con precisión las oscilaciones de Rabi, la del modelo de Rabi presenta errores debido a la no conmutatividad de sus términos hamiltonianos. Se emplean la aproximación de Trotter-Suzuki y la expansión de Zassenhaus para mejorar la precisión, pero esta última introduce ruido en la simulación. Los resultados evidencian las limitaciones actuales en la implementación de modelos cuánticos de interacción luz-materia en hardware cuántico y sugieren la necesidad de métodos más avanzados. |
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2025 |
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2025 |
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D. Braak. “Integrability of the Rabi Model”. En: Phys. Rev. Lett. 107 (10 ago. de 2011), pág. 100401. doi: 10.1103/PhysRevLett.107.100401. url: https://link.aps.org/doi/10.1103/PhysRevLett.107.100401. Katherine L. Brown, William J. Munro y Vivien M. Kendon. “Using Quantum Computers for Quantum Simulation”. En: Entropy 12.11 (nov. de 2010), págs. 2268-2307. issn: 1099-4300. doi: 10.3390/e12112268. url: http://dx.doi.org/10.3390/e12112268. Fernando Casas, Ander Murua y Mladen Nadinic. “Efficient computation of the Zassenhaus formula”. En: Computer Physics Communications 183.11 (2012), págs. 2386-2391. issn: 0010-4655. doi: https : / / doi . org/10.1016/j.cpc.2012.06.006. url: https://www.sciencedirect.com/science/article/pii/ S001046551200207X. Peter L. Knight Christopher C. Gerry. Introductory Quantum Optics. Cambridge: Cambridge University Press, 2005. Toby S. Cubitt, Ashley Montanaro y Stephen Piddock. “Universal quantum Hamiltonians”. En: Proceedings of the National Academy of Sciences 115.38 (2018), págs. 9497-9502. doi: 10.1073/pnas.1804949115. eprint: https://www.pnas.org/doi/pdf/10.1073/pnas.1804949115. url: https://www.pnas.org/doi/abs/10.1073/pnas.1804949115. Richard P. Feynman. “Simulating physics with computers”. En: International Journal of Theoretical Physics 21.6 (jun. de 1982), págs. 467-488. issn: 1572-9575. doi: 10.1007/BF02650179. url: https://doi.org/10.1007/BF02650179. I. M. Georgescu, S. Ashhab y Franco Nori. “Quantum simulation”. En: Rev. Mod. Phys. 86 (1 mar. de 2014), págs. 153-185. doi: 10.1103/RevModPhys.86.153. url: https://link.aps.org/doi/10.1103/RevModPhys.86.153. J. A. Gyamfi. An Introduction to the Holstein-Primakoff Transformation, with Applications in Magnetic Resonance. 2019. arXiv: 1907.07122 [cond-mat.other]. url: https://arxiv.org/abs/1907.07122. Xin-Yu Huang et al. Qubitization of Bosons. 2022. arXiv: 2105.12563 [quant-ph]. url: https://arxiv.org/abs/2105.12563. Ali Javadi-Abhari et al. Quantum computing with Qiskit. 2024. doi: 10.48550/arXiv.2405.08810. arXiv:2405.08810 [quant-ph]. Seth Lloyd. “A Potentially Realizable Quantum Computer”. En: Science 261.5128 (1993), págs. 1569-1571. doi: 10.1126/science.261.5128.1569. eprint: https://www.science.org/doi/pdf/10.1126/science.261.5128.1569. url: https://www.science.org/doi/abs/10.1126/science.261.5128.1569. Michael A. Nielsen e Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. Bo Peng et al. Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Cons truction, and Error Analysis. 2025. arXiv: 2307.06580 [quant-ph]. url: https://arxiv.org/abs/2307.06580. J.J. Sakurai y J. Napolitano. Modern Quantum Mechanics. Cambridge University Press, 2017. isbn: 9781108422413. url: https://books.google.com.co/books?id=010yDwAAQBAJ. Rohit Sarma Sarkar, Sabyasachi Chakraborty y Bibhas Adhikari. Scalable quantum circuits for exponential of Pauli strings and Hamiltonian simulations. 2024. arXiv: 2405.13605 [quant-ph]. url: https://arxiv.org/abs/2405.13605. S Somaroo et al. “Quantum Simulations on a Quantum Computer”. En: Physical Review Letters 82.26 (jun. de 1999). doi: 10.1103/PhysRevLett.82.5381. url: https://www.osti.gov/biblio/349372. R. Somma et al. “Simulating physical phenomena by quantum networks”. En: Phys. Rev. A 65 (4 abr. de 2002), pág. 042323. doi: 10.1103/PhysRevA.65.042323. url: https://link.aps.org/doi/10.1103/PhysRevA.65.042323. Masuo Suzuki. “Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems”. En: Communications in Mathematical Physics 51.2 (jun. de 1976), págs. 183-190. issn: 1432-0916. doi: 10.1007/BF01609348. url: https://doi.org/10.1007/BF01609348. James D. Whitfield, Jacob Biamonte y Alán Aspuru-Guzik. “Simulation of electronic structure Hamiltonians using quantum computers”. En: Molecular Physics 109.5 (mar. de 2011), págs. 735-750. issn: 1362-3028. doi:10.1080/00268976.2011.552441. url: http://dx.doi.org/10.1080/00268976.2011.552441. Leo Zhou y Dorit Aharonov. Strongly Universal Hamiltonian Simulators. 2021. arXiv: 2102.02991 [quant-ph]. url: https://arxiv.org/abs/2102.02991 |
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Sanz Vicaro, José LuisVélez Peña, EsneiderGrupo de Física Atómica y Molecular2025-05-22T12:44:52Z2025https://hdl.handle.net/10495/46047Este trabajo implementa la simulación cuántica del modelo de Jaynes-Cummings y del modelo de Rabi en la plataforma Qiskit de IBM, utilizando los mapeos de Holstein-Primakoff y binario. Mientras que la simulación del modelo de Jaynes-Cummings reproduce con precisión las oscilaciones de Rabi, la del modelo de Rabi presenta errores debido a la no conmutatividad de sus términos hamiltonianos. Se emplean la aproximación de Trotter-Suzuki y la expansión de Zassenhaus para mejorar la precisión, pero esta última introduce ruido en la simulación. Los resultados evidencian las limitaciones actuales en la implementación de modelos cuánticos de interacción luz-materia en hardware cuántico y sugieren la necesidad de métodos más avanzados.COL0008441PregradoFísico31 páginasapplication/pdfspaUniversidad de AntioquiaFísicaMedellín, ColombiaFacultad de Ciencias Exactas y NaturalesCampus Medellín - Ciudad Universitariahttp://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccessAttribution-NonCommercial-ShareAlike 4.0 Internationalhttp://purl.org/coar/access_right/c_abf2Sistemas cuánticosQuantum systemsLuz - DispersiónLight - ScatteringÓptica cuánticaQuantum opticsComputación cuánticaQuantum computingElectrodinámica cuánticaQuantum electrodynamicsTeoría de la aproximaciónApproximation theorySistemas hamiltonianosHamiltonian systemsSimulación por computadorComputer simulationModelo de Jaynes–Cummingshttp://id.loc.gov/authorities/subjects/sh2013002642http://id.loc.gov/authorities/subjects/sh85076877http://id.loc.gov/authorities/subjects/sh85109465http://id.loc.gov/authorities/subjects/sh2014002839http://id.loc.gov/authorities/subjects/sh85109459http://id.loc.gov/authorities/subjects/sh85006190http://id.loc.gov/authorities/subjects/sh85058563http://id.loc.gov/authorities/subjects/sh85029533Simulación cuántica del Modelo de Jaynes-CummingsTrabajo de grado - Pregradohttp://purl.org/coar/resource_type/c_7a1fhttp://purl.org/redcol/resource_type/TPTexthttp://purl.org/coar/version/c_b1a7d7d4d402bcceinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/draftD. Braak. “Integrability of the Rabi Model”. En: Phys. Rev. Lett. 107 (10 ago. de 2011), pág. 100401. doi: 10.1103/PhysRevLett.107.100401. url: https://link.aps.org/doi/10.1103/PhysRevLett.107.100401.Katherine L. Brown, William J. Munro y Vivien M. Kendon. “Using Quantum Computers for Quantum Simulation”. En: Entropy 12.11 (nov. de 2010), págs. 2268-2307. issn: 1099-4300. doi: 10.3390/e12112268. url: http://dx.doi.org/10.3390/e12112268.Fernando Casas, Ander Murua y Mladen Nadinic. “Efficient computation of the Zassenhaus formula”. En: Computer Physics Communications 183.11 (2012), págs. 2386-2391. issn: 0010-4655. doi: https : / / doi . org/10.1016/j.cpc.2012.06.006. url: https://www.sciencedirect.com/science/article/pii/ S001046551200207X.Peter L. Knight Christopher C. Gerry. Introductory Quantum Optics. Cambridge: Cambridge University Press, 2005.Toby S. Cubitt, Ashley Montanaro y Stephen Piddock. “Universal quantum Hamiltonians”. En: Proceedings of the National Academy of Sciences 115.38 (2018), págs. 9497-9502. doi: 10.1073/pnas.1804949115. eprint: https://www.pnas.org/doi/pdf/10.1073/pnas.1804949115. url: https://www.pnas.org/doi/abs/10.1073/pnas.1804949115.Richard P. Feynman. “Simulating physics with computers”. En: International Journal of Theoretical Physics 21.6 (jun. de 1982), págs. 467-488. issn: 1572-9575. doi: 10.1007/BF02650179. url: https://doi.org/10.1007/BF02650179.I. M. Georgescu, S. Ashhab y Franco Nori. “Quantum simulation”. En: Rev. Mod. Phys. 86 (1 mar. de 2014), págs. 153-185. doi: 10.1103/RevModPhys.86.153. url: https://link.aps.org/doi/10.1103/RevModPhys.86.153.J. A. Gyamfi. An Introduction to the Holstein-Primakoff Transformation, with Applications in Magnetic Resonance. 2019. arXiv: 1907.07122 [cond-mat.other]. url: https://arxiv.org/abs/1907.07122.Xin-Yu Huang et al. Qubitization of Bosons. 2022. arXiv: 2105.12563 [quant-ph]. url: https://arxiv.org/abs/2105.12563.Ali Javadi-Abhari et al. Quantum computing with Qiskit. 2024. doi: 10.48550/arXiv.2405.08810. arXiv:2405.08810 [quant-ph].Seth Lloyd. “A Potentially Realizable Quantum Computer”. En: Science 261.5128 (1993), págs. 1569-1571. doi: 10.1126/science.261.5128.1569. eprint: https://www.science.org/doi/pdf/10.1126/science.261.5128.1569. url: https://www.science.org/doi/abs/10.1126/science.261.5128.1569.Michael A. Nielsen e Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010.Bo Peng et al. Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Cons truction, and Error Analysis. 2025. arXiv: 2307.06580 [quant-ph]. url: https://arxiv.org/abs/2307.06580.J.J. Sakurai y J. Napolitano. Modern Quantum Mechanics. Cambridge University Press, 2017. isbn: 9781108422413. url: https://books.google.com.co/books?id=010yDwAAQBAJ.Rohit Sarma Sarkar, Sabyasachi Chakraborty y Bibhas Adhikari. Scalable quantum circuits for exponential of Pauli strings and Hamiltonian simulations. 2024. arXiv: 2405.13605 [quant-ph]. url: https://arxiv.org/abs/2405.13605.S Somaroo et al. “Quantum Simulations on a Quantum Computer”. En: Physical Review Letters 82.26 (jun. de 1999). doi: 10.1103/PhysRevLett.82.5381. url: https://www.osti.gov/biblio/349372.R. Somma et al. “Simulating physical phenomena by quantum networks”. En: Phys. Rev. A 65 (4 abr. de 2002), pág. 042323. doi: 10.1103/PhysRevA.65.042323. url: https://link.aps.org/doi/10.1103/PhysRevA.65.042323.Masuo Suzuki. “Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems”. En: Communications in Mathematical Physics 51.2 (jun. de 1976), págs. 183-190. issn: 1432-0916. doi: 10.1007/BF01609348. url: https://doi.org/10.1007/BF01609348.James D. Whitfield, Jacob Biamonte y Alán Aspuru-Guzik. “Simulation of electronic structure Hamiltonians using quantum computers”. En: Molecular Physics 109.5 (mar. de 2011), págs. 735-750. issn: 1362-3028. doi:10.1080/00268976.2011.552441. url: http://dx.doi.org/10.1080/00268976.2011.552441.Leo Zhou y Dorit Aharonov. 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