We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected. This is a quantum version of the first-passage problem. We focus on the total probability P det that the particle is eventually detected in some target state, for example, on a node r d on a graph, after an arbitrary number of detection attempts. Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P det in terms of the energy eigenstates which is generically τ independent. We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection. Specifically, we give a simple upper bound for P det that is controlled by the number of equivalent (with respect to the detection) states in the system. We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.

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A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it fully explores space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local measurements, made on a target state, the final detection of the system is not guaranteed since the Hilbert space is split into a bright subspace and an orthogonal dark one. Using this we find an uncertainty relation for the deviations of the detection probability from its classical counterpart, in terms of the energy fluctuations.