Survival probability of the Grover walk on the ladder graph

We provide a detailed analysis of the survival probability of the Grover walk on the ladder graph with an absorbing sink. This model was discussed in Mare.s et al (2020 Phys. Rev. A 101 032113), as an example of counter-intuitive behaviour in quantum transport where it was found that the survival probability decreases with the length of the ladder L, despite the fact that the number of dark states increases. An orthonormal basis in the dark subspace is constructed, which allows us to derive a closed formula for the survival probability. It is shown that the course of the survival probability as a function of L can change from increasing and converging exponentially quickly to decreasing and converging like L-1 simply by attaching a loop to one of the corners of the ladder. The interplay between the initial state and the graph configuration is investigated.