# Spin Chains, Graphs and State Revival

## TL;DR

This paper surveys the relationship between spin chain dynamics and quantum walks on graphs, focusing on perfect state transfer, fractional revival, and the role of orthogonal polynomials, especially within the ordered Hamming scheme.

## Contribution

It explores the connection between graph-based quantum walks and spin lattice dynamics, highlighting the role of multivariate Krawtchouk polynomials and the ordered Hamming scheme in state transfer phenomena.

## Key findings

- Analysis of quantum walks on graphs related to the ordered Hamming scheme.
- Identification of conditions for perfect state transfer and fractional revival.
- Demonstration of the role of orthogonal polynomials in understanding quantum state dynamics.

## Abstract

Connections between the 1-excitation dynamics of spin lattices and quantum walks on graphs will be surveyed. Attention will be paid to perfect state transfer (PST) and fractional revival (FR) as well as to the role played by orthogonal polynomials in the study of these phenomena. Included is a discussion of the ordered Hamming scheme, its relation to multivariate Krawtchouk polynomials of the Tratnik type, the exploration of quantum walks on graphs of this association scheme and their projection to spin lattices with PST and FR.