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Quantum Walks on Graphs of the Ordered Hamming Scheme and Spin Networks
by Hiroshi Miki, Satoshi Tsujimoto, Luc Vinet
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Hiroshi Miki 
Submission information  

Preprint Link:  https://arxiv.org/abs/1712.09200v5 (pdf) 
Date accepted:  20190626 
Date submitted:  20190530 02:00 
Submitted by:  Miki, Hiroshi 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
It is shown that the hopping of a single excitation on certain triangular spin lattices with nonuniform 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.
Published as SciPost Phys. 7, 001 (2019)
Author comments upon resubmission
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 2variable 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 rHamming 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 rHamming 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.
Submission & Refereeing History
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Reports on this Submission
Report 1 by Kareljan Schoutens on 2019619 (Invited Report)
Report
I am satisfied with the changes made to the manuscript and recommend that it be published in SciPost.