Quantum Walk on a Line with Absorbing Boundaries

TL;DR

This paper derives formulas for absorption probabilities in a finite quantum walk with absorbing boundaries, analyzing different initial conditions and confirming results with numerical simulations.

Contribution

It provides closed-form expressions for absorption probabilities in quantum walks with boundaries, extending previous work and considering various initial configurations.

Findings

Absorption probability depends on coin parameter and initial state in large systems.

Exponential correction term appears when initial position is fixed at a distance from boundary.

Numerical results agree well with analytical formulas for small system sizes.

Abstract

Absorption of two-state coined quantum walks on a finite line with two sinks located at $N$ and $-N$ is investigated. Elaborating on the results of Konno et al., J. Phys. A: Math. Gen. 36 241 (2003), we derive closed formulas for the absorption probabilities at the boundaries in the limit of large system size $N$. Two limiting cases are considered, with the starting position $k$ being independent of $N$, or kept at a constant distance $\delta$ from one of absorbers. In the first scenario, the absorption probability is determined only by the coin parameter and polar angle of the initial coin state decomposed into the eigenbasis of the coin operator. In the second case, a correction depending exponentially on $\delta$ is introduced. Finally, we perform an extensive numerical investigation for small system size $N$, showing excellent agreement between numerical and analytical results.