Simulation methods for quantum walks on graphs applied to formal language recognition

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.

[1]  Viv Kendon,et al.  Decoherence can be useful in quantum walks , 2002, quant-ph/0209005.

[2]  Andrew M. Childs,et al.  Universal Computation by Multiparticle Quantum Walk , 2012, Science.

[3]  C.H. Papadimitriou,et al.  On selecting a satisfying truth assignment , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[4]  Vivien M. Kendon,et al.  Periodicity and perfect state transfer in quantum walks on variants of cycles , 2012, Quantum Inf. Comput..

[5]  Andris Ambainis,et al.  One-dimensional quantum walks , 2001, STOC '01.

[6]  Viv Kendon,et al.  Where to quantum walk , 2011 .

[7]  John Watrous,et al.  On the power of quantum finite state automata , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[8]  Matthew A. Jaro,et al.  Probabilistic linkage of large public health data files. , 1995, Statistics in medicine.

[9]  John Watrous.  Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity , 2001, J. Comput. Syst. Sci..

[10]  Barry C. Sanders,et al.  Quantum walks in higher dimensions , 2002 .

[11]  Timothy Proctor,et al.  Periodicity and perfect state transfer in quantum walks on three families of graphs , 2012 .

[12]  Andrew M. Childs,et al.  Universal computation by quantum walk. , 2008, Physical review letters.

[13]  Will Flanagan,et al.  Controlling discrete quantum walks: coins and initial states , 2003 .

[14]  Andris Ambainis.  Quantum Walk Algorithm for Element Distinctness , 2004, FOCS.

[15]  Alexander Russell,et al.  Quantum Walks on the Hypercube , 2002, RANDOM.

[16]  S. Bose.  Quantum communication through an unmodulated spin chain. , 2002, Physical review letters.

[17]  Viv Kendon,et al.  Quantum walks on general graphs , 2003, quant-ph/0306140.

[18]  José D. P. Rolim,et al.  Randomization and Approximation Techniques in Computer Science , 2002, Lecture Notes in Computer Science.

[19]  Salvador Elías Venegas-Andraca,et al.  Quantum Walks for Computer Scientists , 2008, Quantum Walks for Computer Scientists.

[20]  Neil B. Lovett,et al.  Universal quantum computation using the discrete-time quantum walk , 2009, 0910.1024.

[21]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[22]  Colin Cooper,et al.  Randomization and Approximation Techniques in Computer Science , 1999, Lecture Notes in Computer Science.

[23]  Andris Ambainis,et al.  Quantum walk algorithm for element distinctness , 2003, 45th Annual IEEE Symposium on Foundations of Computer Science.

[24]  Christino Tamon,et al.  Perfect State Transfer in Quantum Walks on Graphs , 2011 .

[25]  Julia Kempe,et al.  Discrete Quantum Walks Hit Exponentially Faster , 2005 .

[26]  Alex Brodsky,et al.  Characterizations of 1-Way Quantum Finite Automata , 2002, SIAM J. Comput..

[27]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[28]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[29]  Vivien M. Kendon,et al.  Formal languages analysed by quantum walks , 2012, ArXiv.

[30]  Julia Kempe,et al.  Discrete Quantum Walks Hit Exponentially Faster , 2003, RANDOM-APPROX.

[31]  Uwe Schöning.  A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems , 1999, FOCS.

[32]  Matthew A. Jaro,et al.  Advances in Record-Linkage Methodology as Applied to Matching the 1985 Census of Tampa, Florida , 1989 .

[33]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[34]  Simone Severini,et al.  Parameters of Integral Circulant Graphs and Periodic Quantum Dynamics , 2007 .

[35]  Milan B. Tasic,et al.  Perfect state transfer in integral circulant graphs , 2009, Appl. Math. Lett..

[36]  Ashley Montanaro,et al.  Quantum walks on directed graphs , 2005, Quantum Inf. Comput..

[37]  A. C. Cem Say,et al.  Languages Recognized with Unbounded Error by Quantum Finite Automata , 2008, CSR.

[38]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[39]  Svante Janson,et al.  Convergence of coined quantum walks on R^d , 2005 .