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Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation (2405.09277v5)

Published 15 May 2024 in quant-ph, cond-mat.str-el, hep-th, math-ph, math.MP, and math.QA

Abstract: Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

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References (53)
  1. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).
  2. J. Preskill, “Lecture notes for physics 229: Quantum information and computation,”  (1998).
  3. D. E. Deutsch, “Quantum computational networks,” Proceedings of the royal society of London. A. mathematical and physical sciences 425, 73 (1989).
  4. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188 (2001).
  5. R. Raussendorf, D. E. Browne,  and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003), arXiv:quant-ph/0301052 [quant-ph] .
  6. M. A. Nielsen, “Cluster-state quantum computation,” Reports on Mathematical Physics 57, 147 (2006), arXiv:quant-ph/0504097 [quant-ph] .
  7. H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf,  and M. Van den Nest, “Measurement-based quantum computation,” Nature Physics 5, 19 (2009), arXiv:0910.1116 [quant-ph] .
  8. M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Nest,  and H.-J. Briegel, “Entanglement in graph states and its applications,”  (2006), arXiv:quant-ph/0602096 [quant-ph] .
  9. R. Qu, J. Wang, Z.-s. Li,  and Y.-r. Bao, “Encoding hypergraphs into quantum states,” Phys. Rev. A 87, 022311 (2013), arXiv:1211.3911 [quant-ph] .
  10. M. Rossi, M. Huber, D. Bruß,  and C. Macchiavello, “Quantum hypergraph states,” New Journal of Physics 15, 113022 (2013), arXiv:1211.5554 [quant-ph] .
  11. F. E. S. Steinhoff, C. Ritz, N. I. Miklin,  and O. Gühne, “Qudit hypergraph states,” Phys. Rev. A 95, 052340 (2017), arXiv:1612.06418 [quant-ph] .
  12. F.-L. Xiong, Y.-Z. Zhen, W.-F. Cao, K. Chen,  and Z.-B. Chen, “Qudit hypergraph states and their properties,” Phys. Rev. A 97, 012323 (2018), arXiv:1701.07733 [quant-ph] .
  13. S. X. Cui, N. Yu,  and B. Zeng, “Generalized graph states based on hadamard matrices,” Journal of Mathematical Physics 56, 072201 (2015), arXiv:1502.07195 [quant-ph] .
  14. C. G. Brell, “Generalized cluster states based on finite groups,” New Journal of Physics 17, 023029 (2015), arXiv:1408.6237 [quant-ph] .
  15. C. Fechisin, N. Tantivasadakarn,  and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state,”  (2023), arXiv:2312.09272 [cond-mat.str-el] .
  16. M. Walschaers, S. Sarkar, V. Parigi,  and N. Treps, “Tailoring non-gaussian continuous-variable graph states,” Phys. Rev. Lett. 121, 220501 (2018), arXiv:1804.09444 [quant-ph] .
  17. D. W. Moore, “Quantum hypergraph states in continuous variables,” Phys. Rev. A 100, 062301 (2019), arXiv:1909.03871 [quant-ph] .
  18. S. Y. Looi, L. Yu, V. Gheorghiu,  and R. B. Griffiths, “Quantum-error-correcting codes using qudit graph states,” Phys. Rev. A 78, 042303 (2008), arXiv:0712.1979 [quant-ph] .
  19. D. Markham and B. C. Sanders, “Graph states for quantum secret sharing,” Phys. Rev. A 78, 042309 (2008), arXiv:0808.1532 [quant-ph] .
  20. A. Keet, B. Fortescue, D. Markham,  and B. C. Sanders, “Quantum secret sharing with qudit graph states,” Phys. Rev. A 82, 062315 (2010), arXiv:1004.4619 [quant-ph] .
  21. W. Son, L. Amico,  and V. Vedral, “Topological order in 1d cluster state protected by symmetry,” Quantum Information Processing 11, 1961 (2012), arXiv:1111.7173 [quant-ph] .
  22. S. Seifnashri and S.-H. Shao, “Cluster state as a non-invertible symmetry protected topological phase,”  (2024), arXiv:2404.01369 [cond-mat.str-el] .
  23. A. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2 (2003), arXiv:quant-ph/9707021 [quant-ph] .
  24. V. V. Albert, D. Aasen, W. Xu, W. Ji, J. Alicea,  and J. Preskill, “Spin chains, defects, and quantum wires for the quantum-double edge,”  (2021), arXiv:2111.12096 [cond-mat.str-el] .
  25. R. Thorngren and Y. Wang, “Fusion category symmetry. part i. anomaly in-flow and gapped phases,” Journal of High Energy Physics 2024, 1 (2024), arXiv:1912.02817 [hep-th] .
  26. R. Thorngren and Y. Wang, “Fusion category symmetry ii: Categoriosities at c = 1 and beyond,”  (2021), arXiv:2106.12577 [hep-th] .
  27. K. Inamura, “On lattice models of gapped phases with fusion category symmetries,” Journal of High Energy Physics 2022, 1 (2022), arXiv:2110.12882 [cond-mat.str-el] .
  28. K. Inamura, “Fermionization of fusion category symmetries in 1+ 1 dimensions,” Journal of High Energy Physics 2023, 1 (2023), arXiv:2206.13159 [cond-mat.str-el] .
  29. Z. Jia, “Cluster symmetry-protected topological phases from hopf symmetries,” in preparation.
  30. O. Buerschaper, J. M. Mombelli, M. Christandl,  and M. Aguado, “A hierarchy of topological tensor network states,” Journal of Mathematical Physics 54, 012201 (2013a), arXiv:1007.5283 [cond-mat.str-el] .
  31. B. Yan, P. Chen,  and S. Cui, “Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras,” Journal of Physics A: Mathematical and Theoretical  (2022), arXiv:2105.08202 [cond-mat.str-el] .
  32. Z. Jia, D. Kaszlikowski,  and S. Tan, “Boundary and domain wall theories of 2d generalized quantum double model,” Journal of High Energy Physics 2023, 1 (2023a), arXiv:2207.03970 [quant-ph] .
  33. Z. Jia, S. Tan, D. Kaszlikowski,  and L. Chang, “On weak Hopf symmetry and weak Hopf quantum double model,” Communications in Mathematical Physics 402, 3045 (2023b), arXiv:2302.08131 [hep-th] .
  34. E. Abe, Hopf algebras, Cambridge Tracts in Mathematics, Vol. 74 (Cambridge University Press, 2004) pp. xii+284.
  35. T. W. Hungerford, Algebra, Vol. 73 (Springer Science & Business Media, 2012).
  36. V. G. Turaev, Quantum invariants of knots and 3-manifolds, Vol. 18 (De Gruyter, 2016) pp. xii+592.
  37. B. Bakalov and A. A. Kirillov, Lectures on tensor categories and modular functors, Vol. 21 (American Mathematical Soc., 2001).
  38. R. G. Larson and D. E. Radford, “Semisimple cosemisimple hopf algebras,” American Journal of Mathematics 110, 187 (1988).
  39. R. G. Larson, “Characters of hopf algebras,” Journal of algebra 17, 352 (1971).
  40. C. Cordova, T. T. Dumitrescu, K. Intriligator,  and S.-H. Shao, “Snowmass white paper: Generalized symmetries in quantum field theory and beyond,”  (2022), arXiv:2205.09545 [hep-th] .
  41. T. D. Brennan and S. Hong, “Introduction to generalized global symmetries in qft and particle physics,”  (2023), arXiv:2306.00912 [hep-ph] .
  42. J. McGreevy, “Generalized symmetries in condensed matter,” Annual Review of Condensed Matter Physics 14, 57 (2023), arXiv:2204.03045 [cond-mat.str-el] .
  43. R. Luo, Q.-R. Wang,  and Y.-N. Wang, “Lecture notes on generalized symmetries and applications,”  (2023), arXiv:2307.09215 [hep-th] .
  44. S.-H. Shao, “What’s done cannot be undone: Tasi lectures on non-invertible symmetries,”  (2024), arXiv:2308.00747 [hep-th] .
  45. L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. Gould, A. Platschorre,  and H. Tillim, “Lectures on generalized symmetries,” Physics Reports 1051, 1 (2024), lectures on generalized symmetries, arXiv:2307.07547 [hep-th] .
  46. A. F. Bais, B. J. Schroers,  and J. K. Slingerland, “Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory,” Journal of High Energy Physics 2003, 068 (2003), arXiv:hep-th/0205114 [hep-th] .
  47. O. Buerschaper, M. Christandl, L. Kong,  and M. Aguado, “Electric–magnetic duality of lattice systems with topological order,” Nuclear Physics B 876, 619 (2013b), arXiv:1006.5823 [cond-mat.str-el] .
  48. G. Böhm, F. Nill,  and K. Szlachányi, “Weak Hopf algebras: I. Integral theory and C∗superscript𝐶{C}^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-structure,” Journal of Algebra 221, 385 (1999), arXiv:math/9805116 [math.QA] .
  49. A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang,  and M. H. Freedman, “Interacting anyons in topological quantum liquids: The golden chain,” Phys. Rev. Lett. 98, 160409 (2007), arXiv:cond-mat/0612341 [cond-mat.str-el] .
  50. R. Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117 (2014), arXiv:1306.2164 [cond-mat.str-el] .
  51. 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. 93, 045003 (2021), arXiv:2011.12127 [quant-ph] .
  52. X.-G. Wen, “Quantum orders in an exact soluble model,” Phys. Rev. Lett. 90, 016803 (2003), arXiv:quant-ph/0205004 [quant-ph] .
  53. F. Girelli, P. K. Osei,  and A. Osumanu, “Semidual Kitaev lattice model and tensor network representation,” Journal of High Energy Physics 2021, 1 (2021), arXiv:1709.00522 [math.QA] .
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