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An Index for Quantum Cellular Automata on Fusion Spin Chains

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Abstract

Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators of 2D topological codes. We show that for the fusion spin chains built from the fusion category \(\textbf{Fib}\), the index is a complete invariant for the group of QCA modulo finite depth circuits.

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Notes

  1. In some of these settings, other equivalence relations on QCA such as stable equivalence and blending are used.

  2. The condition on \(\alpha ^{-1}\) follows automatically from the condition on \(\alpha \) if we assume some version of Haag duality, see [47].

References

  1. Arano, Y., De Commer, K.: Torsion-freeness for fusion rings and tensor \({\rm C}^*\)-categories. J. Noncommut. Geom. 13(1), 35–58 (2019)

    Article  MathSciNet  Google Scholar 

  2. Aasen, D., Fendley, P., Roger, M.: Dualities and degeneracies, Topological defects on the lattice (2020)

  3. Aasen, D., Haah, J., Li, Z., Mong, R.: Measurement quantum cellular automata and anomalies in floquet codes (2023)

  4. Aasen, D., Mong, R., Fendley, P.: Topological defects on the lattice: I. The Ising model. J. Phys. A Math. Theor. 49(35), 354001 (2016)

    Article  MathSciNet  Google Scholar 

  5. Arrighi, P., Nesme, V., Werner, R.: Unitarity plus causality implies localizability. J. Comput. Syst. Sci. 77(2), 372–378 (2011)

    Article  MathSciNet  Google Scholar 

  6. Arrighi, P.: An overview of quantum cellular automata. Nat. Comput. 18(4), 885–899 (2019)

    Article  MathSciNet  CAS  Google Scholar 

  7. Aasen, D., Wang, Z., Hastings, M.: Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes. Phys. Rev. B 106, 085122085122 (2022)

    Article  ADS  Google Scholar 

  8. Buican, M., Gromov, A.: Anyonic chains, topological defects, and conformal field theory. Comm. Math. Phys. 356(3), 1017–1056 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  9. Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.: Tensor categories and endomorphisms of von Neumann algebras–with applications to quantum field theory. In: SpringerBriefs in Mathematical Physics, vol. 3. Springer, Cham (2015)

  10. Bultinck, N., Mariën, M., Williamson, D., Şahinoğlu, M.B., Haegeman, J., Verstraete, F.: Anyons and matrix product operator algebras. Ann. Phys. 378, 183–233 (2017)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  11. Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras. I. Integral theory and \(C^*\)-structure. J. Algebra 221(2), 385–438 (1999)

    Article  MathSciNet  Google Scholar 

  12. Bratteli, O., Robinson, D.: Operator algebras and quantum statistical mechanics. 1. In: Texts and Monographs in Physics. Springer-Verlag, New York, (2nd edn), 1987. \(C^\ast \)- and \(W^\ast \)-algebras, symmetry groups, decomposition of states

  13. Bratteli, O., Robinson, D.: Operator algebras and quantum statistical mechanics. 2. In: Texts and Monographs in Physics. Springer-Verlag, Berlin, (2nd edn), 1997. Equilibrium states. Models in quantum statistical mechanics

  14. Bratteli, O.: Inductive limits of finite dimensional \(C^{\ast } \)-algebras. Trans. Amer. Math. Soc. 171, 195–234 (1972)

    MathSciNet  Google Scholar 

  15. Chen, X., Zheng-Cheng, G., Wen, X.-G.: Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B 82, 155138 (2010)

    Article  ADS  Google Scholar 

  16. Chen, Q., Hernández Palomares, R., Jones, C., Penneys, D.: Q-system completion for \(\rm C^*\) 2-categories. J. Funct. Anal. 283(3), 59 (2022)

    Article  MathSciNet  Google Scholar 

  17. Ignacio Cirac, J., Perez-Garcia, D., Schuch, N., Verstraete, F.: Matrix product unitaries: structure, symmetries, and topological invariants. J. Stat. Mech. Theory Exp. 2017(8), 083105 (2017)

    Article  MathSciNet  Google Scholar 

  18. Effros, E.: Dimensions and \(C^{\ast } \)-algebras, volume 46. Conference board of the mathematical sciences, Washington (1981)

  19. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. In: Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence (2015)

  20. Evans, D., Kawahigashi, Y.: Subfactors and mathematical physics. Bull. Amer. Math. Soc. (N. S.), 1–24. electronically published on June 1, (to appear in print) (2023)

  21. Evans, D., Kawahigashi, Y.: Quantum symmetries on operator algebras. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1998)

  22. Elliott, G.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38(1), 29–44 (1976)

    Article  MathSciNet  Google Scholar 

  23. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. 162, 581–642 (2002)

    Article  MathSciNet  Google Scholar 

  24. Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. Quant. Topol. 1(3), 209–273 (2010)

    Article  MathSciNet  Google Scholar 

  25. Farrelly, T.: A review of quantum cellular automata. Quantum 4, 368 (2020)

    Article  Google Scholar 

  26. Freedman, M., Hastings, M.: Classification of quantum cellular automata. Commun. Math. Phys. 376, 06 (2020)

    Article  MathSciNet  Google Scholar 

  27. Freedman, M., Haah, J., Hastings, M.: The group structure of quantum cellular automata. Commun. Math. Phys. 389, 1277–1302 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  28. Fidkowski, L., Po, H.C., Potter, A., Vishwanath, A.: Interacting invariants for floquet phases of fermions in two dimensions. Phys. Rev. B 99, 085115 (2019)

    Article  ADS  CAS  Google Scholar 

  29. Feiguin, A., Trebst, S., Ludwig, A., Troyer, M., Kitaev, A., Wang, Z., Freedman, M.: Interacting anyons in topological quantum liquids: the golden chain. Phys. Rev. Lett. 98, 160409 (2007)

    Article  ADS  PubMed  Google Scholar 

  30. Gabbiani, F., Fröhlich, J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  31. Gross, D., Nesme, V., Vogts, H., Werner, R.: Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys. 310, 10 (2009)

    MathSciNet  Google Scholar 

  32. Garre-Rubio, J., Lootens, L., Molnár, A.: Classifying phases protected by matrix product operator symmetries using matrix product states. Quantum 7, 927 (2023)

    Article  Google Scholar 

  33. Gong, Z., Sünderhauf, C., Schuch, N., Ignacio Cirac, J.: Classification of matrix-product unitaries with symmetries. Phys. Rev. Lett. 124, 03 (2020)

    Article  MathSciNet  Google Scholar 

  34. Haah, J.: Clifford quantum cellular automata: trivial group in 2D and Witt group in 3D. J. Math. Phys. 62(9), 092202 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  35. Haah, J.: Topological phases of unitary dynamics: classification in Clifford category. arXiv:2205.09141, (2022)

  36. Haah, J.: Invertible subalgebras. Commun. Math. Phys. 403, 1–38 (2023)

    Article  MathSciNet  Google Scholar 

  37. Haah, J., Fidkowski, L., Hastings, M.: Nontrivial quantum cellular automata in higher dimensions. Commun. Math. Phys. 398, 469–540 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  38. Huang, T.-C., Lin, Y.-H., Ohmori, K., Tachikawa, Y., Tezuka, M.: Numerical evidence for a Haagerup conformal field theory. Phys. Rev. Lett. 128, 10 (2021)

    MathSciNet  Google Scholar 

  39. Hollands, S.: Anyonic chains-\(\alpha \)-induction-CFT-defects-subfactors. Commun. Math. Phys. 399(12), 1549–1621 (2022)

    ADS  MathSciNet  Google Scholar 

  40. Inamura, K.: On lattice models of gapped phases with fusion category symmetries. J. High Energy Phys. 2022, 03 (2022)

    Article  MathSciNet  Google Scholar 

  41. Jones, V., Morrison, S., Snyder, N.: The classification of subfactors of index at most 5. Bull. Amer. Math. Soc. (N.S.) 51(2), 277–327 (2014)

    Article  MathSciNet  Google Scholar 

  42. Jones, C., Naaijkens, P., Penneys, D., Wallick, D.: Local topological order and boundary algebras. arXiv: 2307.12552, (2023)

  43. Jones, V.: Index for subfactors. Invent. Math. 72, 1–26 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  44. Jones, V.: A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.) 12(1), 103–111 (1985)

    Article  MathSciNet  Google Scholar 

  45. Jones, V.: In and around the origin of quantum groups. In: Prospects in Mathematical Physics, volume 437 of Contemp. Math., pp. 101–126. Amer. Math. Soc., Providence, RI, (2007)

  46. Jones, V.: Planar algebras. N. Z. J. Math. 52, 1–107 (2021)

    Article  MathSciNet  Google Scholar 

  47. Jones, C.: DHR bimodules and symmetric quantum cellular automata. arXiv: 2304.00068, (2023)

  48. Vaughan, J., Vaikalathur, S. S.: Introduction to subfactors. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (1997)

  49. Kawahigashi, Y.: A remark on matrix product operator algebras, anyons and subfactors. Lett. Math. Phys. 110(6), 1113–1122 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  50. Kawahigashi, Y.: Two-dimensional topological order and operator algebras. Int. J. Mod. Phys. B 35(8), 2130003–2616 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  51. Kawahigashi, Y.: Projector matrix product operators, anyons and higher relative commutants of subfactors. Math. Ann. (2022)

  52. Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Comm. Math. Phys. 219(3), 631–669 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  53. Lootens, L., Delcamp, C., Ortiz, G., Verstraete, F.: Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners. PRX Quant. 4, 020357 (2023)

    Article  ADS  Google Scholar 

  54. Lootens, L., Delcamp, C., Verstraete, F.: Dualities in one-dimensional quantum lattice models: topological sectors. arXiv:2211.03777, 11 (2022)

  55. Lootens, L., Fuchs, J., Haegeman, J., Schweigert, C., Verstraete, F.: Matrix product operator symmetries and intertwiners in string-nets with domain walls. SciPost Phys. (2020)

  56. Longo, R., Roberts, J.: A theory of dimension. K-Theory 11(02), 103–159 (1997)

    Article  MathSciNet  Google Scholar 

  57. Molnar, A., Alarcón, A., Garre-Rubio, J., Schuch, N., Ignacio Cirac, J., Perez-Garcia, D.: Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states. arXiv:2204.05940, 04 (2022)

  58. Naaijkens, P.: Quantum spin systems on infinite lattices. Lecture Notes in Physics, vol. 933, Springer, Cham, 2017. A concise introduction

  59. Nill, F., Szlachányi, K.: Quantum chains of Hopf algebras with quantum double cosymmetry. Commun. Math. Phys. 187, 159–200 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  60. Neshveyev, S., Yamashita, M.: Categorically Morita equivalent compact quantum groups. Doc. Math. (2017)

  61. Ocneanu, A.: Quantized groups, string algebras and Galois theory for algebras. In: Operator algebras and applications, volume 136 of London Math. Soc. Lecture Note Ser., vol. 2, pp. 119–172. Cambridge Univ. Press, Cambridge, (1988)

  62. Po, H.C., Fidkowski, L., Morimoto, T., Potter, A.C., Vishwanath, A.: Chiral floquet phases of many-body localized bosons. Phys. Rev. X 6, 041070 (2016)

    Google Scholar 

  63. Popa, S.: Classification of subfactors: the reduction to commuting squares. Invent. Math. 101, 19–43 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  64. Popa, S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994)

    Article  MathSciNet  Google Scholar 

  65. Popa, S.: Symmetric enveloping algebras, amenability and AFD properties for subfactors. Math. Res. Lett. 1, 409–425 (1994)

    Article  MathSciNet  Google Scholar 

  66. Popa, S.: Classification of subfactors and their endomorphisms. In: Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, (1995)

  67. Pimsner, M., Popa, S.: Entropy and index for subfactors. Annales scientifiques de l’École Normale Supérieure 19(1), 57–106 (1986)

    Article  MathSciNet  Google Scholar 

  68. Rudner, M., Lindner, N., Berg, E., Levin, M.: Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013)

    CAS  Google Scholar 

  69. Shirley, W., Chen, Y.-A., Dua, A., Ellison, T., Tantivasadakarn, N., Williamson, D.: Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond. PRX Quant. 3, 08 (2022)

    Google Scholar 

  70. Schumacher, B., Werner, R.: Reversible quantum cellular automata. arXiv: Quantum Physics, (2004)

  71. Trebst, S., Troyer, M., Wang, Z., Ludwig, A.: A short introduction to fibonacci anyon models. Prog. Theor. Phys. Suppl. 176(06), 384–407 (2008)

    Article  ADS  CAS  Google Scholar 

  72. Tambara, D., Yamagami, S.: Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209(2), 692–707 (1998)

    Article  MathSciNet  Google Scholar 

  73. Verdon, D.: A covariant Stinespring theorem. arXiv:2108.09872, 08 (2021)

  74. Vanhove, R., Lootens, L., Damme, M., Wolf, R., Osborne, T., Haegeman, J., Verstraete, F.: Critical lattice model for a Haagerup conformal field theory. Phys. Rev. Lett. 128(06), 231602 (2022)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  75. Vogts, H.: Discrete Time Quantum Lattice Systems. PhD thesis, Technische Universität Braunschweig, (2009)

  76. Zeng, B., Chen, X., Zhou, D-L., Wen, X-G.: Quantum information meets quantum matter. London Mathematical Society Lecture Note Series, (1st edn). Springer, Berlin (2019)

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Acknowledgements

The authors would like to thank Dave Aasen, Dietmar Bisch, Jeongwan Haah, Andrew Schopieray and Dominic Williamson for helpful comments and conversations. The first author is supported by NSF Grant DMS- 2247202. The second author is supported by US ARO grant W911NF2310026.

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  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Corey Jones

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Junhwi Lim

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Jones, C., Lim, J. An Index for Quantum Cellular Automata on Fusion Spin Chains. Ann. Henri Poincaré (2024). https://doi.org/10.1007/s00023-024-01429-y

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