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

TL;DR

This paper establishes a connection between discrete quantum walks and continuous Schrödinger scattering, showing how quantum walk paths can represent scattering matrix entries, with the Hadamard walk modeling barrier-top scattering in the semiclassical limit.

Contribution

It introduces a novel correspondence linking quantum walks on integers with Schrödinger operators, enabling path-based representation of scattering matrices and relating barrier-top scattering to Hadamard walks.

Findings

Scattering matrix entries can be expressed as sums over quantum walk paths.

Barrier-top scattering corresponds to the Hadamard walk in the semiclassical limit.

The approach bridges discrete quantum walks and continuous quantum scattering theory.

Abstract

The aim of this article is to relate the discrete quantum walk on $\mathbb{Z}$ with the continuous Schr\"odinger operator on $\mathbb{R}$ in the scattering problem. Each point of $\mathbb{Z}$ 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 Schr\"odinger 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.