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A Gauge-Invariant Reversible Cellular Automaton

  • Pablo Arrighi
  • Giuseppe Di Molfetta
  • Nathanaël Eon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10875)

Abstract

Gauge-invariance is a fundamental concept in physics—known to provide mathematical justifications for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts, directly in terms of Cellular Automata. More precisely, we describe a step-by-step gauging procedure to enforce local symmetries upon a given Cellular Automaton. We apply it to a simple Reversible Cellular Automaton for concreteness. From a Computer Science perspective, discretized gauge theories may be of use in numerical analysis, quantum simulation, fault-tolerant (quantum) computation. From a mathematical perspective, discreteness provides a simple yet rigorous route straight to the core concepts.

Notes

Acknowledgements

The authors would like to thank Cédric Bény, Thomas Krajewski, Terry Farrelly and Pablo Arnault, for very instructive conversations about gauge theories. This work was partially supported by the CNRS PEPS JCJC GQNet and the CNRS PEPS Défi InFinitTI “Lattice Quantum Simulation Theory” LaQuST.

References

  1. 1.
    Quigg, C.: Gauge Theories of the Strong, Weak, and Electromagnetic Interactions. Princeton University Press, Princeton (2013)CrossRefGoogle Scholar
  2. 2.
    Georgescu, I., Ashhab, S., Nori, F.: Quantum simulation. Rev. Mod. Phys. 86(1), 153 (2014)CrossRefGoogle Scholar
  3. 3.
    Hastings, W.K.: Monte carlo sampling methods using markov chains and their applications. Biometrika 57(1), 97–109 (1970)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Harao, M., Noguchi, S.: Fault tolerant cellular automata. J. Comput. Syst. Sci. 11(2), 171–185 (1975)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Toom, A.: Cellular automata with errors: problems for students of probability. In: Topics in Contemporary Probability and Its Applications, pp. 117–157 (1995)Google Scholar
  6. 6.
    Kitaev, A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80(3), 1083 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Toffoli, T., Margolus, N.: Cellular Automata Machine - A New Environment for Modelling. MIT Press, Cambridge (1987)MATHGoogle Scholar
  9. 9.
    Wolf-Gladrow, D.A.: Lattice Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. LNM, vol. 1725. Springer, Heidelberg (2000).  https://doi.org/10.1007/b72010CrossRefMATHGoogle Scholar
  10. 10.
    Arrighi, P., Facchini, S., Forets, M.: Discrete lorentz covariance for quantum walks and quantum cellular automata. New J. Phys. 16(9), 093007 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Formenti, E., Kari, J., Taati, S.: On the hierarchy of conservation laws in a cellular automaton. Nat. Comput. 10(4), 1275–1294 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Salo, V., Törmä, I.: Color blind cellular automata. In: Kari, J., Kutrib, M., Malcher, A. (eds.) AUTOMATA 2013. LNCS, vol. 8155, pp. 139–154. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40867-0_10CrossRefMATHGoogle Scholar
  13. 13.
    Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks in artificial electric and gravitational fields. Phys. A Stat. Mech. Appl. 397, 157–168 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Arnault, P., Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks and non-abelian discrete gauge theory. Phys. Rev. A 94(1), 012335 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Di Molfetta, G., Pérez, A.: Quantum walks as simulators of neutrino oscillations in a vacuum and matter. New J. Phys. 18(10), 103038 (2016)CrossRefGoogle Scholar
  16. 16.
    Willson, S.J.: Computing fractal dimensions for additive cellular automata. Phys. D 24, 190–206 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chandrasekharan, S., Wiese, U.J.: Quantum link models: a discrete approach to gauge theories. Nucl. Phys. B 492(1–2), 455–471 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rico, E., Pichler, T., Dalmonte, M., Zoller, P., Montangero, S.: Tensor networks for lattice gauge theories and atomic quantum simulation. Phys. Rev. Lett. 112(20), 201601 (2014)CrossRefGoogle Scholar
  19. 19.
    Silvi, P., Rico, E., Calarco, T., Montangero, S.: Lattice gauge tensor networks. New J. Phys. 16(10), 103015 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wegner, F.J.: Duality in generalized ising models and phase transitions without local order parameters. J. Math. Phys. 12(10), 22592272 (1971)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kornyak, V.V.: Discrete dynamics: gauge invariance and quantization. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 180–194. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-04103-7_17CrossRefGoogle Scholar
  22. 22.
    Arrighi, P., Nesme, V., Werner, R.: One-dimensional quantum cellular automata over finite, unbounded configurations. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 64–75. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-88282-4_8CrossRefMATHGoogle Scholar
  23. 23.
    Itzykson, C., Zuber, J.B.: Quantum Field Theory. Courier Corporation (2006)Google Scholar
  24. 24.
    Strocchi, F.: An Introduction to Non-Perturbative Foundations of Quantum Field Theory, vol. 158. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  • Pablo Arrighi
    • 1
    • 2
  • Giuseppe Di Molfetta
    • 1
    • 3
  • Nathanaël Eon
    • 1
    • 4
  1. 1.Aix-Marseille Univ, Université de Toulon, CNRS, LISMarseilleFrance
  2. 2.IXXILyonFrance
  3. 3.Departamento de Física Terica and IFICUniversidad de Valencia-CSICBurjassotSpain
  4. 4.École CentraleMarseilleFrance

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