Quantum Walks on Graphs of the Ordered Hamming Scheme and Spin Networks

TL;DR

This paper demonstrates that quantum walks on graphs of the ordered Hamming scheme can model excitation hopping in spin lattices, revealing perfect state transfer and fractional revival phenomena with potential applications in quantum information processing.

Contribution

It introduces a novel connection between quantum walks on ordered Hamming scheme graphs and spin network dynamics, including perfect state transfer and fractional revival.

Findings

Perfect state transfer between lattice summits for specific parameters

Observation of fractional revival in certain configurations

Eigenstates described by bivariate Krawtchouk polynomials

Abstract

It is shown that the hopping of a single excitation on certain triangular spin lattices with non-uniform couplings and local magnetic fields can be described as the projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. For some values of the parameters the models exhibit perfect state transfer between two summits of the lattice. Fractional revival is also observed in some instances. The bivariate Krawtchouk polynomials of the Tratnik type that form the eigenvalue matrices of the ordered Hamming scheme of depth 2 give the overlaps between the energy eigenstates and the occupational basis vectors.