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We connect the Grover walk with sinks to the Grover walk with tails. Thesurvival probability of the Grover walk with sinks in the long time limit ischaracterized by the centered generalized eigenspace of the Grover walk withtails. The centered eigenspace of the Grover walk is the attractor eigenspaceof the Grover walk with sinks. It is described by the persistent eigenspace ofthe underlying random walk whose support has no overlap to the boundaries ofthe graph and combinatorial flow in the graph theory.

Relation between quantum walks with tails and quantum walks with sinks on finite graphs