  We investigate a tight binding quantum walk on a graph. Repeated stroboscopic
measurements of the position of the particle yield a measured "trajectory", and
a combination of classical and quantum mechanical properties for the walk are
observed. We explore the effects of the measurements on the spreading of the
packet on a one dimensional line, showing that except for the Zeno limit, the
system converges to Gaussian statistics similarly to a classical random walk. A
large deviation analysis and an Edgeworth expansion yield quantum corrections
to this normal behavior. We then explore the first passage time to a target
state using a generating function method, yielding properties like the
quantization of the mean first return time. In particular, we study the effects
of certain sampling rates which cause remarkable change in the behavior in the
system, like divergence of the mean detection time in finite systems and a
decomposition of the phase space into mutually exclusive regions, an effect
that mimics ergodicity breaking, whose origin here is the destructive
interference in quantum mechanics. For a quantum walk on a line we show that in
our system the first detection probability decays classically like
$(\text{time})^{-3/2}$, this is dramatically different compared to local
measurements which yield a decay rate of $(\text{time})^{-3}$, indicating that
the exponents of the first passage time depends on the type of measurements
used.
