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Link to original content: https://doi.org/10.1007/S11047-014-9441-X
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Simulation methods for quantum walks on graphs applied to formal language recognition

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Abstract

We describe an algorithm which automates the generation of appropriate shift and coin operators for a discrete time quantum walk, given the adjacency matrix of the graph over which the walk is run. This gives researchers the freedom to numerically investigate any discrete time quantum walk over graphs of a computationally tractable size by greatly reducing the time required to initialise a given walk. We then describe one application in which the swift initialisation of walks has enabled systematic investigations of walks over a large number of structures. New results concerning this application, which is to formal language recognition, are described. The reliability of these results, as well as the general suitability of numerical analysis as a tool for investigating discrete time quantum walks, are briefly discussed. We also mention specific Python packages which facilitate our simulations and analysis, motivating the use of high level programming languages in this context.

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References

  • Ambainis A (2004) Quantum walk algorithm for element distinctness. In: Proceedings of the 45th annual IEEE symposium on foundations of computer science. IEEE Computer Society, Washington, DC, pp 22–31

  • Ambainis A, Bach E, Nayak A, Vishwanath A, Watrous J (2001) One-dimensional quantum walks. In: Proceedings of the thirty-third annual ACM symposium on Theory of computing. ACM, New York, pp 37–49

  • Barr K, Kendon V (2012) Formal languages analysed by discrete time quantum walks. arXiv preprint:1209.5238

  • Barr K, Proctor T, Allen D, Kendon V (2012) Periodicity and perfect state transfer in quantum walks on three families of graphs. arXiv preprint:1204.5937v3

  • Barr K, Felming T, Kendon V (2013) Simulation methods for quantum walks on graphs applied to perfect state transfer and formal language recognition. In: Stepney S, Andrews PS (eds) Proceedings of the 2013 workshop on complex systems modelling and simulation, Milan, Italy, July 2013. Luniver Press, Frome

  • Bašić M, Petković MD, Stevanović D (2009) Perfect state transfer in integral circulant graphs. Appl Math Lett 22(7):1117–1121

    Article  MATH  MathSciNet  Google Scholar 

  • Bose S (2003) Quantum communication through an unmodulated spin chain. Phys Rev Lett 91(20):207901

    Article  Google Scholar 

  • Brodsky A, Pippenger B (2000) Characterizations of quantum finite automata. SIAM J Comput 31:1456–1478

    Article  MathSciNet  Google Scholar 

  • Childs AM (2009) Universal computation by quantum walk. Phys Rev Lett 102(18):180501

    Article  MathSciNet  Google Scholar 

  • Childs AM, Gosset D, Webb Z (2013) Universal computation by multiparticle quantum walk. Science 339(6121):791–794

    Article  MathSciNet  Google Scholar 

  • Gottlieb AD, Janson S, Scudo PF (2005) Convergence of coined quantum walks on \(\mathbb{R}_d\). Infin Dimens Anal Quantum Probab Relat Top 8(1):129–140

    Article  MathSciNet  Google Scholar 

  • Grover LK (1996) A fast quantum mechanical algorithm for database search. In: Proceedings of the twenty-eighth annual ACM symposium on Theory of computing. ACM, New York, pp 212–219

  • Hopcroft J, Ullman J (1979) Introduction to automata theory, languages, and computation. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Jaro MA (1989) Advances in record-linkage methodology as applied to matching the 1985 census of Tampa, Florida. J Am Stat Assoc 84(406):414–420

    Article  Google Scholar 

  • Jaro MA (1995) Probabilistic linkage of large public health data files. Stat Med 14(5–7):491–498

    Article  Google Scholar 

  • Kempe J (2005) Discrete quantum walks hit exponentially faster. Probab Theory Relat Fields 133(2):215–235

    Article  MATH  MathSciNet  Google Scholar 

  • Kendon V (2006) Quantum walks on general graphs. Int J Quantum Inf 4(5):791–805

    Article  MATH  Google Scholar 

  • Kendon V (2011) Where to quantum walk. arXiv preprint:1107.3795

  • Kendon VM, Tamon C (2011) Perfect state transfer in quantum walks on graphs. J Comput Theor Nanosci 8(3):422–433

    Article  Google Scholar 

  • Kendon V, Tregenna B (2003) Decoherence can be useful in quantum walks. Phys Rev A 67(4):042315

    Article  Google Scholar 

  • Kondacs A, Watrous J (1997) On the power of quantum finite state automata. In: FOCS 1997, pp 66–75

  • Lovett NB, Cooper S, Everitt M, Trevers M, Kendon V (2010) Universal quantum computation using the discrete-time quantum walk. Phys Rev A 81(4):042330

    Article  MathSciNet  Google Scholar 

  • Mackay TD, Bartlett SD, Stephenson LT, Sanders BC (2002) Quantum walks in higher dimensions. J Phys A 35:2745–2753

    Article  MATH  MathSciNet  Google Scholar 

  • Montanaro A (2007) Quantum walks on directed graphs. Quantum Inf Comput 7(1):93–102

    MATH  MathSciNet  Google Scholar 

  • Moore C, Russell A (2002) Quantum walks on the hypercube. In: Randomization and approximation techniques in computer science. Springer, Berlin, pp 164–178

  • Papadimitriou CH (1991) On selecting a satisfying truth assignment. In: Proceedings of the 32nd annual IEEE symposium on foundations of computer science, 1991, pp 163–169

  • Saxena N, Severini S, Shparlinski IE (2007) Parameters of integral circulant graphs and periodic quantum dynamics. Int J Quantum Inf 5(03):417–430

    Article  MATH  Google Scholar 

  • Schoning T (1999) A probabilistic algorithm for k-SAT and constraint satisfaction problems. In: 40th Annual IEEE symposium on foundations of computer science, 1999, pp 410–414

  • Shenvi N, Kempe J, Whaley K (2003) Quantum random-walk search algorithm. Phys Rev A 67:052307

    Article  Google Scholar 

  • Shor PW (1994) Algorithms for quantum computation: discrete logarithms and factoring. In: 1994 Proceedings of the 35th annual IEEE symposium on foundations of computer science, pp 124–134

  • Sipser M (2012) Introduction to the theory of computation. Cengage Learning, Boston

    Google Scholar 

  • Tregenna B, Flanagan W, Maile R, Kendon V (2003) Controlling discrete quantum walks: coins and initial states. N J Phys 5(1):83

    Article  Google Scholar 

  • Venegas-Andraca SE (2008) Quantum walks for computer scientists. Synth Lect Quantum Comput 1(1):1–119

    Article  MathSciNet  Google Scholar 

  • Watrous J (1999) Quantum simulations of classical random walks and undirected graph connectivity. In: 1999 Proceedings of the fourteenth annual IEEE conference on computational complexity. IEEE, pp 180–187

  • Yakaryilmaz A, Cem Say AC (2008) Languages recognized with unbounded error by quantum finite automata. arXiv:0809.0073v2

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Authors and Affiliations

  1. School of Physics and Astronomy, E C Stoner Building, University of Leeds, Leeds, LS2 9JT, UK

    K. Barr, T. Fleming & V. Kendon

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  1. K. Barr
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  2. T. Fleming
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  3. V. Kendon
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Correspondence to K. Barr.

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Barr, K., Fleming, T. & Kendon, V. Simulation methods for quantum walks on graphs applied to formal language recognition. Nat Comput 14, 145–156 (2015). https://doi.org/10.1007/s11047-014-9441-x

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