Quantum walking in curved spacetime: discrete metric

Pablo Arrighi1, Giuseppe Di Molfetta2, and Stefano Facchini3

1Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, and IXXI, Lyon, France
2Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France and Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, Dr. Moliner 50, 46100-Burjassot, Spain
3Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators-differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.

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[1] Andre Ahlbrecht, Andrea Alberti, Dieter Meschede, Volkher B Scholz, Albert H Werner, and Reinhard F Werner. Molecular binding in interacting quantum walks. New Journal of Physics, 14(7):073050, 2012. doi:https:/​/​doi.org/​10.1088/​1367-2630/​14/​7/​073050.

[2] Andre Ahlbrecht, Volkher B Scholz, and Albert H Werner. Disordered quantum walks in one lattice dimension. Journal of Mathematical Physics, 52(10):102201, 2011. doi:https:/​/​doi.org/​10.1063/​1.3643768.

[3] Jan Ambjørn, Jerzy Jurkiewicz, and Renate Loll. Emergence of a 4d world from causal quantum gravity. Physical review letters, 93(13):131301, 2004. doi:https:/​/​doi.org/​10.1103/​PhysRevLett.93.131301.

[4] Pablo Arnault and Fabrice Debbasch. Quantum walks and gravitational waves. Annals of Physics, 383:645 - 661, 2017. doi:https:/​/​doi.org/​10.1016/​j.aop.2017.04.003.

[5] Pablo Arnault, Giuseppe Di Molfetta, Marc Brachet, and Fabrice Debbasch. Quantum walks and non-abelian discrete gauge theory. Physical Review A, 94(1):012335, 2016. doi:https:/​/​doi.org/​10.1103/​PhysRevA.94.012335.

[6] Pablo Arrighi and Stefano Facchini. Decoupled quantum walks, models of the klein-gordon and wave equations. EPL (Europhysics Letters), 104(6):60004, 2013. doi:https:/​/​doi.org/​10.1209/​0295-5075/​104/​60004.

[7] Pablo Arrighi, Stefano Facchini, and Marcelo Forets. Quantum walking in curved spacetime. Quantum Information Processing, 15(8):3467-3486, Aug 2016. URL: https:/​/​doi.org/​10.1007/​s11128-016-1335-7.

[8] Pablo Arrighi and Stefano Facchini. Quantum walking in curved spacetime: (3 + 1) dimensions, and beyond. Quantum Info. Comput., 17(9-10):810-824, August 2017. URL: http:/​/​dl.acm.org/​citation.cfm?id=3179561.3179565.

[9] Pablo Arrighi, Vincent Nesme, and Marcelo Forets. The dirac equation as a quantum walk: higher dimensions, observational convergence. Journal of Physics A: Mathematical and Theoretical, 47(46):465302, 2014. doi:https:/​/​doi.org/​10.1088/​1751-8113/​47/​46/​465302.

[10] Jacob D. Bekenstein. Universal upper bound to entropy-to-energy ratio for bounded systems. Phys. Rev. D, 23:287-298, 1981. doi:https:/​/​doi.org/​10.1103/​PhysRevD.23.287.

[11] Iwo Bialynicki-Birula. Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D., 49(12):6920-6927, 1994. doi:https:/​/​doi.org/​10.1103/​PhysRevD.49.6920.

[12] Alessandro Bisio, Giacomo Mauro D'Ariano, and Alessandro Tosini. Quantum field as a quantum cellular automaton i: the dirac free evolution in one dimension. arXiv preprint arXiv:1212.2839, 2012. doi:https:/​/​doi.org/​10.1016/​j.aop.2014.12.016.

[13] C. Cedzich, T Rybár, AH Werner, A Alberti, M Genske, and RF Werner. Propagation of quantum walks in electric fields. Physical review letters, 111(16):160601, 2013. doi:https:/​/​doi.org/​10.1103/​PhysRevLett.111.160601.

[14] CM Chandrashekar, S. Banerjee, and R. Srikanth. Relationship between quantum walks and relativistic quantum mechanics. Phys. Rev. A., 81(6):62340, 2010. doi:https:/​/​doi.org/​10.1103/​PhysRevA.81.062340.

[15] CG De Oliveira and J Tiomno. Representations of dirac equation in general relativity. Il Nuovo Cimento, 24(4):672-687, 1962. doi:https:/​/​doi.org/​10.1007/​BF02816716.

[16] P. J. Dellar, D. Lapitski, S. Palpacelli, and S. Succi. Isotropy of three-dimensional quantum lattice boltzmann schemes. Phys. Rev. E, 83:046706, Apr 2011. doi:https:/​/​doi.org/​10.1103/​PhysRevE.83.046706.

[17] Giuseppe Di Molfetta and Fabrice Debbasch. Discrete-time quantum walks: Continuous limit and symmetries. Journal of Mathematical Physics, 53(12):123302-123302, 2012. doi:https:/​/​doi.org/​10.1063/​1.4764876.

[18] Giuseppe Di Molfetta, Marc Brachet, and Fabrice Debbasch. Quantum walks as massless dirac fermions in curved space-time. Physical Review A, 88(4):042301, 2013. doi:https:/​/​doi.org/​10.1103/​PhysRevA.88.042301.

[19] Giuseppe Di Molfetta, Marc Brachet, and Fabrice Debbasch. Quantum walks in artificial electric and gravitational fields. Physica A: Statistical Mechanics and its Applications, 397:157-168, 2014. doi:https:/​/​doi.org/​10.1016/​j.physa.2013.11.036.

[20] Giuseppe Di Molfetta and Armando Pérez. Quantum walks as simulators of neutrino oscillations in a vacuum and matter. New Journal of Physics, 18(10):103038, 2016. doi:https:/​/​doi.org/​10.1088/​1367-2630/​18/​10/​103038.

[21] Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21(6):467-488, 1982.

[22] Maximilian Genske, Wolfgang Alt, Andreas Steffen, Albert H Werner, Reinhard F Werner, Dieter Meschede, and Andrea Alberti. Electric quantum walks with individual atoms. Physical review letters, 110(19):190601, 2013. doi:https:/​/​doi.org/​10.1103/​PhysRevLett.110.190601.

[23] Alain Joye and Marco Merkli. Dynamical localization of quantum walks in random environments. Journal of Statistical Physics, 140(6):1-29, 2010. doi:https:/​/​doi.org/​10.1007/​s10955-010-0047-0.

[24] T. Konopka, F. Markopoulou, and L. Smolin. Quantum graphity. Arxiv preprint hep-th/​0611197, 2006.

[25] P. Love and B. Boghosian. From Dirac to Diffusion: decoherence in Quantum Lattice gases. Quantum Information Processing, 4(4):335-354, 2005. doi:https:/​/​doi.org/​10.1007/​s11128-005-7852-4.

[26] Ivan Márquez-Martín, Giuseppe Di Molfetta, and Armando Pérez. Fermion confinement via quantum walks in (2+ 1)-dimensional and (3+ 1)-dimensional space-time. Physical Review A, 95(4):042112, 2017. doi:https:/​/​doi.org/​10.1103/​PhysRevA.95.042112.

[27] David A. Meyer. From quantum cellular automata to quantum lattice gases. J. Stat. Phys, 85:551-574, 1996. doi:https:/​/​doi.org/​10.1142/​S0129183197000618.

[28] David A Meyer. Quantum lattice gases and their invariants. International Journal of Modern Physics C, 8(04):717-735, 1997.

[29] P Nicoletopoulous, J Orloff, et al. A two-dimensional model with discrete general coordinate-invariance. Physicalia Magazine, 12:265, 1990.

[30] Carlo Rovelli. Simple model for quantum general relativity from loop quantum gravity. In Journal of Physics: Conference Series, volume 314, page 012006. IOP Publishing, 2011. doi:https:/​/​doi.org/​10.1088/​1742-6596/​314/​1/​012006.

[31] Linda Sansoni, Fabio Sciarrino, Giuseppe Vallone, Paolo Mataloni, Andrea Crespi, Roberta Ramponi, and Roberto Osellame. Two-particle bosonic-fermionic quantum walk via integrated photonics. Phys. Rev. Lett., 108:010502, Jan 2012. doi:https:/​/​doi.org/​10.1103/​PhysRevLett.108.010502.

[32] A. M. Steane. Overhead and noise threshold of fault-tolerant quantum error correction. Phys. Rev. A., 68(4):042322, Oct 2003. doi:https:/​/​doi.org/​10.1103/​PhysRevA.68.042322.

[33] Frederick W Strauch. Relativistic quantum walks. Physical Review A, 73(5):054302, 2006. doi:https:/​/​doi.org/​10.1103/​PhysRevA.73.054302.

[34] Sauro Succi and Roberto Benzi. Lattice boltzmann equation for quantum mechanics. Physica D: Nonlinear Phenomena, 69(3):327-332, 1993.

[35] Salvador Elías Venegas-Andraca. Quantum walks: a comprehensive review. Quantum Information Processing, 11(5):1015-1106, 2012. doi:https:/​/​doi.org/​10.1007/​s11128-012-0432-5.

Cited by

[1] Pablo Arnault, Armando Pérez, Pablo Arrighi, and Terry Farrelly, "Discrete-time quantum walks as fermions of lattice gauge theory", arXiv:1807.08303 (2018).

[2] Arindam Mallick, Sanjoy Mandal, Anirban Karan, and C. M. Chandrashekar, "Simulating Dirac Hamiltonian in Curved Space-time by Split-step Quantum Walk", arXiv:1712.03911 (2017).

[3] Arindam Mallick, Sanjoy Mandal, Anirban Karan, and C M Chandrashekar, "Simulating Dirac Hamiltonian in curved space-time by split-step quantum walk", Journal of Physics Communications 3 1, 015012 (2019).

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