Extreme Events of Quantum Walks on Graphs

TL;DR

This paper demonstrates that extreme events can occur in quantum walks on graphs, exhibiting properties similar to classical random walks, with probabilities influenced by graph structure and degree.

Contribution

It introduces a quantum version of the flux-fluctuation relation and analyzes extreme events in quantum walks on different graph types, revealing their statistical properties.

Findings

Extreme events in quantum walks follow a power-law distribution related to vertex degree.

Extreme event probability is higher for low-degree nodes than hubs.

Probability scales with the threshold defining an extreme event.

Abstract

Due to the unitary evolution, quantum walks display different dynamical features from that of classical random walks. In contrast to this expectation, in this work, we show that extreme events can arise in unitary dynamics and its properties are qualitatively similar to that of random walks. We consider quantum walks on a ring lattice and a scale-free graph. Firstly, we obtain quantum version of flux-fluctuation relation and use this to define to extreme events on vertices of a graph as exceedences above the mean flux. The occurrence probability for extreme events on scale-free graphs displays a power-law with the degree of vertices, in qualitative agreement with corresponding classical random walk result. For both classical and quantum walks, the extreme event probability is larger for small degree nodes compared to hubs on the graph. Further, it is shown that extreme event probability scales with threshold used to define extreme events.