Absorption Probabilities for the Two-Barrier Quantum Walk

Let p_j^(n) be the probability that a Hadamard quantum walk, started at site j on the integer lattice {0,…,n}, is absorbed at 0. We give an explicit formula for p_j^(n). Our formula proves a conjecture of John Watrous, concerning an empirically observed linear fractional recurrence relation for the numbers p_1^(n).