## SciPost Submission Page

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

### by Hiroshi Miki, Satoshi Tsujimoto, Luc Vinet

### Submission summary

As Contributors: | Hiroshi Miki |

Arxiv Link: | https://arxiv.org/abs/1712.09200v5 |

Date submitted: | 2019-05-30 |

Submitted by: | Miki, Hiroshi |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Quantum Physics |

### 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.

###### Current status:

### Author comments upon resubmission

We are grateful to both referees for their valuable comments. Their deep insight help to improve the contents and presentation of our manuscript.

Following their comments, we made several changes and some additions in the manuscript.

-response to report 1:

We fixed typos and added the case alpha=beta/sqrt{2} case which corresponds to the model with SU(3) symmetry.

As pointed out, the FR from corner to base points actually takes place and we gave the corresponding figure.

The 2-variable Krawtchouk polynomials of course have an algebraic interpretation based upon this symmetry and we added the corresponding references.

We tried to put simple figures about ordered Hamming scheme of depth 2. However, for N=1, there are 3 graphs G_{0,0} (4 nodes without edge), G_{0,1} (square), G_{1,0} (4 nodes with 2 edges) and they are too simple. For N=2, we have 6 graphs (each has 16 nodes and several edges) and they are too complicated for this manuscript. Therefore, we did not put the figures of the ordered Hamming graph in the manuscript.

-response to report 2:

We agree with suggestions and changed the terminology from ordered r-Hamming scheme to ordered Hamming scheme of depth r in the manuscript including its title.

We changed the description about |e_{i,j}) from (N+1)x(N+1) matrix to orthonormal basis vector which might be easier to understand.

We also added the references and fixed some typos.

### List of changes

-We corrected some typos.

-We added references [5],[14] and [20].

-We changed the terminology from ordered r-Hamming scheme to ordered Hamming scheme of depth r. We then changed the title according to it.

-We added case alpha=beta/sqrt{2} in p. 7. Some description and one figure about this case were given.