Ground state energies of H2 using variational quantum circuits

ilustraciones, diagramas

Autores:
Cotrino Sandoval, Sergio Andrés
Tipo de recurso:
Fecha de publicación:
2024
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
eng
OAI Identifier:
oai:repositorio.unal.edu.co:unal/86333
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/86333
https://repositorio.unal.edu.co/
Palabra clave:
530 - Física::539 - Física moderna
540 - Química y ciencias afines::541 - Química física
QUIMICA CUANTICA
Quantum chemistry
quantum circuits
Variational Quantum Eigensolver
quantum machine learning
VQE
circuitos cuánticos
autosolucionador cuántico variacional
aprendizaje automático cuántico
PennyLane
computación cuántica
quantum computing
Rights
openAccess
License
Atribución-CompartirIgual 4.0 Internacional
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oai_identifier_str oai:repositorio.unal.edu.co:unal/86333
network_acronym_str UNACIONAL2
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repository_id_str
dc.title.eng.fl_str_mv Ground state energies of H2 using variational quantum circuits
dc.title.translated.spa.fl_str_mv Energías de estado fundamental de H2 usando circuitos cuánticos variacionales
title Ground state energies of H2 using variational quantum circuits
spellingShingle Ground state energies of H2 using variational quantum circuits
530 - Física::539 - Física moderna
540 - Química y ciencias afines::541 - Química física
QUIMICA CUANTICA
Quantum chemistry
quantum circuits
Variational Quantum Eigensolver
quantum machine learning
VQE
circuitos cuánticos
autosolucionador cuántico variacional
aprendizaje automático cuántico
PennyLane
computación cuántica
quantum computing
title_short Ground state energies of H2 using variational quantum circuits
title_full Ground state energies of H2 using variational quantum circuits
title_fullStr Ground state energies of H2 using variational quantum circuits
title_full_unstemmed Ground state energies of H2 using variational quantum circuits
title_sort Ground state energies of H2 using variational quantum circuits
dc.creator.fl_str_mv Cotrino Sandoval, Sergio Andrés
dc.contributor.advisor.none.fl_str_mv Viviescas, Carlos
dc.contributor.author.none.fl_str_mv Cotrino Sandoval, Sergio Andrés
dc.contributor.researchgroup.spa.fl_str_mv Caos y Complejidad
dc.subject.ddc.spa.fl_str_mv 530 - Física::539 - Física moderna
540 - Química y ciencias afines::541 - Química física
topic 530 - Física::539 - Física moderna
540 - Química y ciencias afines::541 - Química física
QUIMICA CUANTICA
Quantum chemistry
quantum circuits
Variational Quantum Eigensolver
quantum machine learning
VQE
circuitos cuánticos
autosolucionador cuántico variacional
aprendizaje automático cuántico
PennyLane
computación cuántica
quantum computing
dc.subject.lemb.none.fl_str_mv QUIMICA CUANTICA
Quantum chemistry
dc.subject.proposal.eng.fl_str_mv quantum circuits
Variational Quantum Eigensolver
quantum machine learning
VQE
dc.subject.proposal.spa.fl_str_mv circuitos cuánticos
autosolucionador cuántico variacional
aprendizaje automático cuántico
dc.subject.proposal.none.fl_str_mv PennyLane
dc.subject.wikidata.none.fl_str_mv computación cuántica
quantum computing
description ilustraciones, diagramas
publishDate 2024
dc.date.accessioned.none.fl_str_mv 2024-06-28T20:12:58Z
dc.date.available.none.fl_str_mv 2024-06-28T20:12:58Z
dc.date.issued.none.fl_str_mv 2024
dc.type.spa.fl_str_mv Trabajo de grado - Maestría
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/masterThesis
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.content.spa.fl_str_mv Text
dc.type.redcol.spa.fl_str_mv http://purl.org/redcol/resource_type/TM
status_str acceptedVersion
dc.identifier.uri.none.fl_str_mv https://repositorio.unal.edu.co/handle/unal/86333
dc.identifier.instname.spa.fl_str_mv Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv https://repositorio.unal.edu.co/
url https://repositorio.unal.edu.co/handle/unal/86333
https://repositorio.unal.edu.co/
identifier_str_mv Universidad Nacional de Colombia
Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv eng
language eng
dc.relation.references.spa.fl_str_mv The quantum state of affairs. Nature Physics, 19(5):605–605, May 2023.
A. Anand, P. Schleich, S. Alperin-Lea, P. W. Jensen, S. Sim, M. Dı́az-Tinoco, J. S. Kottmann, M. Degroote, A. F. Izmaylov, and A. Aspuru-Guzik. A quantum computing view on unitary coupled cluster theory. Chemical Society Reviews, 2022.
G.-L. R. Anselmetti, D. Wierichs, C. Gogolin, and R. M. Parrish. Local, expressive, quantum-number-preserving vqe ansätze for fermionic systems. New Journal of Physics, 23(11):113010, 2021.
A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles. Operator sampling for shot- frugal optimization in variational algorithms. arXiv preprint arXiv:2004.06252, 2020.
J. M. Arrazola, V. Bergholm, K. Brádler, T. R. Bromley, M. J. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. G. Helt, et al. Quantum circuits with many photons on a programmable nanophotonic chip. Nature, 591(7848):54–60, 2021.
J. M. Arrazola, O. Di Matteo, N. Quesada, S. Jahangiri, A. Delgado, and N. Killoran. Universal quantum circuits for quantum chemistry. Quantum, 6:742, 2022.
J. M. Arrazola, S. Jahangiri, A. Delgado, J. Ceroni, J. Izaac, A. Száva, U. Azad, R. A. Lang, Z. Niu, O. Di Matteo, et al. Differentiable quantum computational chemistry with pennylane. arXiv preprint arXiv:2111.09967, 2021.
F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boi- xo, F. G. Brandao, D. A. Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510, 2019.
R. J. Bartlett and M. Musial. Coupled-cluster theory in quantum chemistry. Reviews of Modern Physics, 79(1):291, 2007.
K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. De- groote, H. Heimonen, J. S. Kottmann, T. Menke, et al. Noisy intermediate-scale quan- tum algorithms. Reviews of Modern Physics, 94(1):015004, 2022.
X. Bonet-Monroig, H. Wang, D. Vermetten, B. Senjean, C. Moussa, T. Bäck, V. Dunjko, and T. E. O’Brien. Performance comparison of optimization methods on variational quantum algorithms. Physical Review A, 107(3):032407, 2023.
S. Buchholz, D. Golden, and C. Brown. A business leader’s guide to quantum techno- logy — www2.deloitte.com. https://www2.deloitte.com/us/en/insights/topics/ innovation/quantum-computing-business-applications.html, 2021. [Accessed 21- Jul-2023].
Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. Sawaya, et al. Quantum chemistry in the age of quantum computing. Chemical reviews, 119(19):10856–10915, 2019.
M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3(9):625–644, 2021.
M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature communications, 12(1):1791, 2021.
A. Cervera-Lierta, J. S. Kottmann, and A. Aspuru-Guzik. Meta-variational quantum eigensolver: Learning energy profiles of parameterized hamiltonians for quantum simu- lation. PRX Quantum, 2(2):020329, 2021.
J. Charry, M. T. d. N. Varella, and A. Reyes. Binding matter with antimatter: the covalent positron bond. Angewandte Chemie International Edition, 57(29):8859–8864, 2018.
J. Chen, C. Wolfe, Z. Li, and A. Kyrillidis. Demon: improved neural network trai- ning with momentum decay. In ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 3958–3962. IEEE, 2022.
B. Choy and D. J. Wales. Molecular energy landscapes of hardware-efficient ansätze in quantum computing. Journal of Chemical Theory and Computation, 19(4):1197–1206, 2023.
D. Cremer. Møller–plesset perturbation theory: from small molecule methods to methods for thousands of atoms. Wiley Interdisciplinary Reviews: Computational Mo- lecular Science, 1(4):509–530, 2011.
G. E. Crooks. Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. arXiv preprint arXiv:1905.13311, 2019.
A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta. Validating quantum computers using randomized model circuits. Physical Review A, 100(3):032328, 2019.
A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller. Practical quantum advantage in quantum simulation. Nature, 607(7920):667–676, 2022.
D. P. DiVincenzo. The physical implementation of quantum computation. Fortschritte der Physik: Progress of Physics, 48(9-11):771–783, 2000.
Y. Du, T. Huang, S. You, M.-H. Hsieh, and D. Tao. Quantum circuit architecture search for variational quantum algorithms. npj Quantum Information, 8(1):62, 2022.
J. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of machine learning research, 12(7), 2011.
V. Dunjko and H. J. Briegel. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics, 81(7):074001, 2018.
D. A. Fedorov, B. Peng, N. Govind, and Y. Alexeev. Vqe method: a short survey and recent developments. Materials Theory, 6(1):1–21, 2022.
R. P. Feynman et al. Simulating physics with computers. Int. j. Theor. phys, 21(6/7), 1982.
P. Friederich, M. Krenn, I. Tamblyn, and A. Aspuru-Guzik. Scientific intuition inspired by machine learning-generated hypotheses. Machine Learning: Science and Technology, 2(2):025027, 2021.
A. Graves. Generating sequences with recurrent neural networks. arXiv preprint ar- Xiv:1308.0850, 2013.
H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature communi- cations, 10(1):3007, 2019.
G. Herzberg. The dissociation energy of the hydrogen molecule. Journal of molecular spectroscopy, 33(1):147–168, 1970.
J. R. Johansson, P. D. Nation, and F. Nori. Qutip: An open-source python frame- work for the dynamics of open quantum systems. Computer Physics Communications, 183(8):1760–1772, 2012.
A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature, 549(7671):242–246, 2017.
Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, et al. Evidence for the utility of quantum computing before fault tolerance. Nature, 618(7965):500–505, 2023.
D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver. Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics, 11:369–395, 2020.
M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt. Artificial intelligence and machine learning for quantum technologies. Physical Review A, 107(1):010101, 2023.
J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles. An adaptive optimizer for measurement-frugal variational algorithms. Quantum, 4:263, 2020.
L. Leone, S. F. Oliviero, L. Cincio, and M. Cerezo. On the practical usefulness of the hardware efficient ansatz. arXiv preprint arXiv:2211.01477, 2022.
J. Li, B. A. Jones, and S. Kais. Toward perturbation theory methods on a quantum computer. Science Advances, 9(19):eadg4576, 2023.
S. Lloyd, M. Schuld, A. Ijaz, J. Izaac, and N. Killoran. Quantum embeddings for machine learning. arXiv preprint arXiv:2001.03622, 2020.
I. Loshchilov and F. Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017.
R. T. Marler and J. S. Arora. Survey of multi-objective optimization methods for engineering. Structural and multidisciplinary optimization, 26:369–395, 2004.
S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan. Quantum compu- tational chemistry. Reviews of Modern Physics, 92(1):015003, 2020.
A. McCaskey, E. Dumitrescu, D. Liakh, and T. Humble. Hybrid programming for near- term quantum computing systems. In 2018 IEEE international conference on rebooting computing (ICRC), pages 1–12. IEEE, 2018.
M. Medvidović and G. Carleo. Classical variational simulation of the quantum appro- ximate optimization algorithm. npj Quantum Information, 7(1):101, 2021.
A. A. Melnikov, H. Poulsen Nautrup, M. Krenn, V. Dunjko, M. Tiersch, A. Zeilinger, and H. J. Briegel. Active learning machine learns to create new quantum experiments. Proceedings of the National Academy of Sciences, 115(6):1221–1226, 2018.
A. Montanaro. Quantum algorithms: an overview. npj Quantum Information, 2(1):1–8, 2016.
A. Morvan, B. Villalonga, X. Mi, S. Mandrà, A. Bengtsson, P. Klimov, Z. Chen, S. Hong, C. Erickson, I. Drozdov, et al. Phase transition in random circuit sampling. arXiv e- prints, pages arXiv–2304, 2023.
M. Motta and J. E. Rice. Emerging quantum computing algorithms for quantum che- mistry. Wiley Interdisciplinary Reviews: Computational Molecular Science, 12(3):e1580, 2022.
M. A. Nielsen and I. Chuang. Quantum computation and quantum information. Ame- rican Association of Physics Teachers, 2002.
M. Ostaszewski, E. Grant, and M. Benedetti. Structure optimization for parameterized quantum circuits. Quantum, 5:391, 2021.
M. Ostaszewski, L. M. Trenkwalder, W. Masarczyk, E. Scerri, and V. Dunjko. Reinfor- cement learning for optimization of variational quantum circuit architectures. Advances in Neural Information Processing Systems, 34:18182–18194, 2021.
F. Pavošević and S. Hammes-Schiffer. Multicomponent equation-of-motion coupled cluster singles and doubles: Theory and calculation of excitation energies for positronium hydride. The Journal of Chemical Physics, 150(16), 2019.
F. Pavosevic and S. Hammes-Schiffer. Multicomponent unitary coupled cluster and equation-of-motion for quantum computation. Journal of Chemical Theory and Compu- tation, 17(6):3252–3258, 2021.
A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru- Guzik, and J. L. O’brien. A variational eigenvalue solver on a photonic quantum pro- cessor. Nature communications, 5(1):4213, 2014.
K. Phalak and S. Ghosh. Shot optimization in quantum machine learning architectures to accelerate training. IEEE Access, 2023.
I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Pod- lesnic, M. Meth, V. Negnevitsky, M. Stadler, et al. Compact ion-trap quantum compu- ting demonstrator. PRX Quantum, 2(2):020343, 2021.
J. Preskill. Quantum computing in the nisq era and beyond. Quantum, 2:79, 2018.
B. P. Pritchard, D. Altarawy, B. Didier, T. D. Gibson, and T. L. Windus. New basis set exchange: An open, up-to-date resource for the molecular sciences community. Journal of chemical information and modeling, 59(11):4814–4820, 2019.
J. Ruane, A. McAfee, and W. D. Oliver. Quantum Computing for Business Leaders — hbr.org. https://hbr.org/2022/01/quantum-computing-for-business-leaders, 2022. [Accessed 21-Jul-2023].
D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. Technical report, California Univ San Diego La Jolla Inst for Cognitive Science, 1985.
M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99(3):032331, 2019.
M. Schuld and N. Killoran. Quantum machine learning in feature hilbert spaces. Physical review letters, 122(4):040504, 2019.
M. Schuld and F. Petruccione. Machine learning with quantum computers. Springer, 2021.
P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994.
S. Sim, P. D. Johnson, and A. Aspuru-Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Advanced Quantum Technologies, 2(12):1900070, 2019.
I. O. Sokolov, P. K. Barkoutsos, P. J. Ollitrault, D. Greenberg, J. Rice, M. Pistoia, and I. Tavernelli. Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equiva- lents? The Journal of chemical physics, 152(12):124107, 2020.
J. C. Spall. Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Transactions on aerospace and electronic systems, 34(3):817–823, 1998.
J. C. Spall. An overview of the simultaneous perturbation method for efficient optimi- zation. Johns Hopkins apl technical digest, 19(4):482–492, 1998.
J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al. The variational quantum eigensolver: a review of methods and best practices. Physics Reports, 986:1–128, 2022.
A. Wack, H. Paik, A. Javadi-Abhari, P. Jurcevic, I. Faro, J. M. Gambetta, and B. R. Johnson. Quality, speed, and scale: three key attributes to measure the performance of near-term quantum computers. arXiv preprint arXiv:2110.14108, 2021.
D. Wecker, M. B. Hastings, and M. Troyer. Progress towards practical quantum varia- tional algorithms. Physical Review A, 92(4):042303, 2015.
M. Wiedmann, M. Hölle, M. Periyasamy, N. Meyer, C. Ufrecht, D. D. Scherer, A. Plinge, and C. Mutschler. An empirical comparison of optimizers for quantum machine learning with spsa-based gradients. arXiv preprint arXiv:2305.00224, 2023.
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dc.publisher.faculty.spa.fl_str_mv Facultad de Ciencias
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spelling Atribución-CompartirIgual 4.0 Internacionalhttp://creativecommons.org/licenses/by-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Viviescas, Carlosf5a3bb5922f522614153d8dfb8996953600Cotrino Sandoval, Sergio Andrés076fe3731bcae44cb1f85dd619503b45Caos y Complejidad2024-06-28T20:12:58Z2024-06-28T20:12:58Z2024https://repositorio.unal.edu.co/handle/unal/86333Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, diagramasConsidering the current limitations on size and reliability of Noisy Intermediate Quantum Scale devices, Variational Quantum Circuits offer a way to get useful results from quantum computation. On top of that, Machine Learning methods using quantum data offer a way to process the information, but also use it to learn and extract useful information. Meta- Variational Quantum Eigensolver (meta-VQE) was used to learn the ground energy profile of a molecule using a set of training points. By training an ansatz circuit using a non-linear Gaussian encoding of each circuit parameter and setting the interatomic distance as a free parameter, it was possible to find a good approximation of the ground energy of the system for any interatomic distance within a certain region. This method also has the advantage to produce good starting parameters to train using standard VQE, and obtain even better results (opt-meta-VQE). Meta-VQE was implemented in an analytic noise-free simulation and a 10000 shots-based simulation using the software framework for quantum computing PennyLane. In the analytic simulation, it was possible to accurately describe the potential energy surface of an H2 molecule within chemical accuracy, using a hardware inspired ansatz and the ADAM optimizer. With the 10000 shots-based simulation, the method is capable to approximate the energy profile, but in general its performance is not as good as the analytical approach due to the variability on the samples obtained. Meta-VQE provides a novel way to extract and produce information by learning using quantum data from variational circuits.Teniendo en cuenta las limitaciones actuales de tamaño y confiabilidad de los dispositi- vos de escala cuántica intermedia ruidosa, los circuitos cuánticos variacionales ofrecen una forma de obtener resultados útiles de la computación cuántica. Además de eso, los méto- dos de aprendizaje automático que utilizan datos cuánticos ofrecen una forma de procesar la información, pero también de usarla para aprender y extraer información útil. Se usó el metodo de meta-autosolucionador cuántico variacional (meta-VQE, por sus siglas en inglés) para aprender el perfil de energı́a fundamental de una molécula usando un conjunto de pun- tos de entrenamiento. Al entrenar un circuito usando una codificación gaussiana no lineal de cada parámetro del circuito y estableciendo la distancia interatómica como un paráme- tro libre, fue posible encontrar una buena aproximación de la energı́a mı́nima del sistema para cualquier distancia interatómica dentro de una región determinada. Este método tam- bién tiene la ventaja de producir buenos parámetros de partida para entrenar usando VQE estándar y obtener resultados aún mejores (opt-meta-VQE). Meta-VQE se implementó en una simulación analı́tica sin ruido y una simulación basada en 10000 muestras utilizando el software para computación cuántica PennyLane. En la simulación analı́tica, fue posible describir con precisión la superficie de energı́a potencial de una molécula H2 con precisión quı́mica, utilizando un ansatz inspirado en hardware y el optimizador ADAM. Con la si- mulación basada en 10000 muestras, el método es capaz de aproximar el perfil de energı́a, pero en general no funciona tan bien como el enfoque analı́tico debido a la variabilidad de las muestras obtenidas. Meta-VQE proporciona una forma novedosa de extraer y producir información mediante el aprendizaje utilizando datos cuánticos de circuitos variacionales.MaestríaMagíster en Ciencias - FísicaQuantum Computingxv, 71 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - FísicaFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá530 - Física::539 - Física moderna540 - Química y ciencias afines::541 - Química físicaQUIMICA CUANTICAQuantum chemistryquantum circuitsVariational Quantum Eigensolverquantum machine learningVQEcircuitos cuánticosautosolucionador cuántico variacionalaprendizaje automático cuánticoPennyLanecomputación cuánticaquantum computingGround state energies of H2 using variational quantum circuitsEnergías de estado fundamental de H2 usando circuitos cuánticos variacionalesTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMThe quantum state of affairs. Nature Physics, 19(5):605–605, May 2023.A. Anand, P. Schleich, S. Alperin-Lea, P. W. Jensen, S. Sim, M. Dı́az-Tinoco, J. S. Kottmann, M. Degroote, A. F. Izmaylov, and A. Aspuru-Guzik. A quantum computing view on unitary coupled cluster theory. Chemical Society Reviews, 2022.G.-L. R. Anselmetti, D. Wierichs, C. Gogolin, and R. M. Parrish. Local, expressive, quantum-number-preserving vqe ansätze for fermionic systems. New Journal of Physics, 23(11):113010, 2021.A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles. Operator sampling for shot- frugal optimization in variational algorithms. arXiv preprint arXiv:2004.06252, 2020.J. M. Arrazola, V. Bergholm, K. Brádler, T. R. Bromley, M. J. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. G. Helt, et al. Quantum circuits with many photons on a programmable nanophotonic chip. Nature, 591(7848):54–60, 2021.J. M. Arrazola, O. Di Matteo, N. Quesada, S. Jahangiri, A. Delgado, and N. Killoran. Universal quantum circuits for quantum chemistry. Quantum, 6:742, 2022.J. M. Arrazola, S. Jahangiri, A. Delgado, J. Ceroni, J. Izaac, A. Száva, U. Azad, R. A. Lang, Z. Niu, O. Di Matteo, et al. Differentiable quantum computational chemistry with pennylane. arXiv preprint arXiv:2111.09967, 2021.F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boi- xo, F. G. Brandao, D. A. Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510, 2019.R. J. Bartlett and M. Musial. Coupled-cluster theory in quantum chemistry. Reviews of Modern Physics, 79(1):291, 2007.K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. De- groote, H. Heimonen, J. S. Kottmann, T. Menke, et al. Noisy intermediate-scale quan- tum algorithms. Reviews of Modern Physics, 94(1):015004, 2022.X. Bonet-Monroig, H. Wang, D. Vermetten, B. Senjean, C. Moussa, T. Bäck, V. Dunjko, and T. E. O’Brien. Performance comparison of optimization methods on variational quantum algorithms. Physical Review A, 107(3):032407, 2023.S. Buchholz, D. Golden, and C. Brown. A business leader’s guide to quantum techno- logy — www2.deloitte.com. https://www2.deloitte.com/us/en/insights/topics/ innovation/quantum-computing-business-applications.html, 2021. [Accessed 21- Jul-2023].Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. Sawaya, et al. Quantum chemistry in the age of quantum computing. Chemical reviews, 119(19):10856–10915, 2019.M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3(9):625–644, 2021.M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature communications, 12(1):1791, 2021.A. Cervera-Lierta, J. S. Kottmann, and A. Aspuru-Guzik. Meta-variational quantum eigensolver: Learning energy profiles of parameterized hamiltonians for quantum simu- lation. PRX Quantum, 2(2):020329, 2021.J. Charry, M. T. d. N. Varella, and A. Reyes. Binding matter with antimatter: the covalent positron bond. Angewandte Chemie International Edition, 57(29):8859–8864, 2018.J. Chen, C. Wolfe, Z. Li, and A. Kyrillidis. Demon: improved neural network trai- ning with momentum decay. In ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 3958–3962. IEEE, 2022.B. Choy and D. J. Wales. Molecular energy landscapes of hardware-efficient ansätze in quantum computing. Journal of Chemical Theory and Computation, 19(4):1197–1206, 2023.D. Cremer. Møller–plesset perturbation theory: from small molecule methods to methods for thousands of atoms. Wiley Interdisciplinary Reviews: Computational Mo- lecular Science, 1(4):509–530, 2011.G. E. Crooks. Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. arXiv preprint arXiv:1905.13311, 2019.A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta. Validating quantum computers using randomized model circuits. Physical Review A, 100(3):032328, 2019.A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller. Practical quantum advantage in quantum simulation. Nature, 607(7920):667–676, 2022.D. P. DiVincenzo. The physical implementation of quantum computation. Fortschritte der Physik: Progress of Physics, 48(9-11):771–783, 2000.Y. Du, T. Huang, S. You, M.-H. Hsieh, and D. Tao. Quantum circuit architecture search for variational quantum algorithms. npj Quantum Information, 8(1):62, 2022.J. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of machine learning research, 12(7), 2011.V. Dunjko and H. J. Briegel. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics, 81(7):074001, 2018.D. A. Fedorov, B. Peng, N. Govind, and Y. Alexeev. Vqe method: a short survey and recent developments. Materials Theory, 6(1):1–21, 2022.R. P. Feynman et al. Simulating physics with computers. Int. j. Theor. phys, 21(6/7), 1982.P. Friederich, M. Krenn, I. Tamblyn, and A. Aspuru-Guzik. Scientific intuition inspired by machine learning-generated hypotheses. Machine Learning: Science and Technology, 2(2):025027, 2021.A. Graves. Generating sequences with recurrent neural networks. arXiv preprint ar- Xiv:1308.0850, 2013.H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature communi- cations, 10(1):3007, 2019.G. Herzberg. The dissociation energy of the hydrogen molecule. Journal of molecular spectroscopy, 33(1):147–168, 1970.J. R. Johansson, P. D. Nation, and F. Nori. Qutip: An open-source python frame- work for the dynamics of open quantum systems. Computer Physics Communications, 183(8):1760–1772, 2012.A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature, 549(7671):242–246, 2017.Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, et al. Evidence for the utility of quantum computing before fault tolerance. Nature, 618(7965):500–505, 2023.D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver. Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics, 11:369–395, 2020.M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt. Artificial intelligence and machine learning for quantum technologies. Physical Review A, 107(1):010101, 2023.J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles. An adaptive optimizer for measurement-frugal variational algorithms. Quantum, 4:263, 2020.L. Leone, S. F. Oliviero, L. Cincio, and M. Cerezo. On the practical usefulness of the hardware efficient ansatz. arXiv preprint arXiv:2211.01477, 2022.J. Li, B. A. Jones, and S. Kais. Toward perturbation theory methods on a quantum computer. Science Advances, 9(19):eadg4576, 2023.S. Lloyd, M. Schuld, A. Ijaz, J. Izaac, and N. Killoran. Quantum embeddings for machine learning. arXiv preprint arXiv:2001.03622, 2020.I. Loshchilov and F. Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017.R. T. Marler and J. S. Arora. Survey of multi-objective optimization methods for engineering. Structural and multidisciplinary optimization, 26:369–395, 2004.S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan. Quantum compu- tational chemistry. Reviews of Modern Physics, 92(1):015003, 2020.A. McCaskey, E. Dumitrescu, D. Liakh, and T. Humble. Hybrid programming for near- term quantum computing systems. In 2018 IEEE international conference on rebooting computing (ICRC), pages 1–12. IEEE, 2018.M. Medvidović and G. Carleo. Classical variational simulation of the quantum appro- ximate optimization algorithm. npj Quantum Information, 7(1):101, 2021.A. A. Melnikov, H. Poulsen Nautrup, M. Krenn, V. Dunjko, M. Tiersch, A. Zeilinger, and H. J. Briegel. Active learning machine learns to create new quantum experiments. Proceedings of the National Academy of Sciences, 115(6):1221–1226, 2018.A. Montanaro. Quantum algorithms: an overview. npj Quantum Information, 2(1):1–8, 2016.A. Morvan, B. Villalonga, X. Mi, S. Mandrà, A. Bengtsson, P. Klimov, Z. Chen, S. Hong, C. Erickson, I. Drozdov, et al. Phase transition in random circuit sampling. arXiv e- prints, pages arXiv–2304, 2023.M. Motta and J. E. Rice. Emerging quantum computing algorithms for quantum che- mistry. Wiley Interdisciplinary Reviews: Computational Molecular Science, 12(3):e1580, 2022.M. A. Nielsen and I. Chuang. Quantum computation and quantum information. Ame- rican Association of Physics Teachers, 2002.M. Ostaszewski, E. Grant, and M. Benedetti. Structure optimization for parameterized quantum circuits. Quantum, 5:391, 2021.M. Ostaszewski, L. M. Trenkwalder, W. Masarczyk, E. Scerri, and V. Dunjko. Reinfor- cement learning for optimization of variational quantum circuit architectures. Advances in Neural Information Processing Systems, 34:18182–18194, 2021.F. Pavošević and S. Hammes-Schiffer. Multicomponent equation-of-motion coupled cluster singles and doubles: Theory and calculation of excitation energies for positronium hydride. The Journal of Chemical Physics, 150(16), 2019.F. Pavosevic and S. Hammes-Schiffer. Multicomponent unitary coupled cluster and equation-of-motion for quantum computation. Journal of Chemical Theory and Compu- tation, 17(6):3252–3258, 2021.A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru- Guzik, and J. L. O’brien. A variational eigenvalue solver on a photonic quantum pro- cessor. Nature communications, 5(1):4213, 2014.K. Phalak and S. Ghosh. Shot optimization in quantum machine learning architectures to accelerate training. IEEE Access, 2023.I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Pod- lesnic, M. Meth, V. Negnevitsky, M. Stadler, et al. Compact ion-trap quantum compu- ting demonstrator. PRX Quantum, 2(2):020343, 2021.J. Preskill. Quantum computing in the nisq era and beyond. Quantum, 2:79, 2018.B. P. Pritchard, D. Altarawy, B. Didier, T. D. Gibson, and T. L. Windus. New basis set exchange: An open, up-to-date resource for the molecular sciences community. Journal of chemical information and modeling, 59(11):4814–4820, 2019.J. Ruane, A. McAfee, and W. D. Oliver. Quantum Computing for Business Leaders — hbr.org. https://hbr.org/2022/01/quantum-computing-for-business-leaders, 2022. [Accessed 21-Jul-2023].D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. Technical report, California Univ San Diego La Jolla Inst for Cognitive Science, 1985.M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99(3):032331, 2019.M. Schuld and N. Killoran. Quantum machine learning in feature hilbert spaces. Physical review letters, 122(4):040504, 2019.M. Schuld and F. Petruccione. Machine learning with quantum computers. Springer, 2021.P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994.S. Sim, P. D. Johnson, and A. Aspuru-Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Advanced Quantum Technologies, 2(12):1900070, 2019.I. O. Sokolov, P. K. Barkoutsos, P. J. Ollitrault, D. Greenberg, J. Rice, M. Pistoia, and I. Tavernelli. Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equiva- lents? The Journal of chemical physics, 152(12):124107, 2020.J. C. Spall. Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Transactions on aerospace and electronic systems, 34(3):817–823, 1998.J. C. Spall. An overview of the simultaneous perturbation method for efficient optimi- zation. Johns Hopkins apl technical digest, 19(4):482–492, 1998.J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al. The variational quantum eigensolver: a review of methods and best practices. Physics Reports, 986:1–128, 2022.A. Wack, H. Paik, A. Javadi-Abhari, P. Jurcevic, I. Faro, J. M. Gambetta, and B. R. Johnson. Quality, speed, and scale: three key attributes to measure the performance of near-term quantum computers. arXiv preprint arXiv:2110.14108, 2021.D. Wecker, M. B. Hastings, and M. Troyer. Progress towards practical quantum varia- tional algorithms. Physical Review A, 92(4):042303, 2015.M. Wiedmann, M. Hölle, M. Periyasamy, N. Meyer, C. Ufrecht, D. D. Scherer, A. Plinge, and C. Mutschler. An empirical comparison of optimizers for quantum machine learning with spsa-based gradients. arXiv preprint arXiv:2305.00224, 2023.Público generalLICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/86333/3/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD53ORIGINALscotrino_master_thesis_june_2024.pdfscotrino_master_thesis_june_2024.pdfTesis de Maestría en Ciencias -Físicaapplication/pdf1361573https://repositorio.unal.edu.co/bitstream/unal/86333/4/scotrino_master_thesis_june_2024.pdf9c08168c3b4853e5658ff3f50779b8c0MD54THUMBNAILscotrino_master_thesis_june_2024.pdf.jpgscotrino_master_thesis_june_2024.pdf.jpgGenerated Thumbnailimage/jpeg4272https://repositorio.unal.edu.co/bitstream/unal/86333/5/scotrino_master_thesis_june_2024.pdf.jpg16c604de8b1c935271aa46a98dc1d8b8MD55unal/86333oai:repositorio.unal.edu.co:unal/863332024-06-28 23:05:22.266Repositorio Institucional Universidad Nacional de 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