**Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs**

**Norio Konno 1, Etsuo Segawa 2,\* and Martin Štefa ˇnák 3**


**Abstract:** We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in graph theory.

**Keywords:** quantum walk; survival probability; attractor eigenspace; dressed photon
