Absorption Probabilities of Quantum Walks

Abstract

Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et.\ al.\ \cite{ambainis01} showed that in the $(\mathbb{Z},C_1,H)$ 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 $(\mathbb{Z},C_1,H,\{0,n\})$ satisfy a linear fractional recurrence in $n$ (here $P_n(1,0)$ is the probability that a Hadamard walk particle initialized in $|1⟩|R⟩$ is eventually absorbed at $|0⟩$ and not at $|n⟩$). 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.

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