A decision procedure for well-formed linear quantum cellular automata

  • Christoph Dürr
  • Huong Lê Thanh
  • Miklos Santha
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


In this paper we introduce a new quantum computation model, the linear quantum cellular automaton. Well-formedness is an essential property for any quantum computing device since it enables us to define the probability of a configuration in an observation as the squared magnitude of its amplitude. We give an efficient algorithm which decides if a linear quantum cellular automaton is well-formed. The complexity of the algorithm is O(n2) if the input automaton has continuous neighborhood.

Classification of topics

algorithms automata and formal languages computational complexity 


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  1. [AP72]
    S. Amoroso and Y. Patt, Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures, Journal of Computer and System Sciences 6, 448–464, 1972.Google Scholar
  2. [BB92]
    A. Berthiaume and G. Brassard, The Quantum Challenge to Structural Complexity Theory, Proceeding of the 7th IEEE Conference on Structure in Complexity Theory, 132–137, 1992.Google Scholar
  3. [Bel58]
    R. Bellman, On a routing problem, Quarterly of Applied Mathematics, 16(1):87–90, 1958.Google Scholar
  4. [Bia94]
    M. Biafore, Can Computers Have Simple Hamiltonians? MIT Physics of Computation Seminar ftp://im.lcs.mit.edu/poc/biafore, 1994.Google Scholar
  5. [BV93]
    E. Bernstein and U. Vazirani, Quantum complexity theory, Proceeding of the 25th ACM Symposium on the Theory of Computing, 11–20, 1993.Google Scholar
  6. [CLR90]
    T. Cormen, C. Leiserson and R. Rivest, Introduction to Algorithms, The MIT Press, 1990.Google Scholar
  7. [Deu85]
    D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceeding of the Royal Society of London, A400:97–117, 1985.Google Scholar
  8. [DJ92]
    D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proceeding of the Royal Society of London, A439:553–558, 1992.Google Scholar
  9. [Fey82]
    R. Feynman, Simulating physics with computers, International Journal of Theoretical Physics 21 467–488, 1982.MathSciNetGoogle Scholar
  10. [Fey86]
    R. Feynman, Quantum Mechanical Computers, Foundations of Physics 16, 507, 1986.Google Scholar
  11. [FF62]
    L. Ford and D. Fulkerson, Flows in Networks, Princeton University Press, 1962.Google Scholar
  12. [Jozs91]
    R. Jozsa, Characterizing classes of functions computable by quantum parallelism, Proceeding of the Royal Society of London, A435:563–574, 1991.Google Scholar
  13. [Llo93]
    S. Lloyd, A potentially realizable Quantum Computer, Science 261, 1569–1571, 1993.Google Scholar
  14. [Llo94]
    S. Lloyd, Envisioning a Quantum Supercomputer, Science 263, 695, 1994.Google Scholar
  15. [Mar94]
    N. Margolus, Parallel Quantum Computation, MIT Physics of Computation Seminar ftp://im.lcs.mit.edu/poc/margyolus, 1994.Google Scholar
  16. [Sim94]
    D. Simon, On the Power of Quantum Computation, Proceeding of the 34th IEEE Symposium on Foundations of Computer Science, 116–123, 1994.Google Scholar
  17. [Sho94]
    P. Shor, Algorithms for Quantum Computation: Discrete Log and Factoring Proceeding of the 26th ACM Symposium on the Theory of Computing, 124–134, 1994Google Scholar
  18. [Sut9l]
    K. Sutner, De Bruijn graphs and cellular automata, Complex Systems, 5:19–30, 1991.Google Scholar
  19. [Tar75]
    R. Tarjan, Depth first search and linear graph algorithms, SIAM Journal on Computing, 1(2):146–160, 1972.CrossRefGoogle Scholar
  20. [Wat95]
    J. Watrous, On one dimensional quantum cellular automata, Proceeding of the 36th IEEE Symposium on Foundations of Computer Science, 528–537, 1995.Google Scholar
  21. [Yao93]
    A. Yao, Quantum circuit complexity, Proceeding of the 34th IEEE Symposium on Foundations of Computer Science, 352–361, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Christoph Dürr
    • 1
  • Huong Lê Thanh
    • 1
  • Miklos Santha
    • 2
  1. 1.Université Paris-Sud, LRIOrsay CedexFrance
  2. 2.CNRS, URA 410Université Paris-Sud, LRIOrsay CedexFrance

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