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Published online by Cambridge University Press:  27 February 2026

Yidun Wan
Yidun Wan
Affiliation:
Fudan University, Shanghai
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Quantum Computing Unveiled
A Concise Course with Topological Extensions
 , pp. 331 - 346
Publisher: Cambridge University Press
Print publication year: 2026

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References

Ince, D. The computer: A very short introduction. No. 292 in Very Short Introductions (Oxford University Press, Oxford, 2011). OCLC: ocn761374747.10.1093/actrade/9780199586592.001.0001CrossRefGoogle Scholar
Stokes, J. Inside the machine: An illustrated introduction to microprocessors and computer architecture (No Starch Press, 2015). OCLC: 913792823.Google Scholar
Ifrah, G. The universal history of computing: From the abacus to the quantum computer (Wiley, New York, 2001).Google Scholar
Kurzweil, R. How to create a mind: The secret of human thought revealed (Penguin Books, New York, 2013).Google Scholar
Hyman, A. Charles Babbage: Pioneer of the computer (Oxford University Press, Oxford, 1982).Google Scholar
Menabrea, L. F. Sketch of the Analytical Engine invented by Charles Babbage, Esq. In Ada’s legacy: Cultures of computing from the Victorian to the digital age (Association for Computing Machinery and Morgan & Claypool, 2015). https://doi.org/10.1145/2809523.2809528.Google Scholar
Liu, C. The three-body problem. Book I in Three-Body trilogy (Tor Books, New York, 2014).Google Scholar
Patterson, D. A., Hennessy, J. L. & Alexander, P. Computer organization and design: The hardware/software interface, 5th edn. (Elsevier Morgan Kaufmann, Amsterdam, Heidelberg, 2014).Google Scholar
Silberschatz, A., Galvin, P. B. & Gagne, G. Operating system concepts, 9th edn. (Wiley, Hoboken, NJ, 2013).Google Scholar
MacCartney, S. ENIAC: The triumphs and tragedies of the world’s first computer (Walker, New York, 1999).Google Scholar
Moore, G. E. Cramming more components onto integrated circuits. Reprinted from Electronics, 38(8), April 19, 1965, pp.114 ff. IEEE Solid-State Circuits Society Newsletter 11(3), 33–35 (2006). http://ieeexplore.ieee.org/document/4785860/.Google Scholar
Brynjolfsson, E. & McAfee, A. The second machine age: Work, progress, and prosperity in a time of brilliant technologies (W. W. Norton & Company, New York, London, 2016).Google Scholar
Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information, 10th Anniversary edn. (Cambridge University Press, Cambridge, 2010).Google Scholar
Georgescu, I., Ashhab, S. & Nori, F. Quantum simulation. Reviews of Modern Physics 86, 153–185 (2014). https://link.aps.org/doi/10.1103/RevModPhys.86.153.CrossRefGoogle Scholar
Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum computation by adiabatic evolution (2000). http://arxiv.org/abs/quant-ph/0001106. ArXiv:quant-ph/0001106.Google Scholar
Zanardi, P. & Rasetti, M. Holonomic quantum computation. Physics Letters A 264, 94–99 (1999). https://linkinghub.elsevier.com/retrieve/pii/S0375960199008038.10.1016/S0375-9601(99)00803-8CrossRefGoogle Scholar
Kitaev, A. Fault-tolerant quantum computation by anyons. Annals of Physics 303, 2–30 (2003). https://linkinghub.elsevier.com/retrieve/pii/S0003491602000180.10.1016/S0003-4916(02)00018-0CrossRefGoogle Scholar
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80, 1083–1159 (2008). https://link.aps.org/doi/10.1103/RevModPhys.80.1083.CrossRefGoogle Scholar
Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: An outlook. Science 339, 1169–1174 (2013). https://www.science.org/doi/10.1126/science.1231930.CrossRefGoogle ScholarPubMed
Monroe, C. & Kim, J. Scaling the ion trap quantum processor. Science 339, 1164–1169 (2013). https://www.science.org/doi/10.1126/science.1231298.CrossRefGoogle ScholarPubMed
Nakahara, M. & Ohmi, T. Quantum computing: From linear algebra to physical realizations (CRC Press, Boca Raton, FL, 2008). OCLC: ocn177825572.10.1201/9781420012293CrossRefGoogle Scholar
Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nature Physics 8, 267–276 (2012). https://www.nature.com/articles/nphys2259.10.1038/nphys2259CrossRefGoogle Scholar
Vandersypen, L. M. K. & Chuang, I. L. NMR techniques for quantum control and computation. Reviews of Modern Physics 76, 1037–1069 (2005). https://link.aps.org/doi/10.1103/RevModPhys.76.1037.CrossRefGoogle Scholar
Waldrop, M. M. The chips are down for Moore’s law. Nature 530, 144–147 (2016). https://www.nature.com/articles/530144a.10.1038/530144aCrossRefGoogle Scholar
Rivest, R. L., Shamir, A. & Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21, 120–126 (1978). https://dl.acm.org/doi/10.1145/359340.359342.CrossRefGoogle Scholar
Wardlaw, W. P. The RSA public key cryptosystem. In Joyner, D. (ed.), Coding theory and cryptography, 101–123 (Springer, Berlin, Heidelberg, 2000).Google Scholar
Zhou, X. & Tang, X. Research and implementation of RSA algorithm for encryption and decryption. In Proceedings of 2011 6th International Forum on Strategic Technology, vol. 2, 1118–1121 (2011).10.1109/IFOST.2011.6021216CrossRefGoogle Scholar
NaQi et al. Analysis and research of the RSA algorithm. Information Technology Journal 12, 1818–1824 (2013). https://scialert.net/fulltext/?doi=itj.2013.1818.1824.Google Scholar
Al-Kaabi, S. S. & Belhaouari, S. B. Methods toward enhancing RSA algorithm: A survey (2019). https://papers.ssrn.com/abstract=3412776.Google Scholar
Manin, U. I. & Dyson, F. J. Mathematics as metaphor (American Mathematical Society, Providence, RI, 2007).Google Scholar
Feynman, R. P. Simulating physics with computers. International Journal of Theoretical Physics 21, 467–488 (1982). https://doi.org/10.1007/BF02650179.CrossRefGoogle Scholar
Landauer, R. Irreversibility and heat generation in the computing process. IBM Journal of Research and Development 5, 183–191 (1961). http://ieeexplore.ieee.org/document/5392446/.10.1147/rd.53.0183CrossRefGoogle Scholar
Ikonen, J., Salmilehto, J. & Mottonen, M. Energy-efficient quantum computing. npj Quantum Information 3, 17 (2017). https://www.nature.com/articles/s41534-017-0015-5.10.1038/s41534-017-0015-5CrossRefGoogle Scholar
Deutsch, D. Quantum theory, the Church–Turing principle and the universal quantum computer. Proceedings of the Royal Society of London A 400, 97–117 (1985). https://royalsocietypublishing.org/doi/10.1098/rspa.1985.0070.Google Scholar
Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26, 1484–1509 (1997). http://epubs.siam.org/doi/10.1137/S0097539795293172.CrossRefGoogle Scholar
Grover, L. K. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing: STOC ’96, 212–219 (ACM Press, Philadelphia, PA, 1996). http://portal.acm.org/citation.cfm?doid=237814.237866.Google Scholar
Zhao, Y. et al. Realization of an error-correcting surface code with superconducting qubits. Physical Review Letters 129, 030501 (2022). https://link.aps.org/doi/10.1103/PhysRevLett.129.030501.CrossRefGoogle ScholarPubMed
Ni, Z. et al. Beating the break-even point with a discrete-variable-encoded logical qubit. Nature 616, 56–60 (2023). https://www.nature.com/articles/s41586-023-05784-4.10.1038/s41586-023-05784-4CrossRefGoogle ScholarPubMed
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). https://www.nature.com/articles/s41586-019-1666-5.10.1038/s41586-019-1666-5CrossRefGoogle ScholarPubMed
Kelly, J. A preview of Bristlecone, Google’s new quantum processor (2018). https://blog.research.google/2018/03/a-preview-of-bristlecone-googles-new.html.Google Scholar
Gambetta, J. The hardware and software for the era of quantum utility is here (2023). https://www.ibm.com/quantum/blog/quantum-roadmap-2033.Google Scholar
Neven, H. Meet Willow, our state-of-the-art quantum chip (2024). https://blog.google/technology/research/google-willow-quantum-chip/.Google Scholar
Lin, Rui et al. AI-Enabled Parallel Assembly of Thousands of Defect-Free Neutral Atom Arrays (2025). Phys. Rev. Lett. 135, 06060. https://doi.org/10.1103/2ym8-vs82.CrossRefGoogle ScholarPubMed
Chiu, NC et al. Continuous operation of a coherent 3,000-qubit system (2025). Nature, (2025). https://doi.org/10.1038/s41586-025-09596-6.CrossRefGoogle ScholarPubMed
Quantinuum. Quantinuum Launches Industry-First, Trapped-Ion 56-Qubit Quantum Computer, Breaking Key Benchmark Record. https://www.quantinuum.com/press-releases/quantinuum-launches-industry-first-trapped-ion-56-qubit-quantum-computer-that-challenges-the-worlds-best-supercomputers.Google Scholar
Intel. Introducing Tunnel Falls. https://www.intel.com/content/www/us/en/research/quantum-computing.html.Google Scholar
Petit, L. et al. Universal quantum logic in hot silicon qubits. Nature 580, 355–359 (2020). https://www.nature.com/articles/s41586-020-2170-7.10.1038/s41586-020-2170-7CrossRefGoogle ScholarPubMed
Bolgar, C. Microsoft’s Majorana 1 chip carves new path for quantum computing. https://news.microsoft.com/source/features/innovation/microsofts-majorana-1-chip-carves-new-path-for-quantum-computing.Google Scholar
Zhu, Q. et al. Quantum computational advantage via 60-qubit 24-cycle random circuit sampling. Science Bulletin 67, 240–245 (2022). https://linkinghub.elsevier.com/retrieve/pii/S2095927321006733.10.1016/j.scib.2021.10.017CrossRefGoogle ScholarPubMed
Xinhua. China’s 176-qubit quantum computing platform goes online(2023). https://english.news.cn/20230531/0946675301284c1786b4ee27251c89a3/c.html.Google Scholar
Quafu Group, BAQIS. Quafu-RL: The Cloud quantum computers based quantum reinforcement Learning (sic; 2023). https://arxiv.org/abs/2305.17966. Version Number: 2.Google Scholar
Quafu Group, BAQIS. Quafu-Qcover: Explore combinatorial optimization problems on Cloud-based quantum computers (2023). https://arxiv.org/abs/2305.17979.Google Scholar
BAQIS. BAQIS Unveiled “Quafu” quantum Cloud platform at 2023 ZGC Forum (2023). http://en.baqis.ac.cn/news/detail/?cid=1699.Google Scholar
Pikulin, D. I. et al. Protocol to identify a topological superconducting phase in a three-terminal device (2021). https://arxiv.org/abs/2103.12217.Google Scholar
Aaronson, S. Is quantum mechanics an island in theoryspace? (2004). http://arxiv.org/abs/quant-ph/0401062. ArXiv:quant-ph/0401062.Google Scholar
Watrous, J. The theory of quantum information (Cambridge University Press, Cambridge, 2018). https://www.cambridge.org/core/books/theory-of-quantum-information/AE4AA5638F808D2CFEB070C55431D897.10.1017/9781316848142CrossRefGoogle Scholar
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: Towards practical large-scale quantum computation. Physical Review 86, 032324 (2012). https://link.aps.org/doi/10.1103/PhysRevA.86.032324.Google Scholar
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018). https://quantum-journal.org/papers/q-2018-08-06-79/.10.22331/q-2018-08-06-79CrossRefGoogle Scholar
Gottesman, D. An introduction to quantum error correction and fault-tolerant quantum computation (2009). http://arxiv.org/abs/0904.2557. ArXiv:0904.2557 [quant-ph].Google Scholar
Bacon, D. M. Decoherence, control, and symmetry in quantum computers. Doctoral thesis, University of California, Berkeley (2001).Google Scholar
Bohm, G. Quantum computers (2018). https://group.mercedes-benz.com/company/magazine/technology-innovation/quantum-computers-future-daimler-google-ibm-technology.html.Google Scholar
ExxonMobil. ExxonMobil and IBM to advance energy sector application of quantum computing (2019). https://corporate.exxonmobil.com/news/news-releases/2019/0108exxonmobil-and-ibm-to-advance-energy-sector-application-of-quantum-computing.Google Scholar
CERN. CERN Quantum Technology Initiative unveils strategic roadmap shaping CERN’s role in next quantum revolution (2021). https://home.cern/news/press-release/knowledge-sharing/cern-quantum-technology-initiative-unveils-strategic-roadmap.Google Scholar
Mitsubishi. Establishment of quantum strategic industry alliance for revolution (2021). https://www.m-chemical.co.jp/en/news/2021/_icsFiles/afieldfile/2021/09/29/210901qstar.pdf.Google Scholar
Yin, H.-L. et al. Measurement-device-independent quantum key distribution over a 404 km optical fiber. Physical Review Letters 117, 190501 (2016). https://link.aps.org/doi/10.1103/PhysRevLett.117.190501.CrossRefGoogle Scholar
Gibney, E. Mini-satellite paves the way for quantum messaging anywhere on Earth. https://doi.org/10.1038/d41586-025-00581-7.CrossRefGoogle Scholar
Lu, C.-Y., Cao, Y., Peng, C.-Z. & Pan, J.-W. Micius quantum experiments in space. Reviews of Modern Physics 94, 035001 (2022). https://link.aps.org/doi/10.1103/RevModPhys.94.035001.CrossRefGoogle Scholar
Ursin, R. et al. Entanglement-based quantum communication over 144 km. Nature Physics 3, 481–486 (2007). https://www.nature.com/articles/nphys629.10.1038/nphys629CrossRefGoogle Scholar
Hu, X.-M., Guo, Y., Liu, B.-H., Li, C.-F. & Guo, G.-C. Progress in quantum teleportation. Nature Reviews Physics 5, 339–353 (2023). https://www.nature.com/articles/s42254-023-00588-x.10.1038/s42254-023-00588-xCrossRefGoogle Scholar
Heshami, K. et al. Quantum memories: Emerging applications and recent advances. Journal of Modern Optics 63, 2005–2028 (2016). https://www.tandfonline.com/doi/full/10.1080/09500340.2016.1148212.CrossRefGoogle ScholarPubMed
Ma, L. et al. High-performance cavity-enhanced quantum memory with warm atomic cell. Nature Communications 13, 2368 (2022). https://www.nature.com/articles/s41467-022-30077-1.Google ScholarPubMed
Wang, Y., Craddock, A. N., Sekelsky, R., Flament, M. & Namazi, M. Field-deployable quantum memory for quantum networking. Physical Review Applied 18, 044058 (2022). https://link.aps.org/doi/10.1103/PhysRevApplied.18.044058.CrossRefGoogle Scholar
Hammerer, K., Sørensen, A. S. & Polzik, E. S. Quantum interface between light and atomic ensembles. Reviews of Modern Physics 82, 1041–1093 (2010). https://link.aps.org/doi/10.1103/RevModPhys.82.1041.CrossRefGoogle Scholar
Zhong, T., Kindem, J. M., Miyazono, E. & Faraon, A. Nanophotonic coherent light–matter interfaces based on rare-earth-doped crystals. Nature Communications 6, 8206 (2015). https://www.nature.com/articles/ncomms9206.10.1038/ncomms9206CrossRefGoogle ScholarPubMed
Manninen, J., Blick, R. H. & Massel, F. Hybrid optomechanical superconducting qubit system. Physical Review Research 6, 023029 (2024). https://link.aps.org/doi/10.1103/PhysRevResearch.6.023029.CrossRefGoogle Scholar
Azuma, K. et al. Quantum repeaters: From quantum networks to the quantum internet (2022). https://arxiv.org/abs/2212.10820. arXiv Version Number: 2.Google Scholar
QuantumXC. Quantum Xchange launches “Phio” the first commercial QKD network for quantum communications in the U.S. https://quantumxc.com/blog/quantum-launches-phio/.Google Scholar
Hawking, S. W. Breakdown of predictability in gravitational collapse. Physical Review D 14, 2460–2473 (1976). https://link.aps.org/doi/10.1103/PhysRevD.14.2460.CrossRefGoogle Scholar
Harlow, D. Jerusalem lectures on black holes and quantum information. Reviews of Modern Physics 88, 015002 (2016). https://link.aps.org/doi/10.1103/RevModPhys.88.015002.CrossRefGoogle Scholar
Maldacena, J. The large-N limit of superconformal field theories and supergravity. International Journal of Theoretical Physics 38, 1113–1133 (1999). http://link.springer.com/10.1023/A:1026654312961.10.1023/A:1026654312961CrossRefGoogle Scholar
Ryu, S. & Takayanagi, T. Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence. Physical Review Letters 96, 181602 (2006). https://link.aps.org/doi/10.1103/PhysRevLett.96.181602.CrossRefGoogle ScholarPubMed
Swingle, B. Entanglement renormalization and holography. Physical Review D 86, 065007 (2012). https://link.aps.org/doi/10.1103/PhysRevD.86.065007.CrossRefGoogle Scholar
Verlinde, E. & Verlinde, H. Black hole entanglement and quantum error correction. Journal of High Energy Physics 2013, 107 (2013). http://link.springer.com/10.1007/JHEP10(2013)107.10.1007/JHEP10(2013)107CrossRefGoogle Scholar
Baker, O. K. Quantum information science in high energy physics. In Topics on Quantum Information Science (IntechOpen, 2021). https://www.intechopen.com/chapters/77398.Google Scholar
Almheiri, A., Dong, X. & Harlow, D. Bulk locality and quantum error correction in AdS/CFT. Journal of High Energy Physics 2015, 163 (2015). http://link.springer.com/10.1007/JHEP04(2015)163.10.1007/JHEP04(2015)163CrossRefGoogle Scholar
Harlow, D. The Ryu–Takayanagi formula from quantum error correction. Communications in Mathematical Physics 354, 865–912 (2017). http://link.springer.com/10.1007/s00220-017-2904-z.10.1007/s00220-017-2904-zCrossRefGoogle Scholar
Akers, C. & Rath, P. Holographic Rényi entropy from quantum error correction. Journal of High Energy Physics 2019, 52 (2019). https://link.springer.com/10.1007/JHEP05(2019)052.10.1007/JHEP05(2019)052CrossRefGoogle Scholar
Kharzeev, D. E. & Levin, E. Deep inelastic scattering as a probe of entanglement: Confronting experimental data. Physical Review D 104, L031503 (2021). https://link.aps.org/doi/10.1103/PhysRevD.104.L031503.CrossRefGoogle Scholar
Susskind, L. Computational complexity and black hole horizons (2014). https://arxiv.org/abs/1402.5674.Google Scholar
Stanford, D. & Susskind, L. Complexity and shock wave geometries. Physical Review D 90, 126007 (2014). https://link.aps.org/doi/10.1103/PhysRevD.90.126007.CrossRefGoogle Scholar
Pastawski, F., Yoshida, B., Harlow, D. & Preskill, J. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. Journal of High Energy Physics 2015, 149 (2015). http://link.springer.com/10.1007/JHEP06(2015)149.10.1007/JHEP06(2015)149CrossRefGoogle Scholar
Bennett, C. H. & Landauer, R. The fundamental physical limits of computation. Scientific American 253, 48–56 (1985). https://www.scientificamerican.com/article/the-fundamental-physical-limits-of.10.1038/scientificamerican0785-48CrossRefGoogle Scholar
Lloyd, S. Ultimate physical limits to computation. Nature 406, 1047–1054 (2000). https://www.nature.com/articles/35023282.10.1038/35023282CrossRefGoogle ScholarPubMed
Thomas, H. Physics of computation: From classical to quantum. In Skjeltorp, A. T. & Vicsek, T. (eds.), Complexity from microscopic to macroscopic scales: Coherence and large deviations, NATO Science Series, 1–20 (Springer Netherlands, Dordrecht, 2002). https://doi.org/10.1007/978-94-010-0419-0_1.Google Scholar
Sipser, M. Introduction to the theory of computation (PWS Pub. Co., Boston, 1997).Google Scholar
Petzold, C. The annotated Turing: A guided tour through Alan Turing’s historic paper on computability and the Turing machine (Wiley, Hoboken, NJ, 2008).Google Scholar
Hilbert, D. Mathematical problems. Bulletin of the American Mathematical Society 8, 437–479 (1902). https://www.ams.org/bull/1902-08-10/S0002-9904-1902-00923-3/.10.1090/S0002-9904-1902-00923-3CrossRefGoogle Scholar
Godel, K. Uber formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 37, 173–198 (1931). http://link.springer.com/10.1007/BF01700692.Google Scholar
Church, A. An unsolvable problem of elementary number theory. American Journal of Mathematics 58, 345 (1936). https://www.jstor.org/stable/2371045?origin=crossref.10.2307/2371045CrossRefGoogle Scholar
Turing, A. M. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society s2-42, 230–265 (1937). http://doi.wiley.com/10.1112/plms/s2-42.1.230.CrossRefGoogle Scholar
Motwani, R. & Raghavan, P. Randomized algorithms (Cambridge University Press, Cambridge, 1995).10.1017/CBO9780511814075CrossRefGoogle Scholar
Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983). https://www.science.org/doi/10.1126/science.220.4598.671.CrossRefGoogle ScholarPubMed
Arora, S. & Barak, B. Computational complexity: A modern approach (Cambridge University Press, Cambridge, 2009). OCLC: ocn286431654.10.1017/CBO9780511804090CrossRefGoogle Scholar
Goldreich, O. Computational complexity: A conceptual perspective (Cambridge University Press, Cambridge, 2008). OCLC: ocn192050142.10.1017/CBO9780511804106CrossRefGoogle Scholar
Papadimitriou, C. H. Computational complexity (Addison-Wesley, Reading, MA, 1994).Google Scholar
Jukna, S. Boolean function complexity: Advances and frontiers, vol. 27 of Algorithms and combinatorics (Springer, Berlin, Heidelberg, 2012). https://link.springer.com/10.1007/978-3-642-24508-4.Google Scholar
Wegener, I. Complexity theory: Exploring the limits of efficient algorithms (Springer, Berlin, Heidelberg, 2005). http://link.springer.com/10.1007/3-540-27477-4.Google Scholar
Lenstra, A. K., Lenstra, H. W., Manasse, M. S. & Pollard, J. M. The number field sieve. In Lenstra, A. K. & Lenstra, H. W. (eds.), The development of the number field sieve, Lecture Notes in Mathematics, 11–42 (Springer, Berlin, Heidelberg, 1993).10.1007/BFb0091537CrossRefGoogle Scholar
Boudot, F. et al. Comparing the difficulty of factorization and discrete logarithm: A 240-digit experiment. In Micciancio, D. & Ristenpart, T. (eds.), Advances in Cryptology – CRYPTO 2020, Lecture Notes in Computer Science, 62–91 (Springer International Publishing, Cham, Switzerland, 2020). https://link.springer.com/10.1007/978-3-030-56880-1_3.Google Scholar
West, D. B. Introduction to graph theory, 2nd edn. (Pearson, Upper Saddle River, NJ, 2000).Google Scholar
Kruskal, J. B. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society 7, 48–50 (1956). https://www.ams.org/proc/1956-007-01/S0002-9939-1956-0078686-%7/.10.1090/S0002-9939-1956-0078686-7CrossRefGoogle Scholar
Prim, R. C. Shortest connection networks and some generalizations. Bell System Technical Journal 36, 1389–1401 (1957). https://ieeexplore.ieee.org/document/6773228.10.1002/j.1538-7305.1957.tb01515.xCrossRefGoogle Scholar
Held, M. & Karp, R. M. A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics 10, 196–210 (1962). https://epubs.siam.org/doi/10.1137/0110015.CrossRefGoogle Scholar
Euler, L. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae 8, 128–140 (1741). https://scholarlycommons.pacific.edu/euler-works/53.Google Scholar
Hierholzer, C. & Wiener, C. Ueber die Moglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematischen Annalen 6, 30–32 (1873). https://doi.org/10.1007/BF01442866.Google Scholar
Kwan, M.-K. Graphic programming using odd or even points. Chinese Mathematics 1 1962, 237–277 (1962).Google Scholar
Edmonds, J. & Johnson, E. L. Matching, Euler tours and the Chinese postman. Mathematical Programming 5, 88–124 (1973).10.1007/BF01580113CrossRefGoogle Scholar
Pevzner, P. A., Tang, H. & Waterman, M. S. An Eulerian path approach to DNA fragment assembly. Proceedings of the National Academy of Sciences of the United States of America 98, 9748–9753 (2001). https://pnas.org/doi/full/10.1073/pnas.171285098.Google ScholarPubMed
Tarjan, R. E. & Vishkin, U. An efficient parallel biconnectivity algorithm. SIAM Journal on Computing 14, 862–874 (1985). http://epubs.siam.org/doi/10.1137/0214061.CrossRefGoogle Scholar
Berkman, O. & Vishkin, U. Finding level-ancestors in trees. Journal of Computer and System Sciences 48, 214–230 (1994). https://linkinghub.elsevier.com/retrieve/pii/S002200000580002%9.10.1016/S0022-0000(05)80002-9CrossRefGoogle Scholar
Savage, C. A survey of combinatorial gray codes. SIAM Review 39, 605–629 (1997). http://epubs.siam.org/doi/10.1137/S0036144595295272.CrossRefGoogle Scholar
Roy, K. Optimum gate ordering of CMOS logic gates using Euler path approach: Some insights and explanations. Journal of Computing and Information Technology 15, 85 (2007). http://cit.srce.unizg.hr/index.php/CIT/article/view/1629.10.2498/cit.1000731CrossRefGoogle Scholar
Achlioptas, D. et al. Random constraint satisfaction: A more accurate picture. Constraints 6, 329–344 (2001). https://doi.org/10.1023/A:1011402324562.CrossRefGoogle Scholar
Mezard, M., Parisi, G. & Zecchina, R. Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002). https://www.science.org/doi/10.1126/science.1073287.CrossRefGoogle ScholarPubMed
Lame, G. Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers. Comptes rendus des seances de l’Academie des Sciences 867–870 (1844).Google Scholar
Dijkstra, E. W. A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959). http://link.springer.com/10.1007/BF01386390.10.1007/BF01386390CrossRefGoogle Scholar
Floyd, R. W. Algorithm 97: Shortest path. Communications of the ACM 5, 345 (1962). https://dl.acm.org/doi/10.1145/367766.368168.CrossRefGoogle Scholar
Moore, E. The shortest path through a maze. In Proceedings of the International Symposium on the Theory of Switching 285–292 (Harvard University Press, Cambridge, MA, 1959).Google Scholar
Schoning, U. & Torán, J. The satisfiability problem: Algorithms and analyses. (corrected reprint edn.), vol. 3 of Mathematik für Anwendungen (Lehmanns Media, Berlin, 2020).Google Scholar
Mathews, G. B. On the partition of numbers. Proceedings of the London Mathematical Society s1-28, 486–490 (1896). http://doi.wiley.com/10.1112/plms/s1-28.1.486.CrossRefGoogle Scholar
Chernoff, H. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics 23, 493–507 (1952). http://projecteuclid.org/euclid.aoms/1177729330.10.1214/aoms/1177729330CrossRefGoogle Scholar
Hoare, C. A. R. Algorithm 64: Quicksort. Communications of the ACM 4, 321 (1961). https://dl.acm.org/doi/10.1145/366622.366644.CrossRefGoogle Scholar
Calude, C. Theories of computational complexity. Vol. 35 in Annals of Discrete Mathematics (North-Holland/Elsevier, Amsterdam, New York, 1988).10.1016/S0167-5060(08)70408-4CrossRefGoogle Scholar
Leeuwen, J. v. (ed.) Handbook of theoretical computer science (Elsevier/MIT Press, Amsterdam, New York, Cambridge, MA, 1990).Google Scholar
Gershenfeld, N. A. The physics of information technology. Cambridge Series on Information and the Natural Sciences (Cambridge University Press, Cambridge, 2011).Google Scholar
Leff, H. S. & Rex, A. F. (eds.) Maxwell’s demon: Entropy, information, computing (Adam Hilger, Bristol, 1990).10.1887/0750307595CrossRefGoogle Scholar
Leff, H. S. & Rex, A. F. (eds.) Maxwell’s demon 2: Entropy, classical and quantum information, computing (Institute of Physics, Bristol; Philadelphia, 2003). OCLC: ocm51569169.Google Scholar
Berut, A. et al. Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483, 187–189 (2012). https://www.nature.com/articles/nature10872.10.1038/nature10872CrossRefGoogle ScholarPubMed
Bennett, C. H. Logical reversibility of computation. IBM Journal of Research and Development 17, 525–532 (1973). http://ieeexplore.ieee.org/document/5391327/.10.1147/rd.176.0525CrossRefGoogle Scholar
Fredkin, E., Landauer, R. & Toffoli, T. Physics of computation. International Journal of Theoretical Physics 21, 903–903 (1982). http://link.springer.com/10.1007/BF02084157.Google Scholar
Fredkin, E. & Toffoli, T. Conservative logic. International Journal of Theoretical Physics 21, 219–253 (1982). http://link.springer.com/10.1007/BF01857727.10.1007/BF01857727CrossRefGoogle Scholar
Toffoli, T. & Margolus, N. Cellular automata machines: A new environment for modeling (MIT Press, Cambridge, MA, 1987). https://direct.mit.edu/books/book/4258/Cellular-Automata-Mach%inesA-New-Environment-for.10.7551/mitpress/1763.001.0001CrossRefGoogle Scholar
Toffoli, T. & Margolus, N. H. Invertible cellular automata: A review. Physica D: Nonlinear Phenomena 45, 229–253 (1990). https://linkinghub.elsevier.com/retrieve/pii/016727899090185R%.10.1016/0167-2789(90)90185-RCrossRefGoogle Scholar
Axelsen, H. B., Glück, R. & Yokoyama, T. Reversible machine code and its abstract processor architecture. In Diekert, V., Volkov, M. V. & Voronkov, A. (eds.), Computer Science – Theory and Applications, vol. 4649, 56–69 (Springer, Berlin, Heidelberg, 2007). http://link.springer.com/10.1007/978-3-540-74510-5_9.10.1007/978-3-540-74510-5_9CrossRefGoogle Scholar
Gerlach, W. & Stern, O. Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld. Zeitschrift für Physik 9, 349–352 (1922). http://link.springer.com/10.1007/BF01326983.10.1007/BF01326983CrossRefGoogle Scholar
Talagrand, M. What is a quantum field theory? (Cambridge University Press, Cambridge, 2022).10.1017/9781108225144CrossRefGoogle Scholar
Bennett, C. H. Quantum cryptography using any two nonorthogonal states. Physical Review Letters 68, 3121–3124 (1992). https://link.aps.org/doi/10.1103/PhysRevLett.68.3121.CrossRefGoogle ScholarPubMed
Hou, J.-M. & Chen, W. Hidden antiunitary symmetry behind “accidental” degeneracy and its protection of degeneracy. Frontiers in Physics 13, 130301 (2018). http://link.springer.com/10.1007/s11467-017-0712-8.Google Scholar
Hung, L.-Y. & Wan, Y. Symmetry-enriched phases obtained via pseudo anyon condensation. International Journal of Modern Physics 28, 1450172 (2014). https://www.worldscientific.com/doi/abs/10.1142/S021797921450%1720.CrossRefGoogle Scholar
Gaiotto, D., Kapustin, A., Seiberg, N. & Willett, B. Generalized global symmetries. Journal of High Energy Physics 2015, 172 (2015). http://link.springer.com/10.1007/JHEP02(2015)172.10.1007/JHEP02(2015)172CrossRefGoogle Scholar
Kong, L., Lan, T., Wen, X.-G., Zhang, Z.-H. & Zheng, H. Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry. Physical Review Research 2, 043086 (2020). https://link.aps.org/doi/10.1103/PhysRevResearch.2.043086.CrossRefGoogle Scholar
Ji, W. & Wen, X.-G. Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions. Physical Review Research 2, 033417 (2020). https://link.aps.org/doi/10.1103/PhysRevResearch.2.033417.CrossRefGoogle Scholar
Zhu, G., Sikander, S., Portnoy, E., Cross, A. W. & Brown, B. J. Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries (2023). https://arxiv.org/abs/2310.16982.Google Scholar
Jozsa, R. Fidelity for mixed quantum states. Journal of Modern Optics 41, 2315–2323 (1994). http://www.tandfonline.com/doi/abs/10.1080/09500349414552171.CrossRefGoogle Scholar
Schrodinger, E. Discussion of probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society 31, 555–563 (1935). https://www.cambridge.org/core/product/identifier/S0305004100%013554/type/journal_article.10.1017/S0305004100013554CrossRefGoogle Scholar
Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical description of Physical reality be considered complete? Physical Review 47, 777–780 (1935). https://link.aps.org/doi/10.1103/PhysRev.47.777.CrossRefGoogle Scholar
Bohr, N. Can quantum-mechanical description of physical reality be considered Complete? Physical Review 48, 696–702 (1935). https://link.aps.org/doi/10.1103/PhysRev.48.696.CrossRefGoogle Scholar
Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika 1, 195–200 (1964). https://link.aps.org/doi/10.1103/PhysicsPhysiqueFizika.1.195.CrossRefGoogle Scholar
Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hidden-variable theories. Physical Review Letters 23, 880–884 (1969). https://link.aps.org/doi/10.1103/PhysRevLett.23.880.CrossRefGoogle Scholar
Page, D. N. Information in black hole radiation. Physical Review Letters 71, 3743–3746 (1993). https://link.aps.org/doi/10.1103/PhysRevLett.71.3743.CrossRefGoogle ScholarPubMed
Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). https://ieeexplore.ieee.org/document/6773024.10.1002/j.1538-7305.1948.tb01338.xCrossRefGoogle Scholar
Rényi, A. On measures of entropy and information. Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 547–561 (1960). https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s4_%v1_article-27.pdf.Google Scholar
Dur, W., Vidal, G. & Cirac, J. I. Three qubits can be entangled in two inequivalent ways. Physical Review A 62, 062314 (2000). https://link.aps.org/doi/10.1103/PhysRevA.62.062314.CrossRefGoogle Scholar
Verstraete, F., Dehaene, J., De Moor, B. & Verschelde, H. Four qubits can be entangled in nine different ways. Physical Review A 65, 052112 (2002). https://link.aps.org/doi/10.1103/PhysRevA.65.052112.CrossRefGoogle Scholar
Miyake, A. Classification of multipartite entangled states by multidimensional determinants. Physical Review A 67, 012108 (2003). https://link.aps.org/doi/10.1103/PhysRevA.67.012108.CrossRefGoogle Scholar
Hu, Y. & Wan, Y. Entanglement entropy, quantum fluctuations, and thermal entropy in topological phases. Journal of High Energy Physics 2019, 110 (2019). https://link.springer.com/10.1007/JHEP05(2019)110.10.1007/JHEP05(2019)110CrossRefGoogle Scholar
Bekenstein, J. D. Black holes and entropy. Physical Review D 7, 2333–2346 (1973). https://link.aps.org/doi/10.1103/PhysRevD.7.2333.CrossRefGoogle Scholar
Kaufman, A. M. et al. Quantum thermalization through entanglement in an isolated many-body system. Science 353, 794–800 (2016). https://www.science.org/doi/10.1126/science.aaf6725.CrossRefGoogle Scholar
Chen, B. & Wu, J.-q. Universal relation between thermal entropy and entanglement entropy in conformal field theories. Physical Review D 91, 086012 (2015). https://link.aps.org/doi/10.1103/PhysRevD.91.086012.CrossRefGoogle Scholar
Casini, H., Huerta, M. & Myers, R. C. Towards a derivation of holographic entanglement entropy. Journal of High Energy Physics 2011, 36 (2011). http://link.springer.com/10.1007/JHEP05(2011)036.10.1007/JHEP05(2011)036CrossRefGoogle Scholar
Axler, S. Linear algebra done right. Undergraduate Texts in Mathematics (Springer, Cham, Switzerland, 2015). https://link.springer.com/10.1007/978-3-319-11080-6.Google Scholar
Miatto, F. M., Di Lorenzo Pires, H., Barnett, S. M. & Van Exter, M. P. Spatial Schmidt modes generated in parametric down-conversion. The European Physical Journal D 66, 263 (2012). http://link.springer.com/10.1140/epjd/e2012-30035-3.10.1140/epjd/e2012-30035-3CrossRefGoogle Scholar
Ren, J., Wang, Y. & You, W.-L. Quantum phase transitions in spin-1 XXZ chains with rhombic single-ion anisotropy. Physical Review A 97, 042318 (2018). https://link.aps.org/doi/10.1103/PhysRevA.97.042318.CrossRefGoogle Scholar
Bennett, C. H. et al. Purification of noisy entanglement and faithful teleportation via noisy channels. Physical Review Letters 76, 722–725 (1996). https://link.aps.org/doi/10.1103/PhysRevLett.76.722.CrossRefGoogle ScholarPubMed
Peres, A. Neumark’s theorem and quantum inseparability. Foundations of Physics 20, 1441–1453 (1990). http://link.springer.com/10.1007/BF01883517.10.1007/BF01883517CrossRefGoogle Scholar
Bloch, F. Nuclear induction. Physical Review 70, 460–474 (1946). https://link.aps.org/doi/10.1103/PhysRev.70.460.Google Scholar
Deutsch, D., Barenco, A. & Ekert, A. Universality in quantum computation. Proceedings of the Royal Society of London 449, 669–677 (1995). https://royalsocietypublishing.org/doi/10.1098/rspa.1995.0065%.Google Scholar
DiVincenzo, D. P. Two-bit gates are universal for quantum computation. Physical Review A 51, 1015–1022 (1995). https://link.aps.org/doi/10.1103/PhysRevA.51.1015.CrossRefGoogle ScholarPubMed
Gray, F. Pulse code communication. https://patents.google.com/patent/US2632058A/en.Google Scholar
Barenco, A. et al. Elementary gates for quantum computation. Physical Review A 52, 3457–3467 (1995). https://link.aps.org/doi/10.1103/PhysRevA.52.3457.CrossRefGoogle ScholarPubMed
Kitaev, A. Y. Quantum computations: Algorithms and error correction. Russian Mathematical Surveys 52, 1191–1249 (1997). https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH0%02155.10.1070/RM1997v052n06ABEH002155CrossRefGoogle Scholar
Dawson, C. M. & Nielsen, M. A. The Solovay–Kitaev algorithm. Quantum Information & Computation 6, 81–95 (2006).10.26421/QIC6.1-6CrossRefGoogle Scholar
Deutsch, D. & Jozsa, R. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London 439, 553–558 (1992). https://royalsocietypublishing.org/doi/10.1098/rspa.1992.0167%.Google Scholar
Shor, P. Algorithms for quantum computation: Discrete logarithms and factoring. In Proceedings 35th Annual Symposium on Foundations of Computer Science, 124–134 (IEEE Computer Society Press, Santa Fe, NM, 1994). http://ieeexplore.ieee.org/document/365700/.Google Scholar
Simon, D. R. On the power of quantum computation. SIAM Journal on Computing 26, 1474–1483 (1997). http://epubs.siam.org/doi/10.1137/S0097539796298637.CrossRefGoogle Scholar
Coppersmith, D. An approximate Fourier transform useful in quantum factoring (2002). http://arxiv.org/abs/quant-ph/0201067. ArXiv:quant-ph/0201067.Google Scholar
Ettinger, M., Hoyer, P. & Knill, E. The quantum query complexity of the hidden subgroup problem is polynomial. Information Processing Letters 91, 43–48 (2004). https://linkinghub.elsevier.com/retrieve/pii/S002001900400084%5.10.1016/j.ipl.2004.01.024CrossRefGoogle Scholar
Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Reviews of Modern Physics 74, 145–195 (2002). https://link.aps.org/doi/10.1103/RevModPhys.74.145.CrossRefGoogle Scholar
Jozsa, R. & Linden, N. On the role of entanglement in quantum-computational speed-up. Proceedings: Mathematical, Physical and Engineering Sciences 459, 2011–2032 (The Royal Society, London, 2003). https://www.jstor.org/stable/3560059.Google Scholar
Ekert, A., Jozsa, R., Penrose, R., Ekert, A. & Jozsa, R. Quantum algorithms: Entanglement-enhanced information processing. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 356, 1769–1782 (1998). https://royalsocietypublishing.org/doi/10.1098/rsta.1998.0248%.CrossRefGoogle Scholar
Markov, A. Extension of the limit theorems of probability theory to a sum of variables connected in a chain. In Howard, R. A., Dynamic Probabilistic Systems, Volume I: Markov Models (Courier Corporation, 2007/Wiley, Hoboken, NJ, 1971; reprinted in Appendix B).Google Scholar
Kraus, K., Bohm, A., Dollard, J. D. & Wootters, W. H. (eds.) States, effects, and operations: Fundamental notions of quantum theory, vol. 190 of Lecture Notes in Physics (Springer, Berlin, Heidelberg, 1983). http://link.springer.com/10.1007/3-540-12732-1.10.1007/3-540-12732-1CrossRefGoogle Scholar
Preskill, J. Lecture Notes for Physics 229: Quantum Information and Computation (CreateSpace Independent Publishing Platform, 2017/California Institute of Technology, 1998).Google Scholar
Stinespring, W. F. Positive functions on C*-algebras. Proceedings of the American Mathematical Society 6, 211 (1955). https://www.jstor.org/stable/2032342?origin=crossref.Google Scholar
Choi, M.-D. Completely positive linear maps on complex matrices. Linear Algebra and its Applications 10, 285–290 (1975). https://linkinghub.elsevier.com/retrieve/pii/0024379575900750%.10.1016/0024-3795(75)90075-0CrossRefGoogle Scholar
Jamiołkowski, A. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics 3, 275–278 (1972). https://linkinghub.elsevier.com/retrieve/pii/0034487772900110%.10.1016/0034-4877(72)90011-0CrossRefGoogle Scholar
Lindblad, G. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics 48, 119–130 (1976). http://link.springer.com/10.1007/BF01608499.10.1007/BF01608499CrossRefGoogle Scholar
Gorini, V., Kossakowski, A. & Sudarshan, E. C. G. Completely positive dynamical semigroups of N-level systems. Journal of Mathematical Physics 17, 821–825 (1976). https://pubs.aip.org/jmp/article/17/5/821/225427/Completely-p%ositive-dynamical-semigroups-of-N.10.1063/1.522979CrossRefGoogle Scholar
Aharonov, Y., Albert, D. Z. & Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Physical Review Letters 60, 1351–1354 (1988). https://link.aps.org/doi/10.1103/PhysRevLett.60.1351.CrossRefGoogle ScholarPubMed
Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N. & Boyd, R. W. Colloquium: Understanding quantum weak values: Basics and applications. Reviews of Modern Physics 86, 307–316 (2014). https://link.aps.org/doi/10.1103/RevModPhys.86.307.CrossRefGoogle Scholar
Peres, A. Reversible logic and quantum computers. Physical Review A 32, 3266–3276 (1985). https://link.aps.org/doi/10.1103/PhysRevA.32.3266.CrossRefGoogle ScholarPubMed
Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Physical Review A 52, R2493–R2496 (1995). https://link.aps.org/doi/10.1103/PhysRevA.52.R2493.CrossRefGoogle ScholarPubMed
Calderbank, A. R., Rains, E. M., Shor, P. W. & Sloane, N. J. A. Quantum error correction and orthogonal geometry. Physical Review Letters 78, 405–408 (1997). https://link.aps.org/doi/10.1103/PhysRevLett.78.405.CrossRefGoogle Scholar
Park, J. L. The concept of transition in quantum mechanics. Foundations of Physics 1, 23–33 (1970). http://link.springer.com/10.1007/BF00708652.10.1007/BF00708652CrossRefGoogle Scholar
Yamaguchi, K. & Kempf, A. Encrypted qubits can be cloned (2025). http://arxiv.org/abs/2501.02757. ArXiv:2501.02757 [quant-ph].Google Scholar
Buzek, V. & Hillery, M. Quantum copying: Beyond the no-cloning theorem. Physical Review A 54, 1844–1852 (1996). https://link.aps.org/doi/10.1103/PhysRevA.54.1844.CrossRefGoogle ScholarPubMed
Bruss, D., Ekert, A. & Macchiavello, C. Optimal universal quantum cloning and state estimation. Physical Review Letters 81, 2598–2601 (1998). https://link.aps.org/doi/10.1103/PhysRevLett.81.2598.CrossRefGoogle Scholar
Steane, A. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London 452, 2551–2577 (1996). https://royalsocietypublishing.org/doi/10.1098/rspa.1996.0136%.Google Scholar
Ekert, A. & Macchiavello, C. Error correction in quantum communication (1996). https://arxiv.org/abs/quant-ph/9602022. arXiv Version Number: 1.Google Scholar
Gottesman, D. Class of quantum error-correcting codes saturating the quantum Hamming bound. Physical Review A 54, 1862–1868 (1996). https://link.aps.org/doi/10.1103/PhysRevA.54.1862.CrossRefGoogle ScholarPubMed
Gottesman, D. The Heisenberg representation of quantum computers (1998). https://arxiv.org/abs/quant-ph/9807006. arXiv Version Number: 1.Google Scholar
Gallager, R. Low-density parity-check codes. IEEE Transactions on Information Theory 8, 21–28 (1962). http://ieeexplore.ieee.org/document/1057683/.10.1109/TIT.1962.1057683CrossRefGoogle Scholar
Gottesman, D., Kitaev, A. & Preskill, J. Encoding a qubit in an oscillator. Physical Review A 64, 012310 (2001). https://link.aps.org/doi/10.1103/PhysRevA.64.012310.CrossRefGoogle Scholar
Zanardi, P. & Rasetti, M. Noiseless quantum codes. Physical Review Letters 79, 3306–3309 (1997). https://link.aps.org/doi/10.1103/PhysRevLett.79.3306.CrossRefGoogle Scholar
Lidar, D. A., Chuang, I. L. & Whaley, K. B. Decoherence-free subspaces for quantum computation. Physical Review Letters 81, 2594–2597 (1998). https://link.aps.org/doi/10.1103/PhysRevLett.81.2594.CrossRefGoogle Scholar
Duan, L.-M. & Guo, G.-C. Preserving coherence in quantum computation by pairing quantum bits. Physical Review Letters 79, 1953–1956 (1997). https://link.aps.org/doi/10.1103/PhysRevLett.79.1953.CrossRefGoogle Scholar
Viola, L. & Lloyd, S. Dynamical suppression of decoherence in two-state quantum systems. Physical Review A 58, 2733–2744 (1998). https://link.aps.org/doi/10.1103/PhysRevA.58.2733.CrossRefGoogle Scholar
Wen, X. G. Mean-field theory of spin-liquid states with finite energy gap and topological orders. Physical Review B 44, 2664–2672 (1991). https://link.aps.org/doi/10.1103/PhysRevB.44.2664.CrossRefGoogle ScholarPubMed
Wen, X.-G. Topological orders and edge excitations in fractional quantum Hall states. Advances in Physics 44, 405–473 (1995). http://www.tandfonline.com/doi/abs/10.1080/00018739500101566.CrossRefGoogle Scholar
Landau, L. On the theory of phase transitions. In Collected Papers of L. D. Landau, 193–216 (Elsevier, 1965). https://linkinghub.elsevier.com/retrieve/pii/B978008010586450%0341.Google Scholar
Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45, 494–497 (1980). https://link.aps.org/doi/10.1103/PhysRevLett.45.494.CrossRefGoogle Scholar
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters 48, 1559–1562 (1982). https://link.aps.org/doi/10.1103/PhysRevLett.48.1559.CrossRefGoogle Scholar
Wen, X. G. Vacuum degeneracy of chiral spin states in compactified space. Physical Review B 40, 7387–7390 (1989). https://link.aps.org/doi/10.1103/PhysRevB.40.7387.CrossRefGoogle ScholarPubMed
Wen, X.-G. Topological orders in rigid states. International Journal of Modern Physics 04, 239–271 (1990). https://www.worldscientific.com/doi/abs/10.1142/S021797929000%0139.CrossRefGoogle Scholar
Laughlin, R. B. Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters 50, 1395–1398 (1983). https://link.aps.org/doi/10.1103/PhysRevLett.50.1395.CrossRefGoogle Scholar
Wu, Y.-S. General theory for quantum statistics in two dimensions. Physical Review Letters 52, 2103–2106 (1984). https://link.aps.org/doi/10.1103/PhysRevLett.52.2103.CrossRefGoogle Scholar
Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Physical Review Letters 53, 722–723 (1984). https://link.aps.org/doi/10.1103/PhysRevLett.53.722.CrossRefGoogle Scholar
Arovas, D. P., Schrieffer, R., Wilczek, F. & Zee, A. Statistical mechanics of anyons. Nuclear Physics B 251, 117–126 (1985). https://linkinghub.elsevier.com/retrieve/pii/0550321385902524%.10.1016/0550-3213(85)90252-4CrossRefGoogle Scholar
Haldane, F. D. M. Geometrical interpretation of momentum and crystal momentum of classical and quantum ferromagnetic Heisenberg chains. Physical Review Letters 57, 1488–1491 (1986). https://link.aps.org/doi/10.1103/PhysRevLett.57.1488.CrossRefGoogle ScholarPubMed
Haldane, F. D. M. & Rezayi, E. H. Spin-singlet wave function for the half-integral quantum Hall effect. Physical Review Letters 60, 956–959 (1988). https://link.aps.org/doi/10.1103/PhysRevLett.60.956.Google ScholarPubMed
Kohmoto, M. Topological invariant and the quantization of the Hall conductance. Annals of Physics 160, 343–354 (1985). https://www.sciencedirect.com/science/article/pii/00034916859%01484.10.1016/0003-4916(85)90148-4CrossRefGoogle Scholar
Hatsugai, Y. & Kohmoto, M. Energy spectrum and the quantum Hall effect on the square lattice with next-nearest-neighbor hopping. Physical Review B 42, 8282–8294 (1990). https://link.aps.org/doi/10.1103/PhysRevB.42.8282.CrossRefGoogle ScholarPubMed
Hung, L.-Y. & Wan, Y. String-net models with Z N fusion algebra. Physical Review B 86, 235132 (2012). https://link.aps.org/doi/10.1103/PhysRevB.86.235132.CrossRefGoogle Scholar
Hu, Y., Huang, Z., Hung, L.-Y. & Wan, Y. Anyon condensation: Coherent states, symmetry enriched topological phases, Goldstone theorem, and dynamical rearrangement of symmetry. Journal of High Energy Physics 2022, 26 (2022). https://link.springer.com/10.1007/JHEP03(2022)026.10.1007/JHEP03(2022)026CrossRefGoogle Scholar
Zhao, Y. & Wan, Y. Landau-Ginzburg Paradigm of Topological Phases. arXiv:2506.05319. https://doi.org/10.48550/arXiv.2506.05319.CrossRefGoogle Scholar
Zhao, Y. & Wan, Y. Noninvertible gauge symmetry in (2+1)d topological orders: A string-net model realization. Journal of High Energy Physics 2025, 138 (2025). https://doi.org/10.1007/JHEP11(2025)138.Google Scholar
Kitaev, A. & Preskill, J. Topological entanglement entropy. Physical Review Letters 96, 110404 (2006). https://link.aps.org/doi/10.1103/PhysRevLett.96.110404.CrossRefGoogle ScholarPubMed
Bravyi, S., Hastings, M. B. & Michalakis, S. Topological quantum order: Stability under local perturbations. Journal of Mathematical Physics 51, 093512 (2010). https://pubs.aip.org/jmp/article/51/9/093512/896702/Topologic%al-quantum-order-Stability-under-local.10.1063/1.3490195CrossRefGoogle Scholar
Fierz, M. Uber die relativistische Theorie kraftefreier Teilchen mit beliebigem Spin. Helvetica Physika Acta 12, 3–37 (1939). https://www.e-periodica.ch/digbib/view?pid=hpa-001:1939:12::6%30.Google Scholar
Pauli, W. The connection between spin and statistics. Physical Review 58, 716–722 (1940). https://link.aps.org/doi/10.1103/PhysRev.58.716.CrossRefGoogle Scholar
Freedman, M. H., Kitaev, A., Larsen, M. J. & Wang, Z. Topological quantum Computation (2001). https://arxiv.org/abs/quant-ph/0101025.Google Scholar
Kitaev, A. Anyons in an exactly solved model and beyond. Annals of Physics 321, 2–111 (2006). https://linkinghub.elsevier.com/retrieve/pii/S000349160500238%1.10.1016/j.aop.2005.10.005CrossRefGoogle Scholar
Reshetikhin, N. & Turaev, V. G. Invariants of 3-manifolds via link polynomials and quantum groups. Inventiones Mathematicae 103, 547–597 (1991). http://link.springer.com/10.1007/BF01239527.10.1007/BF01239527CrossRefGoogle Scholar
Chern, S.-S. & Simons, J. Characteristic forms and geometric invariants. The Annals of Mathematics 99, 48 (1974). https://www.jstor.org/stable/1971013?origin=crossref.10.2307/1971013CrossRefGoogle Scholar
Turaev, V. & Viro, O. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31, 865–902 (1992). https://linkinghub.elsevier.com/retrieve/pii/004093839290015A%.10.1016/0040-9383(92)90015-ACrossRefGoogle Scholar
Turaev, V. G. Quantum invariants of knots and 3-manifolds (De Gruyter, Berlin, 2016). https://www.degruyter.com/document/doi/10.1515/9783110435221/%html.CrossRefGoogle Scholar
Dijkgraaf, R. & Witten, E. Topological gauge theories and group cohomology. Communications in Mathematical Physics 129, 393–429 (1990). http://link.springer.com/10.1007/BF02096988.10.1007/BF02096988CrossRefGoogle Scholar
Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. Journal of Mathematical Physics 43, 4452–4505 (2002). https://pubs.aip.org/jmp/article/43/9/4452/230976/Topological%-quantum-memory.10.1063/1.1499754CrossRefGoogle Scholar
Chen, X., Gu, Z.-C. & Wen, X.-G. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Physical Review B 82, 155138 (2010). https://link.aps.org/doi/10.1103/PhysRevB.82.155138.CrossRefGoogle Scholar
Dehn, M. Die Gruppe der Abbildungsklassen: Das arithmetische Feld auf Flachen. Acta Mathematica 69, 135–206 (1938). http://projecteuclid.org/euclid.acta/1485888214.10.1007/BF02547712CrossRefGoogle Scholar
Mignard, M. & Schauenburg, P. Modular categories are not determined by their modular data. Letters in Mathematical Physics 111, 60 (2021). https://link.springer.com/10.1007/s11005-021-01395-0.10.1007/s11005-021-01395-0CrossRefGoogle Scholar
Wen, X. & Wen, X.-G. Distinguish modular categories and 2+1D topological orders beyond modular data: Mapping class group of higher genus manifold (2019). https://arxiv.org/abs/1908.10381.Google Scholar
Raussendorf, R. & Harrington, J. Fault-tolerant quantum computation with high threshold in two dimensions. Physical Review Letters 98, 190504 (2007). https://link.aps.org/doi/10.1103/PhysRevLett.98.190504.CrossRefGoogle ScholarPubMed
Google Quantum, AI et al. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023). https://www.nature.com/articles/s41586-022-05434-1.10.1038/s41586-022-05434-1CrossRefGoogle Scholar
Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature (2023). https://www.nature.com/articles/s41586-023-06927-3.Google ScholarPubMed
Bombin, H. & Martin-Delgado, M. A. Topological quantum distillation. Physical Review Letters 97, 180501 (2006). https://link.aps.org/doi/10.1103/PhysRevLett.97.180501.CrossRefGoogle ScholarPubMed
Bombin, H. & Martin-Delgado, M. A. Topological computation without braiding. Physical Review Letters 98, 160502 (2007). https://link.aps.org/doi/10.1103/PhysRevLett.98.160502.CrossRefGoogle ScholarPubMed
Zhang, J., Wu, Y.-C. & Guo, G.-P. Facilitating practical fault-tolerant quantum computing based on color codes (2023). https://arxiv.org/abs/2309.05222. arXiv Version Number: 4.Google Scholar
Eastin, B. & Knill, E. Restrictions on transversal encoded quantum gate sets. Physical Review Letters 102, 110502 (2009). https://link.aps.org/doi/10.1103/PhysRevLett.102.110502.CrossRefGoogle ScholarPubMed
Berent, L., Burgholzer, L., Derks, P.-J. H. S., Eisert, J. & Wille, R. Decoding quantum color codes with MaxSAT (2023). https://arxiv.org/abs/2303.14237. arXiv Version Number: 2.Google Scholar
Freedman, M. H., Larsen, M. & Wang, Z. A modular functor which is universal for quantum computation. Communications in Mathematical Physics 227, 605–622 (2002). http://link.springer.com/10.1007/s002200200645.10.1007/s002200200645CrossRefGoogle Scholar
Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Il Nuovo Cimento B 37, 1–23 (1977). http://link.springer.com/10.1007/BF02727953.10.1007/BF02727953CrossRefGoogle Scholar
Wilczek, F. Magnetic flux, angular momentum, and statistics. Physical Review Letters 48, 1144–1146 (1982). https://link.aps.org/doi/10.1103/PhysRevLett.48.1144.CrossRefGoogle Scholar
Wilczek, F. Quantum mechanics of fractional-spin particles. Physical Review Letters 49, 957–959 (1982). https://link.aps.org/doi/10.1103/PhysRevLett.49.957.CrossRefGoogle Scholar
Wilczek, F. & Zee, A. Linking numbers, spin, and statistics of solitons. Physical Review Letters 51, 2250–2252 (1983). https://link.aps.org/doi/10.1103/PhysRevLett.51.2250.CrossRefGoogle Scholar
Wilczek, F. Fractional statistics and anyon superconductivity (World Scientific, Singapore, Teaneck, NJ, 1990).10.1142/0961CrossRefGoogle Scholar
Wu, Y.-S. Multiparticle quantum mechanics obeying fractional Statistics. Physical Review Letters 53, 111–114 (1984). https://link.aps.org/doi/10.1103/PhysRevLett.53.111.CrossRefGoogle Scholar
Wu, Y.-S. & Zee, A. Comments on the Hopf lagrangian and fractional statistics of solitons. Physics Letters B 147, 325–329 (1984). https://www.sciencedirect.com/science/article/pii/03702693849%01266.10.1016/0370-2693(84)90126-6CrossRefGoogle Scholar
Wu, Y.-S. Statistical distribution for generalized ideal gas of fractional-statistics particles. Physical Review Letters 73, 922–925 (1994). https://link.aps.org/doi/10.1103/PhysRevLett.73.922.CrossRefGoogle ScholarPubMed
Halperin, B. I. Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Physical Review Letters 52, 1583–1586 (1984). https://link.aps.org/doi/10.1103/PhysRevLett.52.1583.CrossRefGoogle Scholar
Shizuya, K. & Tamura, H. Anyon statistics and its variation with wavelength in Maxwell–Chern–Simons gauge theories. Physics Letters B 252, 412–416 (1990). https://www.sciencedirect.com/science/article/pii/03702693909%0561J.10.1016/0370-2693(90)90561-JCrossRefGoogle Scholar
Haldane, F. D. M. “Fractional statistics” in arbitrary dimensions: A generalization of the Pauli principle. Physical Review Letters 67, 937–940 (1991). https://link.aps.org/doi/10.1103/PhysRevLett.67.937.CrossRefGoogle ScholarPubMed
Nayak, C. & Wilczek, F. Exclusion statistics: Low-temperature properties, fluctuations, duality, and applications. Physical Review Letters 73, 2740–2743 (1994). https://link.aps.org/doi/10.1103/PhysRevLett.73.2740.CrossRefGoogle ScholarPubMed
Levin, M. A. & Wen, X.-G. String-net condensation: A physical mechanism for topological phases. Physical Review B 71, 045110 (2005). https://link.aps.org/doi/10.1103/PhysRevB.71.045110.CrossRefGoogle Scholar
Hu, Y., Wan, Y. & Wu, Y.-S. Boundary Hamiltonian theory for gapped topological orders. Chinese Physics Letters 34, 077103 (2017). https://iopscience.iop.org/article/10.1088/0256-307X/34/7/077%103.10.1088/0256-307X/34/7/077103CrossRefGoogle Scholar
Hu, Y., Luo, Z.-X., Pankovich, R., Wan, Y. & Wu, Y.-S. Boundary Hamiltonian theory for gapped topological phases on an open surface. Journal of High Energy Physics 2018, 134 (2018). http://link.springer.com/10.1007/JHEP01(2018)134.10.1007/JHEP01(2018)134CrossRefGoogle Scholar
Zhao, Y., Wang, H., Hu, Y. & Wan, Y. Symmetry fractionalized (irrationalized) fusion rules and two domain-wall Verlinde formulae (2023). https://arxiv.org/abs/2304.08475. arXiv Version Number: 2.Google Scholar
Hu, Y., Stirling, S. D. & Wu, Y.-S. Emergent exclusion statistics of quasiparticles in two-dimensional topological phases. Physical Review B 89, 115133 (2014). https://link.aps.org/doi/10.1103/PhysRevB.89.115133.CrossRefGoogle Scholar
Li, Y., Wang, H., Hu, Y. & Wan, Y. Anyonic exclusions statistics on surfaces with gapped boundaries. Journal of High Energy Physics 2019, 78 (2019). https://link.springer.com/10.1007/JHEP04(2019)078.10.1007/JHEP04(2019)078CrossRefGoogle Scholar
Artin, E. Theory of braids. The Annals of Mathematics 48, 101 (1947). https://www.jstor.org/stable/1969218?origin=crossref.10.2307/1969218CrossRefGoogle Scholar
Yang, C. N. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Physical Review Letters 19, 1312–1315 (1967). https://link.aps.org/doi/10.1103/PhysRevLett.19.1312.CrossRefGoogle Scholar
Baxter, R. J. Exactly solved models in statistical mechanics. In Integrable systems in statistical mechanics, vol. 1, 5–63 (Series on Advances in Statistical Mechanics; World Scientific, Singapore, 1985). http://www.worldscientific.com/doi/abs/10.1142/9789814415255_%0002.CrossRefGoogle Scholar
Kauffman, L. H. & Lomonaco, S. J. Chapter 14: Quantum computing and quantum topology. In Kauffman, L. & Lomonaco, S. J. (eds.), Mathematics of quantum computation and quantum technology (Chapman and Hall/CRC, 2007). https://www.taylorfrancis.com/books/9781584889007.10.1201/9781584889007CrossRefGoogle Scholar
Xu, H. & Wan, X. Constructing functional braids for low-leakage topological quantum computing. Physical Review A 78, 042325 (2008). https://link.aps.org/doi/10.1103/PhysRevA.78.042325.CrossRefGoogle Scholar
Field, B. & Simula, T. Introduction to topological quantum computation with non-Abelian anyons. Quantum Science and Technology 3, 045004 (2018). https://iopscience.iop.org/article/10.1088/2058-9565/aacad2.10.1088/2058-9565/aacad2CrossRefGoogle Scholar
Cui, S. X., Tian, K. T., Vasquez, J. F., Wang, Z. & Wong, H. M. The search for leakage-free entangling Fibonacci braiding gates. Journal of Physics A: Mathematical and Theoretical 52, 455301 (2019). https://iopscience.iop.org/article/10.1088/1751-8121/ab488e.10.1088/1751-8121/ab488eCrossRefGoogle Scholar
Zhang, Y.-H., Zheng, P.-L., Zhang, Y. & Deng, D.-L. Topological quantum compiling with reinforcement learning. Physical Review Letters 125, 170501 (2020). https://link.aps.org/doi/10.1103/PhysRevLett.125.170501.CrossRefGoogle ScholarPubMed
Bonesteel, N. E., Hormozi, L., Zikos, G. & Simon, S. H. Braid topologies for quantum computation. Physical Review Letters 95, 140503 (2005). https://link.aps.org/doi/10.1103/PhysRevLett.95.140503.CrossRefGoogle ScholarPubMed
Fan, Y.-a. et al. Experimental quantum simulation of a topologically protected Hadamard gate via braiding Fibonacci anyons. Innovation 4 (2023). https://www.cell.com/the-innovation/abstract/S2666-6758(23)00%108-X.Google ScholarPubMed

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  • Yidun Wan, Fudan University, Shanghai
  • Book: Quantum Computing Unveiled
  • Online publication: 27 February 2026
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  • Book: Quantum Computing Unveiled
  • Online publication: 27 February 2026
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