  Staggered quantum walks on graphs are based on the concept of graph
tessellation and generalize some well-known discrete-time quantum walk models.
In this work, we address the class of 2-tessellable quantum walks with the goal
of obtaining an eigenbasis of the evolution operator. By interpreting the
evolution operator as a quantum Markov chain on an underlying multigraph, we
define the concept of quantum detailed balance, which helps to obtain the
eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of
the double discriminant matrix of the quantum Markov chain. To obtain the
remaining eigenvectors, we have to use the quantum detailed balance conditions.
If the quantum Markov chain has a quantum detailed balance, there is an
eigenvector for each fundamental cycle of the underlying multigraph. If the
quantum Markov chain does not have a quantum detailed balance, we have to use
two fundamental cycles linked by a path in order to find the remaining
eigenvectors. We exemplify the process of obtaining the eigenbasis of the
evolution operator using the kagome lattice (the line graph of the hexagonal
lattice), which has symmetry properties that help in the calculation process.
