Dark states of quantum search cause imperfect detection

We consider a quantum walk where a detector repeatedly probes the system with fixed rate $1/\tau$ until the walker is detected. This is a quantum version of the first-passage problem. We focus on the total probability, $P_{\mathrm{det}}$, that the particle is eventually detected in some target state, for example on a node $r_{\mathrm{d}}$ 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 $P_{\mathrm{det}}$ in terms of the energy eigenstates which is generically $\tau$ independent. We find that disorder in the underlying Hamiltonian renders perfect detection: $P_{\mathrm{det}}=1$, and then expose the role of symmetry with respect to sub-optimal detection. Specifically, we give a simple upper bound for $P_{\mathrm{det}}$ 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 $\tau$-dependent and less than half, well below Polya's optimal detection for a classical random walk.