Grover Walks on a Line with Absorbing Boundaries

TL;DR

This paper analyzes quantum Grover walks on a line with absorbing boundaries, providing analytical formulas for absorption probabilities and highlighting their distinct behavior compared to classical and Hadamard walks.

Contribution

It introduces analytical expressions for absorption probabilities of Grover walks with boundaries, using combinatorial methods, and compares quantum walk behavior to classical and Hadamard walks.

Findings

Absorbing probabilities are derived analytically and match numerical simulations.

Grover walks exhibit unique boundary absorption behavior unlike classical or Hadamard walks.

Analytical results reveal significant differences in quantum walk dynamics with boundaries.

Abstract

In this paper, we study Grover walks on a line with one and two absorbing boundaries. In particular, we present some results for the absorbing probabilities both in a semi-finite and finite line. Analytical expressions for these absorbing probabilities are presented by using the combinatorial approach. These results are perfectly matched with numerical simulations. We show that the behavior of Grover walks on a line with absorbing boundaries is strikingly different from that of classical walks and that of Hadamard walks.