Quantum random walk on the line as a Markovian process

We analyze in detail the discrete-time quantum walk on the line by separating the quantum evolution equation into Markovian and interference terms. As a result of this separation, it is possible to show analytically that the quadratic increase in the variance of the quantum walker's position with time is a direct consequence of the coherence of the quantum evolution. If the evolution is decoherent, as in the classical case, the variance is shown to increase linearly with time, as expected. Furthermore, we show that this system has an evolution operator analogous to that of a resonant quantum kicked rotor. As this rotator may be described through a quantum computational algorithm, one may employ this algorithm to describe the time evolution of the quantum walker.

[1]  S. Goldstein.  ON DIFFUSION BY DISCONTINUOUS MOVEMENTS, AND ON THE TELEGRAPH EQUATION , 1951 .

[2]  Felix M. Izrailev,et al.  Simple models of quantum chaos: Spectrum and eigenfunctions , 1990 .

[3]  Godoy,et al.  From the quantum random walk to classical mesoscopic diffusion in crystalline solids. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[5]  R. Siri,et al.  Markovian Behaviour and Constrained Maximization of the Entropy in Chaotic Quantum Systems , 2003 .

[6]  Nayak Ashwin,et al.  Quantum Walk on the Line , 2000 .

[7]  Eric Bach,et al.  One-dimensional quantum walks with absorbing boundaries , 2004, J. Comput. Syst. Sci..

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

[9]  Moore,et al.  Atom optics realization of the quantum delta -kicked rotor. , 1995, Physical review letters.

[10]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[11]  P. Busch.  The Role of Entanglement in Quantum Measurement and Information Processing , 2002, quant-ph/0209090.

[12]  E. Andrade.  Contemporary Physics , 1945, Nature.

[13]  F. Izrailev,et al.  Quantum resonance for a rotator in a nonlinear periodic field , 1980 .

[14]  Norio Konno,et al.  Quantum Random Walks in One Dimension , 2002, Quantum Inf. Process..

[15]  R. Donangelo,et al.  Dynamical localization in quasiperiodic driven systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[17]  R. Feynman.  Simulating physics with computers , 1999 .

[18]  G. J. Milburn,et al.  Implementing the quantum random walk , 2002 .

[19]  Exponential gain in quantum computing of quantum chaos and localization. , 2000, Physical review letters.

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