Quantum random walks in one dimension via generating functions

We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.

[1]  Norio Konno,et al.  A new type of limit theorems for the one-dimensional quantum random walk , 2002, quant-ph/0206103.

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

[3]  Julia Kempe,et al.  Quantum Random Walks Hit Exponentially Faster , 2002, ArXiv.

[4]  Mark C. Wilson,et al.  Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions , 2005, SIAM Rev..

[5]  Yuliy Baryshnikov,et al.  Two-dimensional Quantum Random Walk , 2008, Journal of Statistical Physics.

[6]  Mourad E. H. Ismail,et al.  Three routes to the exact asymptotics for the one-dimensional quantum walk , 2003, quant-ph/0303105.

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

[8]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[9]  D. Meyer.  From quantum cellular automata to quantum lattice gases , 1996, quant-ph/9604003.

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

[11]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[12]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences: I. Smooth Points of the Singular Variety , 2002, J. Comb. Theory, Ser. A.

[13]  Andris Ambainis,et al.  Quantum walks on graphs , 2000, STOC '01.

[14]  Simone Severini.  On the Digraph of a Unitary Matrix , 2003, SIAM J. Matrix Anal. Appl..

[15]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences II: Multiple Points of the Singular Variety , 2004, Combinatorics, Probability and Computing.

[16]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[17]  Gary L. Miller,et al.  Proceedings of the twenty-eighth annual ACM symposium on Theory of computing , 1996, STOC 1996.

[18]  Edward Farhi,et al.  An Example of the Difference Between Quantum and Classical Random Walks , 2002, Quantum Inf. Process..

[19]  Aharonov,et al.  Quantum random walks. , 1993, Physical review. A, Atomic, molecular, and optical physics.