A. Ahibrecht, H. Vogts, A. H. Werner, and R. F. Werner, “Asymptotic Evolution of Quantum Walks with Random Coin,” Journal of Mathematical Physics, Vol. 52, No. 4, 2011, Article ID: 042201. doi:10.1063/1.3575568

has been cited by the following article:

• TITLE: Solution of the Time Dependent Schrödinger Equation and the Advection Equation via Quantum Walk with Variable Parameters

  AUTHORS: Shinji Hamada, Masayuki Kawahata, Hideo Sekino

  KEYWORDS: Quantum Walk; Quantum Cellular Automaton; Time Dependent Schrödinger Equation; Advection Equation

  JOURNAL NAME: Journal of Quantum Information Science, Vol.3 No.3, September 24, 2013

  ABSTRACT: We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the “Time Dependent Schr?dinger Equation with a space dependent imaginary diffusion coefficient” or “the advection equation with space dependent velocity fields”. Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.