Quantum walks on graphs of the ordered Hamming scheme and spin networks

Hiroshi Miki,Luc Vinet,Satoshi Tsujimoto

doi:10.21468/scipostphys.7.1.001

Hiroshi Miki, Luc Vinet + Show 1 more

Open Access

https://doi.org/10.21468/scipostphys.7.1.001

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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.

Highlights

This paper introduces two-dimensional spin lattices that exhibit perfect state transfer between two single locations and multi-site fractional revival on a one-dimensional subset of the lattice
The quantum walk on the ordered Hamming graph Gα,β is equivalent to the oneexcitation dynamics of the spin lattice of triangular shape governed by the following Hamiltonian: H=
Hamming scheme of depth 2 and the single excitation dynamics of certain two-dimensional lattices of triangular shape. This relation has featured the bivariate Krawtchouk polynomials of the Tratnik type that appear as eigenvalue matrices of the scheme and whose recurrence coefficients provide the couplings and Zeeman terms

Summary

Introduction

This paper introduces two-dimensional spin lattices that exhibit perfect state transfer between two single locations and multi-site fractional revival on a one-dimensional subset of the lattice These novel models are obtained by projecting quantum walks on graphs that belong to the ordered Hamming scheme which generalizes the well known Hamming association scheme. In pursuing that question we will identify graphs in a generalization of the Hamming scheme with dynamics that projects to 1-excitation hopping on a triangular lattice exhibiting perfect state transfer and multi-site fractional revival. We suggest that these systems could be realized as photonic lattices and possibly be of use for certain algorithms.

The ordered Hamming scheme of depth 2

Special weighted graphs and their projections

Bivariate Krawtchouk polynomials and energy eigenstates

Transfer properties on the graphs

Concluding Remarks