Quantum error correction and local topological order from Quantum Double model
This thesis presents a detailed analysis of the quantum double model and its twisted version, focusing on their properties as quantum error-correcting codes. In the case of Kitaev's quantum double model for finite Abelian groups, the error correction process is explicitly described. The number...
- Autores:
-
Romero Fonseca, Diego Arturo
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2025
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/76343
- Acceso en línea:
- https://hdl.handle.net/1992/76343
- Palabra clave:
- Quantum Double Model
Quantum codes
Lattice
Error-correction
Ground state
Topological order
Matemáticas
- Rights
- openAccess
- License
- Attribution-NonCommercial-NoDerivatives 4.0 International
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Quantum error correction and local topological order from Quantum Double model |
| title |
Quantum error correction and local topological order from Quantum Double model |
| spellingShingle |
Quantum error correction and local topological order from Quantum Double model Quantum Double Model Quantum codes Lattice Error-correction Ground state Topological order Matemáticas |
| title_short |
Quantum error correction and local topological order from Quantum Double model |
| title_full |
Quantum error correction and local topological order from Quantum Double model |
| title_fullStr |
Quantum error correction and local topological order from Quantum Double model |
| title_full_unstemmed |
Quantum error correction and local topological order from Quantum Double model |
| title_sort |
Quantum error correction and local topological order from Quantum Double model |
| dc.creator.fl_str_mv |
Romero Fonseca, Diego Arturo |
| dc.contributor.advisor.none.fl_str_mv |
Galindo Martínez, Cesar Neyit Cui, Shawn Xingshan |
| dc.contributor.author.none.fl_str_mv |
Romero Fonseca, Diego Arturo |
| dc.contributor.jury.none.fl_str_mv |
Jones, Corey Uribe Jongbloed, Bernardo Ángel Cárdenas, Jairo Andrés |
| dc.subject.keyword.eng.fl_str_mv |
Quantum Double Model Quantum codes Lattice Error-correction Ground state Topological order |
| topic |
Quantum Double Model Quantum codes Lattice Error-correction Ground state Topological order Matemáticas |
| dc.subject.themes.spa.fl_str_mv |
Matemáticas |
| description |
This thesis presents a detailed analysis of the quantum double model and its twisted version, focusing on their properties as quantum error-correcting codes. In the case of Kitaev's quantum double model for finite Abelian groups, the error correction process is explicitly described. The number of correctable errors depends on the lattice and the topology of the underlying surface. Although there exists a theoretical maximum number of errors that can be corrected, it is proven that correcting this number of errors is, in general, an NP-complete problem. As an alternative, a polynomial-time correction algorithm is proposed that corrects a number of errors below the theoretical maximum. In regard to the twisted quantum double model, it is studied within the framework of Local Topological Order (LTO). Originally formulated for square lattices, the definition of LTO is extended here to arbitrary two-dimensional lattices, enabling an explicit characterization of the ground states space through invariant spaces of monomial representations. By reformulating the LTO conditions to include such general lattices, it is proven that the twisted model satisfies all four axioms of LTO on any 2D lattice. As a result, the ground states space of the model is shown to define a quantum error-correcting code, highlighting the role of topological order in fault-tolerant quantum computation. |
| publishDate |
2025 |
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2025-06-19T15:24:57Z |
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2025-06-19T15:24:57Z |
| dc.date.issued.none.fl_str_mv |
2025-05-28 |
| dc.type.none.fl_str_mv |
Trabajo de grado - Doctorado |
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info:eu-repo/semantics/doctoralThesis |
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info:eu-repo/semantics/acceptedVersion |
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http://purl.org/coar/resource_type/c_db06 |
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Text |
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https://purl.org/redcol/resource_type/TD |
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http://purl.org/coar/resource_type/c_db06 |
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acceptedVersion |
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instname:Universidad de los Andes |
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eng |
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S. B. Bravyi and A. Yu. Kitaev. Quantum codes on a lattice with boundary. 1998. arXiv: quant-ph/9811052 [quant-ph]. Sergey Bravyi and Jeongwan Haah. “Magic-state distillation with low overhead”. In: Physical Review A 86.5 (2012), p. 052329. Sergey Bravyi and Jeongwan Haah. “Quantum Self-Correction in the 3D Cubic Code Model”. In: Physical Review Letters 111.20 (Nov. 2013). doi: 10.1103/physrevlett. 111.200501. Sergey Bravyi and Matthew B Hastings. “A short proof of stability of topological order under local perturbations”. In: Communications in mathematical physics 307.3 (2011), p. 609. Sergey Bravyi, Matthew B Hastings, and Spyridon Michalakis. “Topological quantum order: stability under local perturbations”. In: Journal of mathematical physics 51.9 (2010). Ranee K Brylinski and Goong Chen. Mathematics of quantum computation. CRC Press, 2002. Oliver Buerschaper, Juan Martín Mombelli, Matthias Christandl, and Miguel Aguado. “A hierarchy of topological tensor network states”. In: Journal of Mathematical Physics 54.1 (2013), p. 012201. A Robert Calderbank and Peter W Shor. “Good quantum error-correcting codes exist”. In: Physical Review A 54.2 (1996), p. 1098. Brett Hungar, Kyle Kawagoe, David Penneys, Chian Yeong Chuah, Mario Tomba, Daniel Wallick, and Shuqi Wei.. “Boundary algebras of the Kitaev quantum double model”. In: Journal of Mathematical Physics 65.10 (2024). William J. Cook and André Rohe. “Computing Minimum-Weight Perfect Matchings”. In: INFORMS J. Comput. 11 (1999), pp. 138–148. Shawn X Cui, César Galindo, and Diego Romero. “Abelian Group Quantum Error Correction in Kitaev’s Model”. In: arXiv preprint arXiv:2404.08552 (2024). Shawn X. Cui. Topological Quantum Computation. Lecture notes. Lecture notes for a 10-lecture course on Topological Quantum Computation taught at Stanford University, Spring 2018. Aug. 2018. Shawn X. Cui, César Galindo, and Diego Romero. Twisted Kitaev quantum double model as local topological order. 2024. arXiv: 2411.08675 [math.QA]. Shawn X. Cui, Dawei Ding, Xizhi Han, Geoffrey Penington, Daniel Ranard, Bran don C. Rayhaun, and Zhou Shangnan.. “Kitaev’s quantum double model as an error correcting code”. In: Quantum 4 (Sept. 2020), p. 331. Michael H Freedman, Michael Larsen, and ZhenghanWang. “A Modular Functor Which is Universal for Quantum Computation”. In: Communications in Mathematical Physics 227.3 (2002), pp. 605–622. Michel X. Goemans and David P. Williamson. “A General Approximation Technique for Constrained Forest Problems”. In: SIAM Journal on Computing 24.2 (1995), pp. 296– 317. doi: 10.1137/S0097539793242618. D Gottesman. Fault-tolerant computation with higher-dimensional systems. Tech. rep. Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 1998. Daniel Gottesman. “Class of quantum error-correcting codes saturating the quantum Hamming bound”. In: Physical Review A 54.3 (Sept. 1996), pp. 1862–1868. J.L. Gross, J. Yellen, and M. Anderson. Graph Theory and Its Applications. Textbooks in Mathematics. CRC Press, 2018. isbn: 9780429757099. Nili Guttmann-Beck and Refael Hassin. “Approximation Algorithms for Min–Max Tree Partition”. In: Journal of Algorithms 24.2 (1997), pp. 266–286. issn: 0196-6774. doi: https://doi.org/10.1006/jagm.1996.0848. Jeongwan Haah. “Local stabilizer codes in three dimensions without string logical operators”. In: Physical Review A 83.4 (2011), p. 042330. Matthew B Hastings, Jeongwan Haah, and Ryan O’Donnell. “Fiber bundle codes: breaking the n1/2polylog(n) barrier for quantum ldpc codes”. In: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. 2021, pp. 1276–1288. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. Yuting Hu, Yidun Wan, and Yong-Shi Wu. “Twisted quantum double model of topological phases in two dimensions”. In: Physical Review B 87.12 (Mar. 2013). doi: 10.1103/ physrevb.87.125114 Adrian Hutter, Daniel Loss, and James R Wootton. “Improved HDRG decoders for qudit and non-Abelian quantum error correction”. In: New Journal of Physics 17.3 (Mar. 2015), p. 035017. doi: 10.1088/1367-2630/17/3/035017. Corey Jones, Pieter Naaijkens, David Penneys, and Daniel Wallick. “Local topological order and boundary algebras”. In: arXiv preprint arXiv:2307.12552 (2023). G. Karpilovsky. Projective Representations of Finite Groups. Finite groups. M. Dekker, 1985. isbn: 9780824773137. D. G. Kirkpatrick and P. Hell. “On the Complexity of General Graph Factor Problems”. In: SIAM Journal on Computing 12.3 (1983), pp. 601–609. doi: 10.1137/0212040. A.Yu. Kitaev. “Fault-tolerant quantum computation by anyons”. In: Annals of Physics 303.1 (2003), pp. 2–30. Emanuel Knill and Raymond Laflamme. “A Theory of quantum error-correcting codes”. In: Phys. Rev. A 55 (2 Feb. 1997), pp. 900–911. Michael A. Levin and Xiao-Gang Wen. “String-net condensation: A physical mechanism for topological phases”. In: Physical Review B 71.4 (June 2005). doi: 10.1103/ physrevb.71.045110. Pieter Naaijkens. “Anyons in infinite quantum systems: QFT in d = 2+1 and the toric code”. PhD thesis. Radboud Universiteit Nijmegen, 2012. Chetan Nayak, Steven H. Simon, Ady Stern, and Michael Freedman . “Non-Abelian anyons and topological quantum computation”. In: Reviews of Modern Physics 80.3 (2008), pp. 1083–1159. Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information. 2002. Pavel Panteleev and Gleb Kalachev. “Quantum LDPC codes with almost linear minimum distance”. In: IEEE Transactions on Information Theory 68.1 (2021), pp. 213– 229. Yang Qiu and Zhenghan Wang. “Ground subspaces of topological phases of matter as error correcting codes”. In: Annals of Physics 422 (2020), p. 168318. R Raussendorf, J Harrington, and K Goyal. “Topological fault-tolerance in cluster state quantum computation”. In: New Journal of Physics 9.6 (June 2007), pp. 199–199. Alexis Schotte, Lander Burgelman, and Guanyu Zhu. “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”. In: arXiv preprint arXiv:2301.00054 (2022). Alexis Schotte, Guanyu Zhu, Lander Burgelman, and Frank Verstraete. “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”. In: Physical Review X 12.2 (2022), p. 021012. Peter W Shor. “Scheme for reducing decoherence in quantum computer memory”. In: Physical review A 52.4 (1995), R2493. Andrew M Steane. “Simple quantum error-correcting codes”. In: Physical Review A 54.6 (1996), p. 4741 Sagar Vijay, Jeongwan Haah, and Liang Fu. “A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations”. In: Physical Review B 92.23 (2015), p. 235136. David S. Wang, Austin G. Fowler, Ashley Stephens, and Lloyd Christopher Hollenberg. “Threshold Error Rates for the Toric and Planar Codes”. In: Quantum Info. Comput. 10.5 (May 2010), pp. 456–469. Xiao-Gang Wen. Quantum field theory of many-body systems: From the origin of sound to an origin of light and electrons. Oxford university press, 2004. James Wootton. “A Simple Decoder for Topological Codes”. In: Entropy 17.4 (Apr. 2015), pp. 1946–1957. doi: 10.3390/e17041946. James R. Wootton, Jan Burri, Sofyan Iblisdir, and Daniel Loss. “Error correction for non-abelian topological quantum computation”. In: Physical review X 4.1 (2014), p. 011051. Bowen Yan, Penghua Chen, and Shawn X Cui. “Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras”. In: Journal of Physics A: Mathematical and Theoretical 55.18 (2022), p. 185201. |
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Galindo Martínez, Cesar Neyitvirtual::24288-1Cui, Shawn XingshanRomero Fonseca, Diego ArturoJones, CoreyUribe Jongbloed, BernardoÁngel Cárdenas, Jairo Andrés2025-06-19T15:24:57Z2025-06-19T15:24:57Z2025-05-28https://hdl.handle.net/1992/76343instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/This thesis presents a detailed analysis of the quantum double model and its twisted version, focusing on their properties as quantum error-correcting codes. In the case of Kitaev's quantum double model for finite Abelian groups, the error correction process is explicitly described. The number of correctable errors depends on the lattice and the topology of the underlying surface. Although there exists a theoretical maximum number of errors that can be corrected, it is proven that correcting this number of errors is, in general, an NP-complete problem. As an alternative, a polynomial-time correction algorithm is proposed that corrects a number of errors below the theoretical maximum. In regard to the twisted quantum double model, it is studied within the framework of Local Topological Order (LTO). Originally formulated for square lattices, the definition of LTO is extended here to arbitrary two-dimensional lattices, enabling an explicit characterization of the ground states space through invariant spaces of monomial representations. By reformulating the LTO conditions to include such general lattices, it is proven that the twisted model satisfies all four axioms of LTO on any 2D lattice. As a result, the ground states space of the model is shown to define a quantum error-correcting code, highlighting the role of topological order in fault-tolerant quantum computation.Doctorado88 páginasapplication/pdfengUniversidad de los AndesDoctorado en MatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Quantum error correction and local topological order from Quantum Double modelTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttps://purl.org/redcol/resource_type/TDQuantum Double ModelQuantum codesLatticeError-correctionGround stateTopological orderMatemáticasS. B. Bravyi and A. Yu. Kitaev. Quantum codes on a lattice with boundary. 1998. arXiv: quant-ph/9811052 [quant-ph].Sergey Bravyi and Jeongwan Haah. “Magic-state distillation with low overhead”. In: Physical Review A 86.5 (2012), p. 052329.Sergey Bravyi and Jeongwan Haah. “Quantum Self-Correction in the 3D Cubic Code Model”. In: Physical Review Letters 111.20 (Nov. 2013). doi: 10.1103/physrevlett. 111.200501.Sergey Bravyi and Matthew B Hastings. “A short proof of stability of topological order under local perturbations”. In: Communications in mathematical physics 307.3 (2011), p. 609.Sergey Bravyi, Matthew B Hastings, and Spyridon Michalakis. “Topological quantum order: stability under local perturbations”. In: Journal of mathematical physics 51.9 (2010).Ranee K Brylinski and Goong Chen. Mathematics of quantum computation. CRC Press, 2002.Oliver Buerschaper, Juan Martín Mombelli, Matthias Christandl, and Miguel Aguado. “A hierarchy of topological tensor network states”. In: Journal of Mathematical Physics 54.1 (2013), p. 012201.A Robert Calderbank and Peter W Shor. “Good quantum error-correcting codes exist”. In: Physical Review A 54.2 (1996), p. 1098.Brett Hungar, Kyle Kawagoe, David Penneys, Chian Yeong Chuah, Mario Tomba, Daniel Wallick, and Shuqi Wei.. “Boundary algebras of the Kitaev quantum double model”. In: Journal of Mathematical Physics 65.10 (2024).William J. Cook and André Rohe. “Computing Minimum-Weight Perfect Matchings”. In: INFORMS J. Comput. 11 (1999), pp. 138–148.Shawn X Cui, César Galindo, and Diego Romero. “Abelian Group Quantum Error Correction in Kitaev’s Model”. In: arXiv preprint arXiv:2404.08552 (2024).Shawn X. Cui. Topological Quantum Computation. Lecture notes. Lecture notes for a 10-lecture course on Topological Quantum Computation taught at Stanford University, Spring 2018. Aug. 2018.Shawn X. Cui, César Galindo, and Diego Romero. Twisted Kitaev quantum double model as local topological order. 2024. arXiv: 2411.08675 [math.QA].Shawn X. Cui, Dawei Ding, Xizhi Han, Geoffrey Penington, Daniel Ranard, Bran don C. Rayhaun, and Zhou Shangnan.. “Kitaev’s quantum double model as an error correcting code”. In: Quantum 4 (Sept. 2020), p. 331.Michael H Freedman, Michael Larsen, and ZhenghanWang. “A Modular Functor Which is Universal for Quantum Computation”. In: Communications in Mathematical Physics 227.3 (2002), pp. 605–622.Michel X. Goemans and David P. Williamson. “A General Approximation Technique for Constrained Forest Problems”. In: SIAM Journal on Computing 24.2 (1995), pp. 296– 317. doi: 10.1137/S0097539793242618.D Gottesman. Fault-tolerant computation with higher-dimensional systems. Tech. rep. Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 1998.Daniel Gottesman. “Class of quantum error-correcting codes saturating the quantum Hamming bound”. In: Physical Review A 54.3 (Sept. 1996), pp. 1862–1868.J.L. Gross, J. Yellen, and M. Anderson. Graph Theory and Its Applications. Textbooks in Mathematics. CRC Press, 2018. isbn: 9780429757099.Nili Guttmann-Beck and Refael Hassin. “Approximation Algorithms for Min–Max Tree Partition”. In: Journal of Algorithms 24.2 (1997), pp. 266–286. issn: 0196-6774. doi: https://doi.org/10.1006/jagm.1996.0848.Jeongwan Haah. “Local stabilizer codes in three dimensions without string logical operators”. In: Physical Review A 83.4 (2011), p. 042330.Matthew B Hastings, Jeongwan Haah, and Ryan O’Donnell. “Fiber bundle codes: breaking the n1/2polylog(n) barrier for quantum ldpc codes”. In: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. 2021, pp. 1276–1288.Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.Yuting Hu, Yidun Wan, and Yong-Shi Wu. “Twisted quantum double model of topological phases in two dimensions”. In: Physical Review B 87.12 (Mar. 2013). doi: 10.1103/ physrevb.87.125114Adrian Hutter, Daniel Loss, and James R Wootton. “Improved HDRG decoders for qudit and non-Abelian quantum error correction”. In: New Journal of Physics 17.3 (Mar. 2015), p. 035017. doi: 10.1088/1367-2630/17/3/035017.Corey Jones, Pieter Naaijkens, David Penneys, and Daniel Wallick. “Local topological order and boundary algebras”. In: arXiv preprint arXiv:2307.12552 (2023).G. Karpilovsky. Projective Representations of Finite Groups. Finite groups. M. Dekker, 1985. isbn: 9780824773137.D. G. Kirkpatrick and P. Hell. “On the Complexity of General Graph Factor Problems”. In: SIAM Journal on Computing 12.3 (1983), pp. 601–609. doi: 10.1137/0212040.A.Yu. Kitaev. “Fault-tolerant quantum computation by anyons”. In: Annals of Physics 303.1 (2003), pp. 2–30.Emanuel Knill and Raymond Laflamme. “A Theory of quantum error-correcting codes”. In: Phys. Rev. A 55 (2 Feb. 1997), pp. 900–911.Michael A. Levin and Xiao-Gang Wen. “String-net condensation: A physical mechanism for topological phases”. In: Physical Review B 71.4 (June 2005). doi: 10.1103/ physrevb.71.045110.Pieter Naaijkens. “Anyons in infinite quantum systems: QFT in d = 2+1 and the toric code”. PhD thesis. Radboud Universiteit Nijmegen, 2012.Chetan Nayak, Steven H. Simon, Ady Stern, and Michael Freedman . “Non-Abelian anyons and topological quantum computation”. In: Reviews of Modern Physics 80.3 (2008), pp. 1083–1159.Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information. 2002.Pavel Panteleev and Gleb Kalachev. “Quantum LDPC codes with almost linear minimum distance”. In: IEEE Transactions on Information Theory 68.1 (2021), pp. 213– 229.Yang Qiu and Zhenghan Wang. “Ground subspaces of topological phases of matter as error correcting codes”. In: Annals of Physics 422 (2020), p. 168318.R Raussendorf, J Harrington, and K Goyal. “Topological fault-tolerance in cluster state quantum computation”. In: New Journal of Physics 9.6 (June 2007), pp. 199–199.Alexis Schotte, Lander Burgelman, and Guanyu Zhu. “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”. In: arXiv preprint arXiv:2301.00054 (2022).Alexis Schotte, Guanyu Zhu, Lander Burgelman, and Frank Verstraete. “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”. In: Physical Review X 12.2 (2022), p. 021012.Peter W Shor. “Scheme for reducing decoherence in quantum computer memory”. In: Physical review A 52.4 (1995), R2493.Andrew M Steane. “Simple quantum error-correcting codes”. In: Physical Review A 54.6 (1996), p. 4741Sagar Vijay, Jeongwan Haah, and Liang Fu. “A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations”. In: Physical Review B 92.23 (2015), p. 235136.David S. Wang, Austin G. Fowler, Ashley Stephens, and Lloyd Christopher Hollenberg. “Threshold Error Rates for the Toric and Planar Codes”. In: Quantum Info. Comput. 10.5 (May 2010), pp. 456–469.Xiao-Gang Wen. Quantum field theory of many-body systems: From the origin of sound to an origin of light and electrons. Oxford university press, 2004.James Wootton. “A Simple Decoder for Topological Codes”. In: Entropy 17.4 (Apr. 2015), pp. 1946–1957. doi: 10.3390/e17041946.James R. Wootton, Jan Burri, Sofyan Iblisdir, and Daniel Loss. “Error correction for non-abelian topological quantum computation”. In: Physical review X 4.1 (2014), p. 011051.Bowen Yan, Penghua Chen, and Shawn X Cui. “Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras”. 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