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A quantum walk with both a continuous-time limit and a continuous-spacetime limit

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Abstract

Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a quantum walks or quantum cellular automata based) enjoying a relativistic continuous-spacetime limit. We provide a first example of a quantum simulation scheme that unifies both approaches. The proposed scheme supports both a continuous-time discrete-space limit, leading to lattice fermions, and a continuous-spacetime limit, leading to the Dirac equation. The transition between the two can be thought of as a general relativistic change of coordinates, pushed to an extreme. As an emergent by-product of this procedure, we obtain a Hamiltonian for lattice fermions in curved spacetime with synchronous coordinates.

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Data Availability Statement

No data sets were generated or analysed during the current study.

Notes

  1. Notice that \(H_L\) commutes with \(\sigma _x\), thus preserving the chiral symmetry w.r.t the components of the spinor \(\Psi = (\psi ^+,\psi ^-)^\intercal \). This will no longer be true when we introduce the mass, which notoriously breaks chirality.

  2. In order to recover a massless Hamiltonian commuting with \(\sigma _y\), we can choose the operator

    $$\begin{aligned} U'=\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad e^{-i \zeta }\sin \theta &{}\quad -\cos \theta &{}\quad 0\\ 0 &{}\quad \cos \theta &{}\quad -e^{i \zeta }\sin \theta &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 \end{pmatrix} \end{aligned}$$

References

  1. Ahlbrecht, A., Alberti, A., Meschede, D., Scholz, V.B., Werner, A.H., Werner, R.F.: Molecular binding in interacting quantum walks. New J. Phys. 14(7), 073050 (2012)

    Article  ADS  Google Scholar 

  2. Arnault, P., Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks and non-abelian discrete gauge theory. Phys. Rev. A 94(1), 012335 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  3. Arnault, P., Pérez, A., Arrighi, P., Farrelly, T.: Discrete-time quantum walks as fermions of lattice gauge theory. Phys. Rev. A 99, 032110 (2019)

    Article  ADS  Google Scholar 

  4. Arrighi, P., Patricot, C.: A note on the correspondence between qubit quantum operations and special relativity. J. Phys. A Math. Gen. 36(20), L287–L296 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  5. Arrighi, P., Grattage, J.: A quantum game of life. In: Second Symposium on Cellular Automata “Journées Automates Cellulaires” (JAC 2010), Turku, December 2010. TUCS Lecture Notes 13, pp. 31–42 (2010)

  6. Arrighi, P., Grattage, J.: Partitioned quantum cellular automata are intrinsically universal. Nat. Comput. 11, 13–22 (2012)

    Article  MathSciNet  Google Scholar 

  7. Arrighi, P., Facchini, F.: Quantum walking in curved spacetime: \((3+1)\) dimensions, and beyond. Quantum Inf. Comput. 17(9–10), 0810–0824 (2017). arXiv:1609.00305

    MathSciNet  Google Scholar 

  8. Arrighi, P., Nesme, V., Werner, R.: Unitarity plus causality implies localizability. J. Comput. Syst. Sci. 77, 372–378 (2010)

    Article  MathSciNet  Google Scholar 

  9. Arrighi, P., Nesme, V., Forets, M.: The dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A Math. Theor. 47(46), 465302 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  10. Arrighi, P., Facchini, S., Forets, M.: Quantum walking in curved spacetime. Quantum Inf. Process. 15, 3467–3486 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  11. Arrighi, P., Bény, C., Farrelly, T.: A quantum cellular automaton for one-dimensional qed. (2019). arXiv:1903.07007

  12. Bialynicki-Birula, I.: Weyl, dirac, and maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D 49, 6920–6927 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  13. Bibeau-Delisle, A., Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Doubly special relativity from quantum cellular automata. EPL (Europhys. Lett.) 109(5), 50003 (2015)

    Article  ADS  Google Scholar 

  14. Bisio, A., D’Ariano, G.M., Perinotti, P.: Quantum walks, weyl equation and the lorentz group. Found. Phys. 47(8), 1065–1076 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Thirring quantum cellular automaton. Phys. Rev. A 97(3), 032132 (2018)

    Article  ADS  Google Scholar 

  16. Cedzich, C., Geib, T., Werner, A.H., Werner, R.F.: Quantum walks in external gauge fields. (2018). arXiv:1808.10850v1

  17. Debbasch, F.: Action principles for quantum automata and lorentz invariance of discrete time quantum walks. ArXiv preprint arXiv:1806.02313 (2018)

  18. Di Molfetta, G., Pérez, A.: Quantum walks as simulators of neutrino oscillations in a vacuum and matter. New J. Phys. 18(10), 103038 (2016)

    Article  Google Scholar 

  19. Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks as massless Dirac fermions in curved space-time. Phys. Rev. A 88(4), 042301 (2013)

    Article  ADS  Google Scholar 

  20. Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks in artificial electric and gravitational fields. Physica A 397, 157–168 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  21. Eisert, J., Gross, D.: Supersonic quantum communication. Phys. Rev. Lett. 102(24), 240501 (2009)

    Article  ADS  Google Scholar 

  22. Feynman, H.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965). (Feynman relativistic chessboard)

    MATH  Google Scholar 

  23. Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  24. Feynman, R.P.: Quantum mechanical computers. Found. Phys. (Hist. Arch.) 16(6), 507–531 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  25. Genske, M., Alt, W., Steffen, A., Werner, A.H., Werner, R.F., Meschede, D., Alberti, A.: Electric quantum walks with individual atoms. Phys. Rev. Lett. 110(19), 190601 (2013)

    Article  ADS  Google Scholar 

  26. Georgescu, I.M., Ashhab, S., Nori, F.: Quantum simulation. Rev. Mod. Phys. 86(1), 153 (2014)

    Article  ADS  Google Scholar 

  27. Gerritsma, R., Kirchmair, G., Florian Zähringer, E., Solano, R.B., Roos, C.F.: Quantum simulation of the dirac equation. Nature 463(7277), 68–71 (2010)

    Article  ADS  Google Scholar 

  28. Jordan, S.P., Lee, K.S.M., Preskill, J.: Quantum algorithms for fermionic quantum field theories. (2014). arXiv:1404.7115v1

  29. Klco, N., Dumitrescu, E.F., McCaskey, A.J., Morris, T.D., Pooser, R.C., Sanz, M., Solano, E., Lougovski, P., Savage, M.J.: Quantum-classical computation of schwinger model dynamics using quantum computers. Phys. Rev. A 98, 032331 (2018)

    Article  ADS  Google Scholar 

  30. Klco, N., Dumitrescu, E.F., McCaskey, A.J., Morris, T.D., Pooser, R.C., Sanz, M., Solano, E., Lougovski, P., Savage, M.J.: Quantum-classical computation of schwinger model dynamics using quantum computers. Phys. Rev. A 98(3), 032331 (2018)

    Article  ADS  Google Scholar 

  31. Kogut, J., Susskind, L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11(2), 395 (1975)

    Article  ADS  Google Scholar 

  32. Mallick, A., Mandal, S., Karan, A., Chandrashekar, C.M.: Simulating Dirac Hamiltonian in curved space-time by split-step quantum walk. J. Phys. Commun. 3(1) (2019). https://doi.org/10.1088/2399-6528/aafe2f

  33. Márquez-Martín, I., Di Molfetta, G., Pérez, A.: Fermion confinement via quantum walks in \((2+ 1)\)-dimensional and \((3+ 1)\)-dimensional space-time. Phys. Rev. A 95(4), 042112 (2017)

    Article  ADS  Google Scholar 

  34. Martinez, E.A., Muschik, C.A., Schindler, P., Nigg, D., Erhard, A., Heyl, M., Hauke, P., Dalmonte, M., Monz, T., Zoller, P., et al.: Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Nature 534(7608), 516–519 (2016)

    Article  ADS  Google Scholar 

  35. Meyer, D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85(5–6), 551–574 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  36. Osborne, T.J.: Continuum limits of quantum lattice systems. (2019). arXiv:1901.06124v1

  37. Sansoni, L., Sciarrino, F., Vallone, G., Mataloni, P., Crespi, A., Ramponi, R., Osellame, R.: Two-particle bosonic–fermionic quantum walk via integrated photonics. Phys. Rev. Lett. 108, 010502 (2012)

    Article  ADS  Google Scholar 

  38. Schumacher, B., Werner, R.: Reversible quantum cellular automata. ArXiv preprint arXiv:quant-ph/0405174 (2004)

  39. Strauch, F.W.: Connecting the discrete-and continuous-time quantum walks. Phys. Rev. A 74(3), 030301 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  40. Succi, S., Benzi, R.: Lattice Boltzmann equation for quantum mechanics. Physica D 69(3), 327–332 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  41. Villegas, K.H., Esguerra, J.P.: Lattice gauge theory and gluon color-confinement in curved spacetime. Mod. Phys. Lett. A 30(05), 1550020 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  42. Yamamoto, A.: Lattice qcd in curved spacetimes. Phys. Rev. D 90(5), 054510 (2014)

    Article  ADS  Google Scholar 

  43. Zohar, E., Cirac, J.I., Reznik, B.: Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices. Rep. Prog. Phys. 79(1), 014401 (2015)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank Pablo Arnault and Cédric Bény for motivating discussions.

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Authors and Affiliations

  1. CNRS, LIS, Aix-Marseille University, Marseille, France

    Giuseppe Di Molfetta

  2. CNRS, LIS, and INRIA, ENS Paris-Saclay, LSV, Université Paris-Saclay, Aix-Marseille University, Marseille, France

    Pablo Arrighi

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  1. Giuseppe Di Molfetta
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  2. Pablo Arrighi
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GDM and PA contributed equally to the main results of the manuscript.

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Correspondence to Giuseppe Di Molfetta.

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Di Molfetta, G., Arrighi, P. A quantum walk with both a continuous-time limit and a continuous-spacetime limit. Quantum Inf Process 19, 47 (2020). https://doi.org/10.1007/s11128-019-2549-2

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