Absorption probabilities of quantum walks

Abstract

Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et al. (in: Proceedings of the thirty-third annual ACM symposium on theory of computing, 2001) showed that in the $$(\mathbb {Z},C_1,H)$$ quantum walk (one-dimensional Hadamard walk) an absorbing boundary partially reflects information. These authors also conjectured that the left absorption probabilities $$P_n^{(1)}(1,0)$$ related to the finite absorbing Hadamard walks $$(\mathbb {Z},C_1,H,\{ 0,n\} )$$ satisfy a linear fractional recurrence in n (here $$P_n(1,0)$$ is the probability that a Hadamard walk particle initialized in $$|1\rangle |R\rangle $$ is eventually absorbed at $$|0\rangle $$ and not at $$|n\rangle $$ ). This result, as well as a third-order linear recurrence in initial position m of $$P_n^{(m)}(1,0)$$ , was later proved by Bach and Borisov (Absorption probabilities for the two-barrier quantum walk, 2009, arXiv:0901.4349 ) using techniques from complex analysis. In this paper, we extend these results to general two-state quantum walks and three-state Grover walks, while providing a partial calculation for absorption in d-dimensional Grover walks by a $$d-1$$ -dimensional wall. In the one-dimensional cases, we prove partial reflection of information, a linear fractional recurrence in lattice size, and a linear recurrence in the initial position.