**1. Introduction**

A simple random walker on a finite and connected graph starting from any vertex hits an arbitrary vertex in a finite time. This fact implies that, if we consider a subset of the vertices of this graph as sinks, where the random walker is absorbed, then the survival probability of the random walk in the long time limit converges to zero. However, for quantum walks (QW) [1], the situation is more complicated and the survival probability depends in general on the graph, coin operator, and the initial state of the walk. For a two-state quantum walk on a finite line with sinks on both ends and a non-trivial coin, the survival probability is also zero, as shown by the studies of the corresponding absorption problem [2–5]. However, for a three-state quantum walk with the Grover coin [6], the survival probability on a finite line is non-vanishing [7] due to the existence of trapped states. These are the eigenstates of the unitary evolution operator which do not have a support on the sinks. Trapped states crucially affect the efficiency of quantum transport [8] and lead to counter-intuitive effects, e.g., the transport efficiency can be improved by increasing the distance between the initial vertex and the sink [9,10]. We find a similar phenomena to this quantum walk model in the experiment on the energy transfer of the dressed photon [11] through the nanoparticles distributed in a finite threedimensional grid [12]. The output signal intensity increases when the depth direction is larger. Although, when the depth is deeper, a lot of "detours" newly appear to reach to the position of the output from the classical point of view, the output signal intensity of the dressed photon becomes stronger. The existence of trapped states also results in infinite hitting times [13,14].

In this paper, we analyze such counter-intuitive phenomena for the Grover walk on a general connected graph using spectral analysis. The Grover walk is an induced quantum walk of the random walk from the viewpoint of the spectral mapping theorem [15].

To this end, first, we connect the Grover walk with sinks to the Grover walk with tails. The tails are the semi-infinite paths attached to a finite and connected graph. We

**Citation:** Konno, N.; Segawa, E.; Štefa ˇnák, M. Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs. *Symmetry* **2021**, *13*, 1169. https://doi.org/10.3390/sym13071169

Academic Editor: Motoichi Ohtsu

Received: 7 May 2021 Accepted: 24 June 2021 Published: 29 June 2021

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call the set of vertices connecting to the tails the boundary. The Grover walk with tail was introduced by [16,17] in terms of the scattering theory. If we set some appropriate bounded initial state so that the support is included in the tail, the existence of the fixed point of the dynamical system induced by the Grover walk with tails is shown, and the stable generalized eigenspace H*<sup>s</sup>*, in which the dynamical system lives, is orthogonal to the centered generalized eigenspace H*c* [18] at every time step [19]. The centered generalized eigenspace is generated by the generalized eigenvectors of the principal submatrix of the time evolution operator of the Grover walk with respect to the internal graph, and all the corresponding absolute values of the eigenvalues are 1. This eigenstate is equivalent to the attractor space [8] of the Grover walk with sink. Indeed, we show that the stationary state of the Grover walk with sink is attracted to this centered generalized eigenstate. Secondly, we characterize this centered generalized eigenspace using the persistent eigenspace of the underlying random walk whose supports have no overlaps to the boundary, also using the concept of "flow" from graph theory. From this result, we see that the existence of the persistent eigenspace of the underlying random walk significantly influences the asymptotic behavior of the corresponding Grover walk, although it has little effect on the asymptotic behavior of the random walk itself. Moreover, we clarify that the graph structure which constructs the symmetric or anti-symmetric flow satisfying the Kirchhoff's law contributes to the non-zero survival probability of the Grover walk, as suggested in [8,15].

This paper is organized as follows. In Section 2, we prepare the notations of graphs and give the definition of the Grover walk and the boundary operators which are related to the chain. In Section 3, we give the definition of the Grover walk on a graph with sinks. In Section 4, a necessary and sufficient condition for the surviving of the Grover walk is described. In Section 5, we give an example. Section 6 is devoted to the relation between the Grover walk with sink and the Grover walk with tail. In Section 7, we partially characterize the centered generalized eigenspace using the concept of flow from graph theory.
