DOI:
https://doi.org/10.14483/23448393.22292Published:
2025-03-31Issue:
Vol. 30 No. 1 (2025): January-AprilSection:
Education in EngineeringBibliometric Analysis and Overview of Matrix Product States in the Bose-Hubbard Model
Estudio bibliométrico y revisión del uso de estados producto de matrices en el modelo de Bose-Hubbard
Keywords:
matrix product states, density matrix renormalization group, strongly correlated systems, Bose-Hubbard model, tensor networks (en).Keywords:
estados producto de matrices, grupo de renormalización de matriz de densidad, sistemas fuertemente correlacionados, modelo de Bose-Hubbard, redes de tensor (es).Downloads
Abstract (en)
Context: Quantum many-body systems have been a prominent topic over the past two decades, underpinning advancements in superconductors, ultracold atoms, and quantum computing, among other fields. This bibliometric analysis explores key concepts, influential authors, and the
current significance of a powerful family of algorithms in computational physics, i.e., density matrix renormalization group (DMRG) algorithms. Special emphasis is placed on the use of tensor product states in developing classical simulations of quantum systems.
Method: This paper presents a literature review sourced from the SCOPUS database. It analyzes trends and approaches related to uncertainty in numerical developments for quantum many-body systems, with a focus on the Bose-Hubbard Model, in order to better understand the imposition of additional constraints to ensure the validity of the results.
Results: The increasing number of publications on this topic over the last decade indicates a growing interest in solutions for many-body quantum systems, driven by promising advances in superconductive materials, quantum computing, and other impactful areas.
Conclusions: This work explored essential foundational works to help beginners understand a well-established technique that aims to overcome the limitations of classical computing. The use of matrix product states in DMRG algorithms is gaining significant traction in various fields, including quantum computing, machine learning, and statistical mechanics, with the purpose of addressing the challenges related to quantum many-body systems.
Abstract (es)
Contexto: Los sistemas cuánticos de muchos cuerpos han sido un tema prominente durante las últimas dos décadas, sustentando avances en superconductores, átomos ultrafríos y computación cuántica. Este análisis bibliométrico explora conceptos clave, autores influyentes y la significancia actual de una poderosa familia de algoritmos en física computacional, i.e., los algoritmos del grupo de renormalización de matriz de densidad (DMRG). Se pone un énfasis especial en el uso de estados producto de tensor en el desarrollo de simulaciones clásicas de sistemas cuánticos.
Métodos: Este artículo presenta una revisión de la literatura obtenida de la base de datos de SCOPUS. Analiza tendencias y enfoques relacionados con el concepto de incertidumbre dentro del marco de desarrollos numéricos para sistemas cuánticos de muchos cuerpos, con énfasis en el modelo de Bose-Hubbard. Esto, con el objetivo de entender mejor la imposición de restricciones adicionales para asegurar la validez de los resultados.
Resultados: El número creciente de publicaciones sobre este tema en la última década indica un mayor interés en soluciones para sistemas cuánticos de muchos cuerpos, impulsadas por avances promisorios en materiales superconductores, computación cuántica y otras áreas de impacto.
Conclusiones: Este trabajo exploró los trabajos fundamentales para ayudar a los principiantes a comprender una técnica bien establecida que promete sobrepasar las limitaciones de la computación clásica. El uso de los estados de producto de matrices en los algoritmos DMRG está ganando una tracción significativa en numerosos campos, incluyendo la computación cuántica, el aprendizaje automático y la mecánica estadística, con el fin de abordar los desafíos asociados con los sistemas
cuánticos de muchos cuerpos.
References
W. Wang, “Linear limit continuation: Theory and an application to two-dimensional bose–einstein condensates,” Chaos Sol. Frac., vol. 182, p. 114735, 2024. https://doi.org/10. 1016/j.chaos.2024.114735
A. Jahn and J. Eisert, “Holographic tensor network models and quantum error correction: A topical review,” Quantum Sci. Technol., vol. 6, no. 3, p. 033002, 2021. https://doi.org/10.1088/2058-9565/ac0293
A. J. Ferris and D. Poulin, “Tensor networks and quantum error correction,” Phys. Rev. Lett., vol. 113, p. 030501, 2014. https://doi.org/10.1103/PhysRevLett.113.030501
A. S. Darmawan and D. Poulin, “Tensor-network simulations of the surface code under realistic noise,” Phys. Rev. Lett., vol. 119, p. 040502, 2017. https://doi.org/10.1103/PhysRevLett.119.040502
S. F. Caballero-Benítez, “Materia cuántica en cavidades de alta reflectancia (many-body CQED),” 2022. [Online]. Available: https://doi.org/10.48550/arXiv.2201.06641
H. A. Gersch and G. C. Knollman, “Quantum cell model for bosons,” Phys. Rev., vol. 129, pp. 959–967, Jan 1963. https://doi.org/10.1103/PhysRev.129.959
Y. Nakamura, Y. Takasu, J. Kobayashi, H. Asaka, Y. Fukushima, K. Inaba, M. Yamashita, and Y. Takahashi, “Experimental determination of bose-hubbard energies,” Phys. Rev. A, vol. 99, no. 3, p. 033609, Mar 2019. http://dx.doi.org/10.1103/PhysRevA.99.033609
G. Sangiovanni, A. Toschi, E. Koch, K. Held, M. Capone, C. Castellani, O. Gunnarsson, S.-K. Mo, J. W. Allen, H.-D. Kim, A. Sekiyama, A. Yamasaki, S. Suga, and P. Metcalf, “Static versus dynamical mean-field theory of mott antiferromagnets,” Phys. Rev. B, vol. 73, no. 20, May 2006. http://dx.doi.org/10.1103/PhysRevB.73.205121
G. Co’, “Introducing the random phase approximation theory,” Universe, vol. 9, no. 3, p. 141, 2023. https://doi.org/10.3390/universe9030141
R. T. Scalettar, “An introduction to the Hubbard Hamiltonian,” in Quantum materials: Experiments and theory, E. Pavarini, E. Koch, J. van den Brink, and G. Sawatzky, Eds. Jülich, Germany: Forschungszentrum Jülich, 2016, pp. 4.1–4.27. [Online]. Available: http://www.cond-mat.de/events/correl16
U. Schollwöck, “The density-matrix renormalization group,” Rev. Mod. Phys., vol. 77, pp. 259–315, Apr 2005. https://doi.org/10.1103/RevModPhys.77.259
M. Iskin, “Mean-field theory for the mott-insulator-paired-superfluid phase transition in the two-species bose-hubbard model,” Phys. Rev. A, vol. 82, p. 055601, 2010. https://doi.org/10.1103/PhysRevA.82.055601
G. Evenbly and G. Vidal, “Tensor network states and geometry,” J. Stat. Phys., vol. 145, no. 4, p. 891–918, Jun 2011. http://dx.doi.org/10.1007/s10955-011-0237-4
H. Tang, J. L. Simancas-García, J. Mai, M. Cheng, I. Iqbal, and L. Y. Por, “Overview of digital quantum simulator: Applications and comparison with latest methods,” SPIN, p. 2440004, 2024. https://doi.org/10.1142/S2010324724400046
M. Brenes, J. J. Mendoza-Arenas, A. Purkayastha, M. T. Mitchison, S. R. Clark, and J. Goold, “Tensor-network method to simulate strongly interacting quantum thermal machines,” Phys. Rev. X, vol. 10, p. 031040, 2020. https://doi.org/10.1103/PhysRevX.10.031040
J. J. Mendoza-Arenas, “Dynamical quantum phase transitions in the one-dimensional extended fermi–hubbard model,” J. Stat. Mech., vol. 2022, p. 043101, 2022. [Online]. Available: https://doi.org/10.1088/1742-5468/ac6031
J. Reslen, “Entanglement at the interplay between single- and many-bodyness,” J. Phys. A, vol. 56, p. 155302, 2023. https://doi.org/10.1088/1751-8121/acc291
D. Jaschke, M. L. Wall, and L. D. Carr, “Open source matrix product states: Opening ways to simulate entangled many-body quantum systems in one dimension,” Comput. Phys. Commun., vol. 225, pp. 59–91, 2018. https://doi.org/10.1016/j.cpc.2017.12.015
V. M. F. Verstraete and J. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Adv. Phys., vol. 57, no. 2, pp. 143–224, 2008. https://doi.org/10.1080/14789940801912366
M. Zheng, Y. Qiao, Y. Wang, J. Cao, and S. Chen, “Exact solution of the bose-hubbard model with unidirectional hopping,” Phys. Rev. Lett., vol. 132, p. 086502, Feb 2024. http://dx.doi.org/10.1103/PhysRevLett.132.086502
V. Colussi, F. Caleffi, C. Menotti, and A. Recati, “Quantum gutzwiller approach for the two-component bose-hubbard model,” SciPost Phys., vol. 12, no. 3, p. 111, Mar 2022. http://dx.doi.org/10.21468/SciPostPhys.12.3.111
G. Catarina and B. Murta, “Density-matrix renormalization group: a pedagogical introduction,” Eur. Phys. J. B, vol. 96, p. 111, Aug 2023. http://dx.doi.org/10.1140/epjb/s10051-023-00575-2
I. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys., vol. 86, no. 1, p. 153–185, Mar 2014. http://dx.doi.org/10.1103/RevModPhys.86.153
S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett., vol. 69, pp. 2863–2866, Nov 1992. https://doi.org/10.1103/PhysRevLett.69. 2863
A. Haller, “Matrix product states: A variational approach to strongly correlated systems,” Bachelor’s thesis, Johannes Gutenberg University Mainz, 2014. [Online]. Available: https: //www.tmqs.lu/images/haller_bsc.pdf
S. Keller, M. Dolfi, M. Troyer, and M. Reiher, “An efficient matrix product operator representation of the quantum chemical hamiltonian,” J. Chem. Phys., vol. 143, p. 244118, 2015. https://doi.org/10.1063/1.4939000
J. Cui, J. I. Cirac, and M. C. Bañuls, “Variational matrix product operators for the steady state of dissipative quantum systems,” Phys. Rev. Lett., vol. 114, p. 220601, 2015. https://doi.org/10.1103/PhysRevLett.114.220601
A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. Lovett, “Efficient non-markovian quantum dynamics using time-evolving matrix product operators,” Nat. Commun., vol. 9, p. 3322, 2018. https://doi.org/10.1038/s41467-018-05617-3
S. Moudgalya, N. Regnault, and B. A. Bernevig, “Entanglement of exact excited states of affleck-kennedy-lieb-tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis,” Phys. Rev. B, vol. 98, p. 235156, 2018. [Online]. Available: https://doi.org/10.1103/PhysRevB.98.235156
B. Pirvu, V. Murg, J. Cirac, and F. Verstraete, “Matrix product operator representations,” New J. Phys., vol. 12, p. 025012, 2010. https://doi.org/10.1088/1367-2630/12/2/ 025012
N. Nakatani and G. K.-L. Chan, “Efficient tree tensor network states (ttns) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm,” J. Chem. Phys., vol. 138, p. 134113, 2013. https://doi.org/10.1063/1.4798639
S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and O. Legeza, “Tensor product methods and entanglement optimization for ab initio quantum chemistry,” Int. J. Quantum Chem., vol. 115, no. 19, p. 1342 – 1391, 2015. https://doi.org/10.1002/qua.24898
D. Liu, S.-J. Ran, P. Wittek, C. Peng, R. B. Garcia, G. Su, and M. Lewenstein, “Machine learning by unitary tensor network of hierarchical tree structure,” New J. Phys., vol. 21, p. 073059, 2019. https://doi.org/10.1088/1367-2630/ab31ef
M. Niedermeier, M. Nairn, C. Flindt, and J. L. Lado, “Quantum computing topological invariants of two-dimensional quantum matter,” Phys. Rev. Res., vol. 6, p. 043288, 2024. https://doi.org/10.1103/PhysRevResearch.6.043288
A. Wulff, B. Chen, M. Steinberg, Y. Tang, M. Möller, and S. Feld, “Quantum computing and tensor networks for laminate design: A novel approach to stacking sequence retrieval,” Comput. Methods Appl. Mech. Eng., vol. 432, part A, p. 117380, 2024. https://doi.org/10.1016/j.cma.2024.117380
A. Termanova, A. Melnikov, E. Mamenchikov, N. Belokonev, S. Dolgov, A. Berezutskii, R. Ellerbrock, C. Mansell, and M. Perelshtein, “Tensor quantum programming,” New J. Phys., vol. 26, p. 123019, 2024. https://doi.org/10.1088/1367-2630/ad985b
A. Singh Bhatia and D. E. Bernal Neira, “Federated learning with tensor networks: a quantum ai framework for healthcare,” Mach. Learn. Sci. Technol., vol. 5, p. 045035, 2024. https://doi.org/10.1088/2632-2153/ad8c11
G. Evenbly, “Hyperinvariant tensor networks and holography,” Phys. Rev. Lett., vol. 119, p. 141602, 2017. https://doi.org/10.1103/PhysRevLett.119.141602
A. Peach and S. F. Ross, “Tensor network models of multiboundary wormholes,” Class. Quantum Gravity, vol. 34, no. 10, p. 105011, 2017. https://doi.org/10.1088/1361-6382/aa6b0f
A. Jahn, M. Gluza, F. Pastawski, and J. Eisert, “Holography and criticality in matchgate tensor networks,” Sci. Adv., vol. 5, no. 8, p. eaaw0092, 2019. https://doi.org/10.1126/sciadv.aaw0092
F. Verstraete, V. Murg, and J. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Adv. Phys., vol. 57, no. 2, p. 143 – 224, 2008. https://doi.org/10.1080/14789940801912366
J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, “Matrix product states and projected entangled pair states: Concepts, symmetries, theorems,” Rev. Mod. Phys., vol. 93, p. 045003, 2021. https://doi.org/10.1103/RevModPhys.93.045003
N. Schuch, D. Pérez-García, and I. Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states,” Phys. Rev. B Cond. Mat. Mater. Phys., vol. 84, p. 165139, 2011. https://doi.org/10.1103/PhysRevB.84.165139
R. Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Ann. Phys., vol. 349, pp. 117–158, 2014. https://doi.org/10.1016/j.aop.2014.06.013
J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. Cirac, “Classical simulation of infinite-size quantum lattice systems in two spatial dimensions,” Phys. Rev. Lett., vol. 101, p. 250602, 2008. https://doi.org/10.1103/PhysRevLett.101.250602
G. Vidal, “Class of quantum many-body states that can be efficiently simulated,” Phys. Rev. Lett., vol. 101, no. 11, 2008. https://doi.org/10.1103/PhysRevLett.101.110501
Y.-Y. Shi, L.-M. Duan, and G. Vidal, “Classical simulation of quantum many-body systems with a tree tensor network,” Phys. Rev. A, vol. 74, p. 022320, 2006. https://doi.org/10.1103/PhysRevA.74.022320
S. Holtz, T. Rohwedder, and R. Schneider, “The alternating linear scheme for tensor optimization in the tensor train format,” SIAM J. Sci. Comput., vol. 34, no. 2, pp. A683–A713, 2012. https://doi.org/10.1137/100818893
V. Murg, F. Verstraete, O. Legeza, and R. Noack, “Simulating strongly correlated quantum systems with tree tensor networks,” Phys. Rev. B, vol. 82, p. 205105, 2010. https://doi.org/10.1103/PhysRevB.82.205105
L. Tagliacozzo, G. Evenbly, and G. Vidal, “Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law,” Phys. Rev. B, vol. 80, p. 235127, 2009. https://doi.org/10.1103/PhysRevB.80.235127
M. L. Wall, A. Reilly, J. S. Van Dyke, C. Broholm, and P. Titum, “Quantum tensor network algorithms for evaluation of spectral functions on quantum computers,” Phys. Rev. B, vol. 110, p. 214402, 2024. https://doi.org/10.1103/PhysRevB.110.214402
R. Huang, H. Yi, and P. H. Li, “Variational quantum recurrent neural networks for multi-feature forecasting,” Int. J. Quantum Inf., vol. 22, no. 7, p. 2450029, 2024. https://doi.org/10.1142/S0219749924500291
How to Cite
APA
ACM
ACS
ABNT
Chicago
Harvard
IEEE
MLA
Turabian
Vancouver
Download Citation
License
Copyright (c) 2025 Juan Sebastián Gómez, Karen Cecilia Rodríguez
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
From the edition of the V23N3 of year 2018 forward, the Creative Commons License "Attribution-Non-Commercial - No Derivative Works " is changed to the following:
Attribution - Non-Commercial - Share the same: this license allows others to distribute, remix, retouch, and create from your work in a non-commercial way, as long as they give you credit and license their new creations under the same conditions.
2.jpg)
