×

Three routes to the exact asymptotics for the one-dimensional quantum walk. (English) Zbl 1049.82025

This paper is concerned with the dynamics of a test particle performing an unbiased quantum walk on the integer parts on the line. There are usually two approaches to analyse the system, namely, (i) the path integral, and (ii) Schrödinger approaches, which reflect two complementary ways of formulating quantum mechanics. The authors want to refine the asymptotic analysis on quantum walks of the paper written by A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous [Proc.ACM Symp.on Theory of Computation, Assoc.Computing Machinery, 37–49 (2001)]. To this aim, the authors propose an alternative third method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path integral representation.
In fact, they have succeeded in analysing in a new way the quantum walk in terms of Airy functions, which has the advantage of being able to handle the dramatic changes in the asymptotic behaviour of the system in a uniform manner. More precisely, a single integral representation for the wavefunction is provided, that works over the full range of positions, including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of the system have difficulties especially in this transitional range, because the approximations on which they were based break down there. The authors have also proved the mathematical relationship between the path integral and Schrödinger approaches to solving this system.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
33C90 Applications of hypergeometric functions
82B10 Quantum equilibrium statistical mechanics (general)
PDFBibTeX XMLCite
Full Text: DOI arXiv