  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.
