Towards a Quantum Game of Life

  • Adrian P. FlitneyEmail author
  • Derek Abbott


Cellular automata provide a means of obtaining complex behaviour from a simple array of cells and a deterministic transition function. They supply a method of computation that dispenses with the need for manipulation of individual cells and they are computationally universal. Classical cellular automata have proved of great interest to computer scientists but the construction of quantum cellular automata pose particular difficulties. We present a version of John Conway’s famous two-dimensional classical cellular automata Life that has some quantum-like features, including interference effects. Some basic structures in the new automata are given and comparisons are made with Conway’s game.


Cellular Automaton Quantum Walk Single Qubit Quantum Game Hadamard Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amoroso, S., Patt, Y.N.: Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. J. Comput. Syst. Sci. 6, 448–464 (1972) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Auon, B., Tarifi, M.: Introduction to quantum cellular automata. Eprint: arXiv:quant-ph/0401123 (2004)
  3. 3.
    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, J., Smolin J., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995) CrossRefGoogle Scholar
  4. 4.
    Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964) Google Scholar
  5. 5.
    Benjamin, S.C.: Schemes for parallel quantum computation without local control of qubits. Phys. Rev. A 61, 020301 (2000) CrossRefGoogle Scholar
  6. 6.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathematical Plays, vol. 2. Academic Press, London (1982) zbMATHGoogle Scholar
  7. 7.
    Dumke, R., Volk, M., Muether, T., Buchkremer, F.B.J., Birkl, G., Ertmer, W.: Microoptical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits. Phys. Rev. Lett. 89, 097903 (2002) CrossRefGoogle Scholar
  8. 8.
    Dürr, C., Santha, M.: A decision procedure for well-formed unitary linear quantum cellular automata. SIAM J. Comput. 31, 1076–1089 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935) zbMATHCrossRefGoogle Scholar
  10. 10.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gardner, M.: Mathematical games: The fantastic combinations of John Conway’s new solitaire game of “Life”. Sci. Am. 223(10), 120 (1970) CrossRefGoogle Scholar
  12. 12.
    Gardner, M.: Mathematical games: On cellular automata, self-reproduction, the Garden of Eden and the game of “Life”. Sci. Am. 224(2), 116 (1971) Google Scholar
  13. 13.
    Gardner, M.: Wheels, Life and Other Mathematical Amusements. Freeman, New York (1983) zbMATHGoogle Scholar
  14. 14.
    Grössing, G., Zeilinger, A.: Quantum cellular automata. Complex Syst. 2, 197–208 (1988) zbMATHGoogle Scholar
  15. 15.
    Grössing, G., Zeilinger, A.: Structures in quantum cellular automata. Physica B 151, 366–370 (1988) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gruska, J.: Quantum Computing. McGraw Hill, Maidenhead (1999) Google Scholar
  17. 17.
    Kari, J.: Reversibility of two-dimensional cellular automata is undecidable. Physica D 45, 379–385 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003) CrossRefGoogle Scholar
  19. 19.
    Konno, N.: Quantum Walks and Quantum Cellular Automata. Lecture Notes in Computer Science. Springer, Berlin/Heidelberg (2008) Google Scholar
  20. 20.
    Konno, N., Mistuda, K., Soshi, T., Yoo, H.J.: Quantum walks and reversible cellular automata. Phys. Lett. A 330, 408–417 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lloyd, S.: Obituary: Rolf Laundauer. Nature 400, 720 (1999) CrossRefGoogle Scholar
  22. 22.
    Mandel, D., Greiner, M., Widera, A., Rom, T., Hänsch, T.W., Bloch, I.: Coherent transport of neutral atoms in spin-dependent optical lattice potentials. Phys. Rev. Lett. 91, 010407 (2003) CrossRefGoogle Scholar
  23. 23.
    Meyer, D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996) zbMATHCrossRefGoogle Scholar
  24. 24.
    Morita, K.: Reversible simulation of one-dimensional irreversible cellular automata. Theor. Comput. Sci. 148, 157–163 (1995) zbMATHCrossRefGoogle Scholar
  25. 25.
    Morita, K., Harao, M.: Computation universality of one-dimensional reversible (injective) cellular automata. Trans. IEICE 72, 758–762 (1989) Google Scholar
  26. 26.
    Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) zbMATHGoogle Scholar
  27. 27.
    Schumacher, B., Werner, R.F.: Reversible quantum cellular automata. Eprint: arXiv:quant-ph/0405174 (2004)
  28. 28.
  29. 29.
    Toffoli, T.: Cellular automata mechanics. PhD thesis, The University of Michigan (1977) Google Scholar
  30. 30.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wolfram, S.: Mathematica: A System for Doing Mathematics by Computer. Addison–Wesley, Redwood City (1988) zbMATHGoogle Scholar
  32. 32.
    Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.School of PhysicsUniversity of MelbourneParkvilleAustralia

Personalised recommendations