In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like Grover’s walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let $S$ be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form $iS$. We show that the Hamiltonian can be written as $iS$ if and only if the adjacency matrix of the bipartite graph is invertible. We show that Grover’s walks and vertex-face walks are special cases of bipartite walks. Via the Hamiltonians, phenomena of bipartite walks lead to phenomena of continuous walks. We show in detail how we use bipartite walks on paths to construct universal perfect state transfer in continuous walks.
@article{10_37236_12040,
author = {Qiuting Chen and Chris Godsil and Mariia Sobchuk and Hanmeng Zhan},
title = {Hamiltonians of bipartite walks},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12040},
zbl = {1551.05388},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12040/}
}
TY - JOUR
AU - Qiuting Chen
AU - Chris Godsil
AU - Mariia Sobchuk
AU - Hanmeng Zhan
TI - Hamiltonians of bipartite walks
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12040/
DO - 10.37236/12040
ID - 10_37236_12040
ER -
%0 Journal Article
%A Qiuting Chen
%A Chris Godsil
%A Mariia Sobchuk
%A Hanmeng Zhan
%T Hamiltonians of bipartite walks
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12040/
%R 10.37236/12040
%F 10_37236_12040
Qiuting Chen; Chris Godsil; Mariia Sobchuk; Hanmeng Zhan. Hamiltonians of bipartite walks. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12040
