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Classification of Quantum Cellular Automata

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Abstract

There exists an index theory to classify strictly local quantum cellular automata in one dimension (Fidkowski et al. in Interacting invariants for Floquet phases of fermions in two dimensions, 2017. arXiv:1703.07360; Gross et al. in Commun Math Phys 310(2):419–454, 2012; Po et al. in Phys Rev B 96: 245116, 2017). We consider two classification questions. First, we study to what extent this index theory can be applied in higher dimensions via dimensional reduction, finding a classification by the first homology group of the manifold modulo torsion. Second, in two dimensions, we show that an extension of this index theory (including torsion) fully classifies quantum cellular automata, at least in the absence of fermionic degrees of freedom. This complete classification in one and two dimensions by index theory is not expected to extend to higher dimensions due to recent evidence of a nontrivial automaton in three dimensions (Haah et al. in Nontrivial quantum cellular automata in higher dimensions, 2018. arXiv:1812.01625). Finally, we discuss some group theoretical aspects of the classification of quantum cellular automata and consider these automata on higher dimensional real projective spaces.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Browder, W.: Structures on m\(\times \) r. Math. Proc. Camb. Philos. Soc. 61, 337–345 (1965)

    Article  ADS  Google Scholar 

  3. Brown, M.: Locally flat imbeddings of topological manifolds. Ann. Math. 75, 331–341 (1962)

    Article  MathSciNet  Google Scholar 

  4. Fidkowski, L., Po, H.C., Potter, A.C., Vishwanath, A.: Interacting invariants for Floquet phases of fermions in two dimensions (2017). arXiv:1703.07360

  5. Freedman, M.H., Haah, J., Hastings, M.B.: The group structure of quantum cellular automata (2019). arXiv:1910.07998

  6. Gross, D., Nesme, V., Vogts, H., Werner, R.F.: Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys. 310(2), 419–454 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  7. Haah, J., Fidkowski, L., Hastings, M.B.: Nontrivial quantum cellular automata in higher dimensions (2018). arXiv:1812.01625

  8. Hastings, M.B.: Classifying quantum phases with the Kirby torus trick. Phys. Rev. B 88, 16 (2013)

    Article  Google Scholar 

  9. Kitaev, A., Lebedev, V., Feigel’man, M.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22 (2009)

    Article  ADS  Google Scholar 

  10. Milnor, J.: On manifolds homeomorphic to the 7-sphere. Ann. Math. 64(2), 399–405 (1956)

    Article  MathSciNet  Google Scholar 

  11. Po, H.C., Fidkowski, L., Vishwanath, A., Potter, A.C.: Radical chiral floquet phases in a periodically driven Kitaev model and beyond. Phys. Rev. B 96, 245116 (2017)

    Article  ADS  Google Scholar 

  12. Selberg, A.: On discontinuous groups in higher-dimensional symmetric space. In: Contribution to Function Theory, Bombay (1960)

Download references

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Authors and Affiliations

  1. Microsoft Research, Station Q, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

    Michael Freedman & Matthew B. Hastings

  2. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

    Michael Freedman

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  1. Michael Freedman
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  2. Matthew B. Hastings
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Correspondence to Matthew B. Hastings.

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Communicated by H. T. Yau

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Freedman, M., Hastings, M.B. Classification of Quantum Cellular Automata. Commun. Math. Phys. 376, 1171–1222 (2020). https://doi.org/10.1007/s00220-020-03735-y

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  • DOI: https://doi.org/10.1007/s00220-020-03735-y

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