Periodicity and perfect state transfer in quantum walks on variants of cycles

TL;DR

This paper explores the rarity and conditions for perfect state transfer in quantum walks on cycle variants, identifying specific graph modifications that enable interesting quantum dynamics like periodicity and high-fidelity transfer.

Contribution

It introduces new families of graphs that support perfect or high-fidelity state transfer in quantum walks, extending known structures to larger, more complex networks.

Findings

Perfect state transfer is rare and occurs only in specific cycle modifications.

Certain graph modifications enable periodicity and high-fidelity transfer.

Robustness of quantum dynamics is tested under various perturbations.

Abstract

We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles C_4, C_6 and C_8 for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of C_4 could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.