Asymptotic dynamics of coined quantum walks on percolation graphs

TL;DR

This paper develops an analytical method to study the long-term behavior of quantum walks affected by percolation-induced imperfections on various graphs, revealing diverse asymptotic states including mixed, periodic, and quasiperiodic regimes.

Contribution

It introduces a general analytical approach to determine the asymptotic dynamics of percolated quantum walks on arbitrary graphs, extending understanding beyond simple cases.

Findings

Explicit asymptotic states for circle and linear graphs

Discovery of stable periodic and quasiperiodic oscillations

Identification of conditions leading to mixed or stationary states

Abstract

Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation. Openness of the system's dynamics creates decoherence, leading to strong mixing. We present a method to analytically solve the asymptotic dynamics of coined, percolated quantum walks for a general graph structure. For the case of a circle and a linear graph we derive the explicit form of the asymptotic states. We find that a rich variety of asymptotic evolutions occur: not only the fully mixed state, but other stationary states; stable periodic and quasiperiodic oscillations can emerge, depending on the coin operator, the initial state, and the topology of the underlying graph.