Feynman Integral in Quantum Walk, Barrier-top Scattering and Hadamard Walk

Abstract

The aim of this article is to relate the discrete quantum walk on Z with the continuous Schrödinger operator on R in the scattering problem. Each point of Z is associated with a barrier of the potential, and the coin operator of the quantum walk is determined by the transfer matrix between bases of WKB solutions on the classically allowed regions of both sides of the barrier. This correspondence enables us to represent each entry of the scattering matrix of the Schrödinger operator as a countable sum of probability amplitudes associated with the paths of the quantum walker. In particular, the barrier-top scattering corresponds to the Hadamard walk in the semiclassical limit.

A discontinuity of the energy of quantum walk in impurities

“…There is no doubt that studies on scattering theory is one of the interesting topic of the Schrödinger equation. Recently, it is revealed that the scatterings of some fundamental stationary Schrödinger equations on the real line with not only delta potentials [26,7,18] but also continuous potential [6] can be recovered by discrete-time quantum walks. These induced quantum walks are given by the following setting: the non-trivial quantum coins are assigned to some vertices in a finite region on the one-dimensional lattice as the impurities and the freequantum coins are assigned at the other vertices.…”

“…The initial state is given so that quantum walkers inflows into the perturbed region at every time step. It is shown that the scattering matrix of the quantum walk on the one-dimensional lattice can be explicitly described by using a path counting in [11] and this path counting method can be described by a discrete analogue of the Feynmann path integral [6]. There are some studies for the scattering theory of quantum walks under slightly general settings and related topics [25,23,24,20,21,27,12].…”