Abstract
A quantum walker on a graph, prepared in the state | ψ_ in , e.g. initially localized at node r_ in, is repeatedly probed, with fixed frequency 1/τ, to test its presence at some target node r_ d until the first successful detection. This is a quantum version of the first-passage problem. We investigate the total detection probability P_ det, i.e. the probability to eventually detect the particle after an arbitrary number of detection attempts. It is demonstrated that this total detection probability is less than unity in symmetric systems, where it is possible to find initial states which are shielded from the detector by destructive interference, so-called dark states. The identification of physically equivalent initial states yields an upper bound for P_ det in terms of the reciprocal of the number ν of physically equivalent states. The relevant subgroup of the system's symmetry operations is found to be the stabilizer of the detection state. Using this, we prove that all bright, i.e. surely detectable, states are symmetric with respect to the stabilizer. This implies that P_ det can be obtained from a diagonalization of the "symmetrized" Hamiltonian, instead of having to find all eigenstates of the Hamiltonian.