# Absorption Probabilities of Quantum Walks

## TL;DR

This paper extends the analysis of absorption probabilities in quantum walks, demonstrating partial reflection and recurrence relations for various types of quantum walks, including general two-state and three-state Grover walks.

## Contribution

It generalizes previous results on absorption probabilities to broader classes of quantum walks, including two-state and three-state Grover walks, and provides partial results for higher-dimensional cases.

## Key findings

- Partial reflection of information in quantum walks
- Linear fractional recurrence in lattice size for absorption probabilities
- Linear recurrence in initial position for absorption probabilities

## Abstract

Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et.\ al.\ \cite{ambainis01} showed that in the $(\Z ,C_1,H)$ quantum walk (one-dimensional Hadamard walk) an absorbing boundary partially reflects information. These authors also conjectured that the left absorption probabilities $P_n^{(1)}(1,0)$ related to the finite absorbing Hadamard walks $(\Z ,C_1,H,\{ 0,n\} )$ satisfy a linear fractional recurrence in $n$ (here $P_n(1,0)$ is the probability that a Hadamard walk particle initialized in $|1\rangle |R\rangle$ is eventually absorbed at $|0\rangle$ and not at $|n\rangle$). This result, as well as a third order linear recurrence in initial position $m$ of $P_n^{(m)}(1,0)$, was later proved by Bach and Borisov \cite{bach09} using techniques from complex analysis. In this paper we extend these results to general two state quantum walks and three-state Grover walks, while providing a partial calculation for absorption in $d$-dimensional Grover walks by a $d-1$-dimensional wall. In the one-dimensional cases, we prove partial reflection of information, a linear fractional recurrence in lattice size, and a linear recurrence in initial position.