Experiments with Schrödinger Cellular Automata

Kees van Berkel; Jan de Graaf; Kees van Hee

doi:10.22331/q-2025-07-23-1811

Experiments with Schrödinger Cellular Automata

Kees van Berkel, Jan de Graaf, and Kees van Hee

TU Eindhoven, Dept. of Mathematics & Computer Science, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Published:	2025-07-23, volume 9, page 1811
Editor:	Leon Loveridge
Eprint:	arXiv:2406.08586v2
Doi:	https://doi.org/10.22331/q-2025-07-23-1811
Citation:	Quantum 9, 1811 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We derive a class of cellular automata for the Schrödinger Hamiltonian, including scalar and vector potentials. It is based on a multi-split of the Hamiltonian, resulting in a multi-step unitary evolution operator in discrete time and space. Experiments with one-dimensional automata offer quantitative insight in phase and group velocities, energy levels, related approximation errors, and the evolution of a time-dependent harmonic oscilator. The apparent effects of spatial waveform aliasing are intriguing. Interference experiments with two-dimensional automata include refraction, Davisson-Germer, Mach-Zehnder, single & double slit, and Aharonov-Bohm.

Featured image: The Aharonov-Bohm effect: a magnetic potential (shown by gray arrows) distorts a double-slit diffraction pattern. The potential is caused by a solenoid placed just behind the screen, between the two slits.

Experiments with Schrödinger Cellular Automata.

Popular summary

We derive a class of cellular automata for the Schrödinger Hamiltonian, including scalar and vector potentials. It is based on a multi-split of the Hamiltonian, resulting in a multi-step unitary evolution operator in discrete time and space. Experiments with one-dimensional automata offer quantitative insight in phase and group velocities, energy levels, related approximation errors, and the evolution of a time-dependent harmonic oscillator. The apparent effects of spatial waveform aliasing are intriguing. Interference experiments with two-dimensional automata include refraction, Davisson-Germer, Mach-Zehnder, single & double slit, and Aharonov-Bohm.

► BibTeX data

@article{vanBerkel2025experiments,
  doi = {10.22331/q-2025-07-23-1811},
  url = {https://doi.org/10.22331/q-2025-07-23-1811},
  title = {Experiments with {S}chr{\"{o}}dinger {C}ellular {A}utomata},
  author = {van Berkel, Kees and de Graaf, Jan and van Hee, Kees},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1811},
  month = jul,
  year = {2025}
}

► References

[1] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover Publications, 1983.

[2] Y. Aharonov and D. Bohm. Significance of electromagnetic potentials in the quantum theory. Phys. Rev., 115: 485–491, Aug 1959. 10.1103/PhysRev.115.485. URL https://link.aps.org/doi/10.1103/PhysRev.115.485.
https://doi.org/10.1103/PhysRev.115.485

[3] Yakir Aharonov, Liema Davidovich, and N. Zagury. Quantum random walks. Physical review. A, 48: 1687–1690, 09 1993. 10.1007/BFb0083545.
https://doi.org/10.1007/BFb0083545

[4] Andris Ambainis, Julia Kempe, and Alexander Rivosh. Coins make quantum walks faster. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '05, page 1099–1108, 2005. ISBN 0898715857.

[5] Pablo Arnault and Fabrice Debbasch. Landau levels for discrete-time quantum walks in artificial magnetic fields. Physica A: Statistical Mechanics and its Applications, 443: 179–191, 2016. ISSN 0378-4371. https://doi.org/10.1016/j.physa.2015.08.011. URL https://www.sciencedirect.com/science/article/pii/S0378437115006664.
https://doi.org/10.1016/j.physa.2015.08.011
https://www.sciencedirect.com/science/article/pii/S0378437115006664

[6] P. Arrighi. An overview of quantum cellular automata. Natural Computing, 18 (4): 885–899, 2019. 10.1007/s11047-019-09762-6. URL https://doi.org/10.1007/s11047-019-09762-6.
https://doi.org/10.1007/s11047-019-09762-6

[7] 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, nov 2014. 10.1088/1751-8113/47/46/465302. URL https://dx.doi.org/10.1088/1751-8113/47/46/465302.
https://doi.org/10.1088/1751-8113/47/46/465302

[8] Iwo Bialynicki-Birula. Weyl, dirac, and maxwell equations on a lattice as unitary cellular automata. Physical review D: Particles and fields, 49: 6920–6927, 07 1994. 10.1103/PhysRevD.49.6920.
https://doi.org/10.1103/PhysRevD.49.6920

[9] Alessandro Bisio, Giacomo Mauro D'Ariano, and Alessandro Tosini. Dirac quantum cellular automaton in one dimension: $\mathit{Zitterbewegung}$ and scattering from potential. Phys. Rev. A, 88: 032301, Sep 2013. 10.1103/PhysRevA.88.032301. URL https://link.aps.org/doi/10.1103/PhysRevA.88.032301.
https://doi.org/10.1103/PhysRevA.88.032301

[10] Alessandro Bisio, Giacomo Mauro D’Ariano, and Alessandro Tosini. Quantum field as a quantum cellular automaton: The dirac free evolution in one dimension. Annals of Physics, 354: 244–264, 2015. ISSN 0003-4916. https://doi.org/10.1016/j.aop.2014.12.016. URL https://www.sciencedirect.com/science/article/pii/S0003491614003546.
https://doi.org/10.1016/j.aop.2014.12.016
https://www.sciencedirect.com/science/article/pii/S0003491614003546

[11] Bruce M. Boghosian and Washington Taylor. Simulating quantum mechanics on a quantum computer. Physica D: Nonlinear Phenomena, 120 (1): 30–42, 1998. ISSN 0167-2789. https://doi.org/10.1016/S0167-2789(98)00042-6. URL https://www.sciencedirect.com/science/article/pii/S0167278998000426. Proceedings of the Fourth Workshop on Physics and Consumption.
https://doi.org/10.1016/S0167-2789(98)00042-6
https://www.sciencedirect.com/science/article/pii/S0167278998000426

[12] Luis A. Bru, Germán J. de Valcárcel, Giuseppe Di Molfetta, Armando Pérez, Eugenio Roldán, and Fernando Silva. Quantum walk on a cylinder. Phys. Rev. A, 94: 032328, Sep 2016. 10.1103/PhysRevA.94.032328. URL https://link.aps.org/doi/10.1103/PhysRevA.94.032328.
https://doi.org/10.1103/PhysRevA.94.032328

[13] C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner. Quantum walks in external gauge fields. Journal of Mathematical Physics, 60 (1): 012107, 01 2019. ISSN 0022-2488. 10.1063/1.5054894. URL https://doi.org/10.1063/1.5054894.
https://doi.org/10.1063/1.5054894

[14] Pedro C. S. Costa, Renato Portugal, and Fernando de Melo. Quantum walks via quantum cellular automata. Quantum Information Processing, 17 (17:226), 2018.

[15] Pedro C.S. Costa. Quantum-to-classical transition via quantum cellular automata. Quantum Information Processing, 20 (7), 2021. ISSN 1573-1332. 10.1007/s11128-021-03175-0. URL http://dx.doi.org/10.1007/s11128-015-1149-z.
https://doi.org/10.1007/s11128-021-03175-0

[16] J. Crank and P. Nicolson. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Advances in Computational Math., 6: 207–226, 1947. URL https://api.semanticscholar.org/CorpusID:16676040.
https://api.semanticscholar.org/CorpusID:16676040

[17] Giacomo Mauro D'Ariano. The quantum field as a quantum computer. Physics Letters A, 376 (5): 697–702, 2012. ISSN 0375-9601. https://doi.org/10.1016/j.physleta.2011.12.021. URL https://www.sciencedirect.com/science/article/pii/S0375960111014836.
https://doi.org/10.1016/j.physleta.2011.12.021
https://www.sciencedirect.com/science/article/pii/S0375960111014836

[18] C. Davisson and L. H. Germer. Diffraction of electrons by a crystal of nickel. Phys. Rev., 30: 705–740, Dec 1927. 10.1103/PhysRev.30.705. URL https://link.aps.org/doi/10.1103/PhysRev.30.705.
https://doi.org/10.1103/PhysRev.30.705

[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. ISSN 0378-4371. https://doi.org/10.1016/j.physa.2013.11.036. URL https://www.sciencedirect.com/science/article/pii/S0378437113011059.
https://doi.org/10.1016/j.physa.2013.11.036
https://www.sciencedirect.com/science/article/pii/S0378437113011059

[20] Giacomo Mauro D’Ariano, Nicola Mosco, Paolo Perinotti, and Alessandro Tosini. Discrete time dirac quantum walk in 3+1 dimensions. Entropy, 18 (6), 2016. ISSN 1099-4300. URL.
https://doi.org/10.3390/e18060228

[21] Terry Farrelly. A review of Quantum Cellular Automata. Quantum, 4: 368, November 2020. ISSN 2521-327X. 10.22331/q-2020-11-30-368. URL https://doi.org/10.22331/q-2020-11-30-368.
https://doi.org/10.22331/q-2020-11-30-368

[22] Richard P. Feynman, Robert B. Leighton, and Matthew Sands. The Feynman lectures on physics: The Definitive Edition (Vol. 2 & 3). Pearson, 2009. ISBN 9788131721698. URL http://www.worldcat.org/isbn/9788131721704.
http://www.worldcat.org/isbn/9788131721704

[23] Gerard 't Hooft. The Cellular Automaton Interpretation of Quantum Mechanics. Springer, 2015.

[24] David J. Griffiths and Darrell F. Schroeter. Introduction to quantum mechanics. Cambridge University Press, Cambridge ; New York, NY, third edition edition, 2018. ISBN 978-1-107-18963-8.

[25] Gerhard Grössing and Anton Zeilinger. Quantum cellular automata. Complex Syst., 2 (2): 197–208, apr 1988. ISSN 0891-2513.

[26] Yang Ji, Yunchul Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and Hadas Shtrikman. An electronic mach–zehnder interferometer. Nature, 422 (6930): 415–418, March 2003. ISSN 1476-4687. 10.1038/nature01503. URL http://dx.doi.org/10.1038/nature01503.
https://doi.org/10.1038/nature01503

[27] Jolly, Nicolas and Di Molfetta, Giuseppe. Twisted quantum walks, generalised dirac equation and fermion doubling. Eur. Phys. J. D, 77 (5): 80, 2023. 10.1140/epjd/s10053-023-00659-9. URL https://doi.org/10.1140/epjd/s10053-023-00659-9.
https://doi.org/10.1140/epjd/s10053-023-00659-9

[28] T. Kiss, J. Janszky, and P. Adam. Time evolution of harmonic oscillators with time-dependent parameters: A step-function approximation. Phys. Rev. A, 49: 4935–4942, Jun 1994. 10.1103/PhysRevA.49.4935. URL https://link.aps.org/doi/10.1103/PhysRevA.49.4935.
https://doi.org/10.1103/PhysRevA.49.4935

[29] Iván Márquez-Martín, Pablo Arnault, Di Molfetta Giuseppe, and Armando Pérez. Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks. Physical Review A, 98 (3): 032333, September 2018. 10.1103/PhysRevA.98.032333. URL https://amu.hal.science/hal-03594739.
https://doi.org/10.1103/PhysRevA.98.032333
https://amu.hal.science/hal-03594739

[30] Daniel Martínez-Tibaduiza, Luis Pires, and Carlos Farina. Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes. Journal of Physics B: Atomic, Molecular and Optical Physics, 54 (20): 205401, nov 2021. 10.1088/1361-6455/ac36ba. URL https://dx.doi.org/10.1088/1361-6455/ac36ba.
https://doi.org/10.1088/1361-6455/ac36ba

[31] Mena, A. Solving the 2D Schrödinger equation using the Crank-Nicolson method. https://artmenlope.github.io/solving-the-2d-schrodinger-equation-using-the-crank-nicolson-method/, 2023. [Online; accessed 23-Mar-2024].
https://artmenlope.github.io/solving-the-2d-schrodinger-equation-using-the-crank-nicolson-method/

[32] P. G. Merli, G. F. Missiroli, and G. Pozzi. On the statistical aspect of electron interference phenomena. American Journal of Physics, 44 (3): 306–307, March 1976. 10.1119/1.10184.
https://doi.org/10.1119/1.10184

[33] David A. Meyer. From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics, 85 (5–6): 551–574, December 1996. ISSN 1572-9613. 10.1007/bf02199356. URL http://dx.doi.org/10.1007/BF02199356.
https://doi.org/10.1007/bf02199356

[34] Mauro E. S. Morales, Pedro C. S. Costa, Giacomo Pantaleoni, Daniel K. Burgarth, Yuval R. Sanders, and Dominic W. Berry. Greatly improved higher-order product formulae for quantum simulation, 2024. URL.
https://doi.org/10.2478/qic-2025-0001

[35] Amanda C. Oliveira, Renato Portugal, and Raul Donangelo. Simulation of the single- and double-slit experiments with quantum walkers, 2007. URL https://arxiv.org/abs/0706.3181.
arXiv:0706.3181

[36] R. Portugal, R. A. M. Santos, T. D. Fernandes, and D. N. Gonçalves. The staggered quantum walk model. Quantum Information Processing, 15 (1): 85–101, October 2015. ISSN 1573-1332. 10.1007/s11128-015-1149-z. URL http://dx.doi.org/10.1007/s11128-015-1149-z.
https://doi.org/10.1007/s11128-015-1149-z

[37] A. T. Sornborger and E. D. Stewart. Higher-order methods for simulations on quantum computers. Phys. Rev. A, 60: 1956–1965, Sep 1999. 10.1103/PhysRevA.60.1956. URL https://link.aps.org/doi/10.1103/PhysRevA.60.1956.
https://doi.org/10.1103/PhysRevA.60.1956

[38] Frederick W. Strauch. Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A, 74: 030301, Sep 2006. 10.1103/PhysRevA.74.030301. URL https://link.aps.org/doi/10.1103/PhysRevA.74.030301.
https://doi.org/10.1103/PhysRevA.74.030301

[39] Gerard 't Hooft. Quantization of discrete deterministic theories by hilbert space extension. Nuclear Physics B, 342 (3): 471—485, 1990.

[40] Toffoli, Tommaso and Margolus, Norman. Cellular Automata Machines: A New Environment for Modeling. MIT Press, 04 1987. ISBN 978-0-262-20060-8. 10.7551/mitpress/1763.001.0001.
https://doi.org/10.7551/mitpress/1763.001.0001

[41] Akira Tonomura, Tsuyoshi Matsuda, Ryo Suzuki, Akira Fukuhara, Nobuyuki Osakabe, Hiroshi Umezaki, Junji Endo, Kohsei Shinagawa, Yutaka Sugita, and Hideo Fujiwara. Observation of aharonov-bohm effect by electron holography. Phys. Rev. Lett., 48: 1443–1446, May 1982. 10.1103/PhysRevLett.48.1443. URL https://link.aps.org/doi/10.1103/PhysRevLett.48.1443.
https://doi.org/10.1103/PhysRevLett.48.1443

[42] Kees van Berkel. UCAlab. https://github.com/kees-van-berkel/UCAlab, 2025.
https://github.com/kees-van-berkel/UCAlab

[43] Kees van Berkel, Jan de Graaf, and Kees van Hee. Experiments with Schrodinger Cellular Automata. https://www.youtube.com/watch?v=tDiOHPf98ic, 2025.
https://www.youtube.com/watch?v=tDiOHPf98ic

[44] Wikipedia contributors. Fraunhofer diffraction equation, 2025. URL https://en.wikipedia.org/wiki/Fraunhofer_diffraction_equation. [Online; accessed 27-Jan-2025].
https://en.wikipedia.org/wiki/Fraunhofer_diffraction_equation

Cited by

[1] Kees van Hee and Kees van Berkel, Lecture Notes in Computer Science 16480, 262 (2026) ISBN:978-3-032-17617-2.

[2] Leonardo Lavagna, Sandra Carillo, and Massimo Panella, "A topical review on time-independent perturbation theory in one-dimensional quantum systems", Physica Scripta 100 10, 102001 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 06:16:31) and SAO/NASA ADS (last updated successfully 2026-08-11 15:45:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-12 06:16:31: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.