Natural Computing

, Volume 18, Issue 4, pp 885–899 | Cite as

An overview of quantum cellular automata

  • P. ArrighiEmail author


Quantum cellular automata are arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and universality results; and quantum simulation results.


Quantum walks Dynamical systems 



I was lucky to have, as regular co-authors, great researchers such as Pablo Arnault, Cédric Bény, Gilles Dowek, Giuseppe Di Molfetta, Stefano Facchini, Terry Farrelly, Marcelo Forets, Jon Grattage, Iván Márquez, Vincent Nesme, Armando Péres, Zizhu Wang, Reinhard Werner. I would like to thank Jarkko Kari and Grzegorz Rozenberg for inviting me to write this overview, a task which I had been postponing for too long.


  1. Ahlbrecht A, Scholz VB, Werner AH (2011) Disordered quantum walks in one lattice dimension. J Math Phys 52(10):102201MathSciNetzbMATHGoogle Scholar
  2. Ahlbrecht A, Alberti A, Meschede D, Scholz VB, Werner AH, Werner RF (2012) Molecular binding in interacting quantum walks. New J Phys 14(7):073050Google Scholar
  3. Ambainis A, Childs AM, Reichardt BW, Špalek R, Zhang S (2010) Any and-or formula of size n can be evaluated in time n\(^{\wedge }\)1/2+o(1) on a quantum computer. SIAM J Comput 39(6):2513–2530MathSciNetzbMATHGoogle Scholar
  4. Andreu A, Pablo A, Pablo A, Di Molfetta G, Iván M, Dirac PM (2019) Lindblad and telegraph equations. Manuscript, Open quantum walksGoogle Scholar
  5. Arnault P, Fabrice D (2017) Quantum walks and gravitational waves. Ann Phys 383:645–661MathSciNetzbMATHGoogle Scholar
  6. Arnault P, Di Molfetta G, Brachet M, Debbasch F (2016) Quantum walks and non-Abelian discrete gauge theory. Phys Rev A 94(1):012335MathSciNetGoogle Scholar
  7. Arnault P, Pérez A, Arrighi P, Farrelly T (2019) Discrete-time quantum walks as fermions of lattice Gauge theory. Phys Rev A 99:032110Google Scholar
  8. Arrighi P, Fargetton R (2007) Intrinsically universal one-dimensional quantum cellular automata. In: Proceedings of DCMGoogle Scholar
  9. Arrighi P, Grattage J (2010) A simple \(n\)-dimensional intrinsically universal quantum cellular automaton. Lang Autom Theory Appl 6031:70–81MathSciNetzbMATHGoogle Scholar
  10. Arrighi P, Dowek G (2010) On the completeness of quantum computation models. In: Programs, Proofs, Processes: 6th Conference on Computability in Europe, CIE, 2010, Ponta Delgada, Azores, Portugal, June 30–July 4, 2010, Proceedings, vol 6158, pp 21–30Google Scholar
  11. Arrighi P, Dowek G (2012) The physical Church–Turing thesis and the principles of quantum theory. Int J Found Comput Sci 23:1131–1145MathSciNetzbMATHGoogle Scholar
  12. Arrighi P, Grattage J (2010) A quantum game of life. In: Second symposium on cellular automata “Journées Automates Cellulaires” (JAC 2010), Turku, 2010. TUCS Lecture Notes, vol 13, pp 31–42Google Scholar
  13. Arrighi P, Nesme V (2010) The block neighborhood. In: TUCS (ed) Proceedings of JAC 2010, Turku, Finlande, pp 43–53Google Scholar
  14. Arrighi P, Nesme V (2011) A simple block representation of reversible cellular automata with time-symmetry. In: 17th international workshop on cellular automata and discrete complex systems, (AUTOMATA 2011), Santiago de ChileGoogle Scholar
  15. Arrighi P, Grattage J (2012a) Intrinsically universal \(n\)-dimensional quantum cellular automata. J Comput Syst Sci 78:1883–1898MathSciNetzbMATHGoogle Scholar
  16. Arrighi P, Grattage J (2012b) Partitioned quantum cellular automata are intrinsically universal. Nat Comput 11:13–22MathSciNetzbMATHGoogle Scholar
  17. Arrighi P, Facchini S (2013) Decoupled quantum walks, models of the klein-gordon and wave equations. EPL (Europhys Lett) 104(6):60004Google Scholar
  18. Arrighi P, Facchini F (2017) Quantum walking in curved spacetime: (3+1) dimensions, and beyond. Quantum Inf Comput 17(9–10):0810–0824 arXiv:1609.00305 MathSciNetGoogle Scholar
  19. Arrighi P, Martiel S (2017) Quantum causal graph dynamics. Phys Rev D 96(2):024026 arXiv:1607.06700 MathSciNetGoogle Scholar
  20. Arrighi P, Nesme V, Werner RF (2008) Quantum cellular automata over finite, unbounded configurations. In: Proceedings of LATA, Lecture Notes in Computer Science, vol 5196. Springer, Berlin, pp 64–75Google Scholar
  21. Arrighi P, Fargetton R, Wang Z (2009) Intrinsically universal one-dimensional quantum cellular automata in two flavours. Fundam Inform 21:1001–1035MathSciNetzbMATHGoogle Scholar
  22. Arrighi P, Nesme V, Werner R (2010) Unitarity plus causality implies localizability. J Comput Syst Sci 77:372–378MathSciNetzbMATHGoogle Scholar
  23. Arrighi P, Nesme V, Werner R (2011a) Unitarity plus causality implies localizability (full version). J Comput Syst Sci 77(2):372–378zbMATHGoogle Scholar
  24. Arrighi P, Nesme V, Werner RF (2011b) One-dimensional quantum cellular automata. IJUC 7(4):223–244zbMATHGoogle Scholar
  25. Arrighi P, Fargetton R, Nesme V, Thierry E (2011c) Applying causality principles to the axiomatization of Probabilistic Cellular Automata. In: Proceedings of CiE 2011, Sofia, June 2011, LNCS, vol 6735, pp 1–10zbMATHGoogle Scholar
  26. Arrighi P, Nesme V, Forets M (2014a) The dirac equation as a quantum walk: higher dimensions, observational convergence. J Phys A Math Theor 47(46):465302MathSciNetzbMATHGoogle Scholar
  27. Arrighi P, Stefano F, Marcelo F (2014b) Discrete lorentz covariance for quantum walks and quantum cellular automata. New J Phys 16(9):093007MathSciNetGoogle Scholar
  28. Arrighi P, Facchini S, Forets M (2016) Quantum walking in curved spacetime. Quantum Inf Process 15:3467–3486MathSciNetzbMATHGoogle Scholar
  29. Arrighi P, Bény C, Farrelly T. (2019) A quantum cellular automaton for one-dimensional qed. ArXiv preprint arXiv:1903.07007
  30. Avalle M, Genoni MG, Serafini A (2015) Quantum state transfer through noisy quantum cellular automata. J Phys A Math Theor 48(19):195304MathSciNetzbMATHGoogle Scholar
  31. Benjamin SC (2000) Schemes for parallel quantum computation without local control of qubits. Phys Rev A 61(2):020301Google Scholar
  32. Bialynicki-Birula I (1994) Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys Rev D 49(12):6920–6927MathSciNetGoogle Scholar
  33. Bibeau-Delisle A, Bisio A, D’Ariano GM, Perinotti P, Tosini A (2015) Doubly special relativity from quantum cellular automata. EPL (Europhys Lett) 109(5):50003Google Scholar
  34. Bisio A, D’Ariano GM, Tosini A (2012) Quantum field as a quantum cellular automaton i: the dirac free evolution in one dimension. ArXiv preprint arXiv:1212.2839
  35. Bisio A, D’Ariano GM, Perinotti P (2017) Quantum walks, Weyl equation and the Lorentz group. Found Phys 47(8):1065–1076MathSciNetzbMATHGoogle Scholar
  36. Bisio A, D’Ariano GM, Perinotti P, Tosini A (2018) Thirring quantum cellular automaton. Phys Rev A 97(3):032132Google Scholar
  37. Bloch I (2005) Ultracold quantum gases in optical lattices. Nat Phys 1(1):23–30Google Scholar
  38. Boghosian BM, Taylor W (1998) Quantum lattice-gas model for the many-particle Schrödinger equation in d-dimensions. Phys Rev E 57(1):54–66MathSciNetGoogle Scholar
  39. Bose S (2007) Quantum communication through spin chain dynamics: an introductory overview. Contemp Phys 48(1):13–30Google Scholar
  40. Bratteli O, Robinson D (1987) Operators algebras and quantum statistical mechanics. Springer, New YorkzbMATHGoogle Scholar
  41. Brennen GK, Williams JE (2003) Entanglement dynamics in one-dimensional quantum cellular automata. Phys Rev A 68(4):042311MathSciNetGoogle Scholar
  42. Cedzich C, Rybár T, Werner AH, Alberti A, Genske M, Werner RF (2013) Propagation of quantum walks in electric fields. Phys Rev Lett 111(16):160601Google Scholar
  43. Chandrashekar CM, Banerjee S, Srikanth R (2010) Relationship between quantum walks and relativistic quantum mechanics. Phys Rev A 81(6):62340Google Scholar
  44. Cirac JI, Perez-Garcia D, Schuch N, Verstraete F (2017) Matrix product unitaries: structure, symmetries, and topological invariants. J Stat Mech Theory Exp 8(2017):083105MathSciNetGoogle Scholar
  45. Debbasch F (2018) Action principles for quantum automata and lorentz invariance of discrete time quantum walks. ArXiv preprint arXiv:1806.02313
  46. Destri C, de Vega HJ (1987) Light cone lattice approach to fermionic theories in 2-d: the massive thirring model. Nucl Phys B 290:363MathSciNetGoogle Scholar
  47. di Molfetta G, Debbasch F (2012) Discrete-time quantum walks: continuous limit and symmetries. J Math Phys 53(12):123302–123302MathSciNetzbMATHGoogle Scholar
  48. Di Molfetta G, Pérez A (2016) Quantum walks as simulators of neutrino oscillations in a vacuum and matter. New J Phys 18(10):103038Google Scholar
  49. Di Molfetta G, Arrighi P (2019) A quantum walk with both a continuous-time discrete-space limit and a continuous spacetime limit. ManuscriptGoogle Scholar
  50. Di Molfetta G, Brachet M, Debbasch F (2014) Quantum walks in artificial electric and gravitational fields. Phys A Stat Mech Appl 397:157–168MathSciNetzbMATHGoogle Scholar
  51. Durand-Lose J (2001) Representing reversible cellular automata with reversible block cellular automata. Discret Math Theor Comput Sci 145:154zbMATHGoogle Scholar
  52. Dürr C, Santha M (1996) A decision procedure for unitary linear quantum cellular automata. In: Proceedings of the 37th IEEE symposium on foundations of computer science. IEEE, pp 38–45Google Scholar
  53. Dürr C, Le Thanh H, Santha M (1996) A decision procedure for well-formed linear quantum cellular automata. In: Proceedings of STACS 96, Lecture Notes in Computer Science. Springer, pp 281–292Google Scholar
  54. D’Ariano GM, Perinotti P (2013) Derivation of the Dirac equation from principles of information processing. Pre-print arXiv:1306.1934
  55. Eisert J, Gross D (2009) Supersonic quantum communication. Phys Rev Lett 102(24):240501Google Scholar
  56. Farrelly T (2019) A review of quantum cellular automaton (To appear on the arXiv)Google Scholar
  57. Farrelly TC, Short AJ (2014) Causal fermions in discrete space–time. Phys Rev A 89(1):012302Google Scholar
  58. Farrelly TC (2015) Insights from quantum information into fundamental physics. PhD thesis, University of Cambridge arXiv:1708.08897
  59. Feynman RP (1982) Simulating physics with computers. Int J Theor Phys 21(6):467–488MathSciNetGoogle Scholar
  60. Feynman RP (1986) Quantum mechanical computers. Found Phys (Hist Arch) 16(6):507–531MathSciNetGoogle Scholar
  61. Fitzsimons J, Twamley J (2006) Globally controlled quantum wires for perfect qubit transport, mirroring, and computing. Phys Rev Lett 97(9):90502Google Scholar
  62. Freedman M, Hastings MB (2019) Classification of quantum cellular automata. ArXiv preprint arXiv:1902.10285
  63. Gandy R (1980) Church’s thesis and principles for mechanisms. In: The Kleene Symposium, North-Holland Publishing Company, AmsterdamGoogle Scholar
  64. Genske M, Alt W, Steffen A, Werner AH, Werner RF, Meschede D, Alberti A (2013) Electric quantum walks with individual atoms. Phys Rev Lett 110(19):190601Google Scholar
  65. Gross D, Nesme V, Vogts H, Werner RF (2012) Index theory of one dimensional quantum walks and cellular automata. Commun Math Phys 310(2):419–454MathSciNetzbMATHGoogle Scholar
  66. Gu M, Weedbrook C, Perales A, Nielsen MA (2009) More really is different. Phys D Nonlinear Phenom 238(9–10):835–839MathSciNetzbMATHGoogle Scholar
  67. Gütschow J (2010) Entanglement generation of Clifford quantum cellular automata. Appl Phys B 98:623–633zbMATHGoogle Scholar
  68. Gütschow J, Uphoff S, Werner RF, Zimborás Z (2010) Time asymptotics and entanglement generation of Clifford quantum cellular automata. J Math Phys 51(1):015203MathSciNetzbMATHGoogle Scholar
  69. Gütschow J, Nesme V, Werner RF (2012) Self-similarity of cellular automata on abelian groups. J Cell Autom 7(2):83–113MathSciNetzbMATHGoogle Scholar
  70. Haah J (2019) Clifford quantum cellular automata: Trivial group in 2D and witt group in 3D. ArXiv preprint arXiv:1907.02075
  71. Haah J, Fidkowski L, Hastings MB (2018) Nontrivial quantum cellular automata in higher dimensions. ArXiv preprint arXiv:1812.01625
  72. Ibarra OH, Jiang T (1987) On the computing power of one-way cellular arrays. In: Proceedings of ICALP 87. Springer, London, pp 550–562Google Scholar
  73. Inokuchi S, Mizoguchi Y (2005) Generalized partitioned quantum cellular automata and quantization of classical CA. Int J Unconv Comput 1(2):149–160Google Scholar
  74. Joye A, Merkli M (2010) Dynamical localization of quantum walks in random environments. J Stat Phys 140(6):1–29MathSciNetzbMATHGoogle Scholar
  75. Karafyllidis IG (2004) Definition and evolution of quantum cellular automata with two qubits per cell. Phys Rev A 70:044301Google Scholar
  76. Kari J (1991) Reversibility of 2D cellular automata is undecidable. In: Cellular automata: theory and experiment, vol 45. MIT Press, pp 379–385Google Scholar
  77. Kari J (1996) Representation of reversible cellular automata with block permutations. Theory Comput Syst 29(1):47–61MathSciNetzbMATHGoogle Scholar
  78. Kari J (1999) On the circuit depth of structurally reversible cellular automata. Fundam Inform 38(1–2):93–107MathSciNetzbMATHGoogle Scholar
  79. Kari K (2005) Theory of cellular automata: a survey. Theor Comput Sci 334:2005MathSciNetGoogle Scholar
  80. Kieu TD (2003) Computing the non-computable. Contemp Phys 44(1):51–71Google Scholar
  81. Love P, Boghosian B (2005) From Dirac to diffusion: decoherence in quantum lattice gases. Quantum Inf Process 4(4):335–354zbMATHGoogle Scholar
  82. Mallick A, Chandrashekar CM (2016) Dirac cellular automaton from split-step quantum walk. Sci Rep 6:25779Google Scholar
  83. Mallick A, Mandal S, Karan A, Chandrashekar CM (2019) Simulating dirac hamiltonian in curved space-time by split-step quantum walk. J Phys Commun 3(1):015012Google Scholar
  84. Marcos D, Widmer P, Rico E, Hafezi M, Rabl P, Wiese U-J, Zoller P (2014) Two-dimensional lattice gauge theories with superconducting quantum circuits. Ann Phys 351:634–654MathSciNetzbMATHGoogle Scholar
  85. Mauro DAG, Franco M, Paolo P, Alessandro T (2014) The feynman problem and fermionic entanglement: fermionic theory versus qubit theory. Int J Mod Phys A 29(17):1430025MathSciNetzbMATHGoogle Scholar
  86. Mazoyer J (1987) A six-state minimal time solution to the firing squad synchronization problem. Theor Comput Sci 50:183–238MathSciNetzbMATHGoogle Scholar
  87. Meyer DA (1996) From quantum cellular automata to quantum lattice gases. J Stat Phys 85:551–574MathSciNetzbMATHGoogle Scholar
  88. Meyer DA, Shakeel A (2016) Quantum cellular automata without particles. Phys Rev A 93(1):012333Google Scholar
  89. Márquez-Martín I, Di Molfetta G, Pérez A (2017) Fermion confinement via quantum walks in (2+ 1)-dimensional and (3+ 1)-dimensional space-time. Phys Rev A 95(4):042112Google Scholar
  90. Nagaj D, Wocjan P (2008) Hamiltonian quantum cellular automata in one dimension. Phys Rev A 78(3):032311Google Scholar
  91. Nielsen MA (1997) Computable functions, quantum measurements, and quantum dynamics. Phys Rev Lett 79(15):2915–2918Google Scholar
  92. Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, CambridgezbMATHGoogle Scholar
  93. Paz JP, Zurek WH (2002) Environment-induced decoherence and the transition from quantum to classical. In: Fundamentals of quantum information, Lecture Notes in Physics. Springer, Berlin, pp 77–148Google Scholar
  94. Pérez-Delgado CA, Cheung D (2007) Local unreversible cellular automaton ableitary quantum cellular automata. Phys Rev A 76(3):32320MathSciNetGoogle Scholar
  95. Raussendorf R (2005) Quantum cellular automaton for universal quantum computation. Phys Rev A 72(2):22301MathSciNetGoogle Scholar
  96. Raynal P (2017) Simple derivation of the Weyl and Dirac quantum cellular automata. Phys Rev A 95:062344Google Scholar
  97. Robens C, Brakhane S, Meschede D, Alberti A (2017) Quantum walks with neutral atoms: quantum interference effects of one and two particles. In: Laser spectroscopy: XXII international conference on laser spectroscopy (ICOLS2015). World Scientific, pp 1–15Google Scholar
  98. Sansoni L, Sciarrino F, Vallone G, Mataloni P, Crespi A, Ramponi R, Osellame R (2012) Two-particle Bosonic-Fermionic quantum walk via integrated photonics. Phys Rev Lett 108:010502Google Scholar
  99. Schaeffer L (2015) A physically universal quantum cellular automaton. In: Jarkko K (ed) Cellular automata and discrete complex systems. Springer, Berlin, pp 46–58zbMATHGoogle Scholar
  100. Schlingemann DM, Vogts H, Werner RF (2008) On the structure of Clifford quantum cellular automata. J Math Phys 49:112104MathSciNetzbMATHGoogle Scholar
  101. Schumacher B, Werner R (2004) Reversible quantum cellular automata. arXiv pre-print quant-ph/0405174,Google Scholar
  102. Shakeel A (2019) The equivalence of Schrödinger and Heisenberg pictures in quantum cellular automata. arXiv:1807.01192
  103. Shakeel A, Love PJ (2013) When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)? J Math Phys 54(9):092203MathSciNetzbMATHGoogle Scholar
  104. Strauch FW (2006a) Connecting the discrete-and continuous-time quantum walks. Phys Rev A 74(3):030301MathSciNetGoogle Scholar
  105. Strauch FW (2006b) Relativistic quantum walks. Phys Rev A 73(5):054302MathSciNetGoogle Scholar
  106. Strauch FW (2007) Relativistic effects and rigorous limits for discrete-and continuous-time quantum walks. J Math Phys 48:082102MathSciNetzbMATHGoogle Scholar
  107. Subrahmanyam V (2004) Entanglement dynamics and quantum-state transport in spin chains. Phys Rev A 69:034304Google Scholar
  108. Subrahmanyam V, Lakshminarayan A (2006) Transport of entanglement through a Heisenberg-XY spin chain. Phys Lett A 349(1–4):164–169Google Scholar
  109. Succi S, Benzi R (1993) Lattice boltzmann equation for quantum mechanics. Phys D Nonlinear Phenom 69(3):327–332MathSciNetzbMATHGoogle Scholar
  110. t’Hooft G (2016) The cellular automaton interpretation of quantum mechanics, vol 185. Fundamental theories of physics. Springer, BerlinGoogle Scholar
  111. Twamley J (2003) Quantum cellular automata quantum computing with endohedral fullerenes. Phys Rev A 67(5):52318–52500MathSciNetGoogle Scholar
  112. Vallejo A, Romanelli A, Donangelo R (2018) Initial-state-dependent thermalization in open qubits. Phys Rev A 98(3):032319Google Scholar
  113. Van Dam W (1996) A Universal Quantum Cellular Automaton. In: Proceedings of PhysComp96, Inter Journal manuscript 91. New England Complex Systems Institute, pp 323–331Google Scholar
  114. Van Dam W (1996) Quantum cellular automata. Masters thesis, University of Nijmegen, The NetherlandsGoogle Scholar
  115. Vollbrecht KGH, Cirac JI (2006) Reversible universal quantum computation within translation-invariant systems. Phys Rev A 73(1):012324Google Scholar
  116. von Neumann J (1955) Mathematical foundations of quantum mechanics. Princeton University Press, PrincetonzbMATHGoogle Scholar
  117. von Neumann J (1966) Theory of self-reproducing automata. University of Illinois Press, ChampaignGoogle Scholar
  118. Wang G (2017) Efficient quantum algorithms for analyzing large sparse electrical networks. Quantum Inf Comput 17(11–12):987–1026MathSciNetGoogle Scholar
  119. Watrous J (1995) On one-dimensional quantum cellular automata. In: Annual IEEE symposium on foundations of computer science, pp 528–537Google Scholar
  120. Weinstein YS, Hellberg CS (2004) Quantum cellular automata pseudorandom maps. Phys Rev A 69(6):062301MathSciNetzbMATHGoogle Scholar
  121. Wiesner K (2008) Quantum cellular automata. ArXiv preprint arXiv:0808.0679

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Aix-Marseille Univ, CNRS, LISand INRIA, ENS Paris-Saclay, LSV, Université Paris-SaclayParis-SaclayFrance

Personalised recommendations