Asymptotic Behavior and Limiting Distribution of Quantum Walk on Cycles with General U(2) Coin by Reduced Characteristic Matrix Method

TL;DR

This paper develops a method to analyze the long-term behavior of quantum walks on cycles with general U(2) coins, providing new analytical tools and insights into their asymptotic properties and limiting distributions.

Contribution

It introduces the reduced density characteristic matrix (RDCM) method for quantum walks on cycles with general U(2) coins, enabling analytical derivation of limiting distributions and entanglement properties.

Findings

Calculated entanglement temperature for general initial states.

Derived analytical expressions for the limiting distribution.

Compared results with previous studies to validate the approach.

Abstract

We calculated reduced density characteristic matrix (RDCM) for quantum walk on cycles (QWC) to study asymptotic properties of the most general form of quantum walk on cycles with general $U(2)$ coin operator. As an example, entanglement temperature for general initial state has been calculated and compared to previous results. Also, we have modified RDCM to derive analytical expression for general form of limiting distribution (LD).