bibkey: chenetal2024bipartitewalkhamiltonians
authors: Qiuting Chen; Chris Godsil; Mariia Sobchuk; Hanmeng Zhan
year: 2024
title: "Hamiltonians of bipartite walks"
doi: 10.37236/12040
url: https://arxiv.org/abs/2207.01673v1
claim: "The paper studies bipartite discrete quantum walks with transition matrix U=(2P-I)(2Q-I) built from the two vertex-class partitions of the edges of a bipartite graph, and in its open questions displays the transition matrix of a seven-edge example with perfect state transfer from e1 to e6 at step 1 and states the suspicion that there is no perfect state transfer from e6 to e1."
strata_touched:
  - D5/S3/Quantum/Walks/BipartiteWalkReverseStateTransfer
license: citation-only
triage: anchor

Hamiltonians of bipartite walks

Q. Chen, C. Godsil, M. Sobchuk and H. Zhan, arXiv:2207.01673v1; The Electronic Journal of Combinatorics 31(4) (2024), P4.10. Quotations are from the v1 TeX source.

The walk (Section 2):

If the initial state is $z$, then after $k$ steps of the walk, the state is $U^{k}z$.

Hence we can define a unitary operator $U$ by [ U := (2P-I)(2Q-I). ] This the transition matrix of the bipartite walk on $G$.

The open question (Section “Open questions”), after the displayed $7\times 7$ transition matrix of the walk on the graph of Figure not return pst graph:

State $e_i$ is the characteristic vector of $i$. It is easy to see that there is perfect state transfer from state $e_1$ to $e_6$ at step $k=1$. But up to $k=300000$ steps, there is no perfect state transfer observed from $e_6$ to $e_1$. We suspect that there is no perfect state transfer from $e_6$ to $e_1$.

The displayed matrix has entry $(3,5)$ equal to $\frac{1}{3}$; the product $(2P-I)(2Q-I)$ formed from the displayed projections $P$ and $Q$ has $-\frac{1}{3}$ there, and every other entry agrees.

Verified locator

Source: https://arxiv.org/abs/2207.01673v1

• URL: https://arxiv.org/abs/2207.01673v1 (source Hamiltonian_7.tex, md5 2e3f9a1a91cb695794df49b0123dfc37, 1587 lines):
  • the walk and $U=(2P-I)(2Q-I)$, l. 135–163;
  • the graph of Figure not return pst graph, l. 237–294;
  • its projections ${\hat{P}}_0$, ${\hat{P}}_1$, $P$ and $Q$, l. 296–339, and its displayed transition matrix, l. 340–351;
  • the open question on reverse perfect state transfer, l. 1276–1290.
• DOI: 10.37236/12040 (https://doi.org/10.37236/12040; Crossref lists The Electronic Journal of Combinatorics, volume 31, issue 4, article P4.10, published 4 October 2024).