  We introduce sequential analysis in quantum information processing, by
focusing on the fundamental task of quantum hypothesis testing. In particular
our goal is to discriminate between two arbitrary quantum states with a
prescribed error threshold, $\epsilon$, when copies of the states can be
required on demand. We obtain ultimate lower bounds on the average number of
copies needed to accomplish the task. We give a block-sampling strategy that
allows to achieve the lower bound for some classes of states. The bound is
optimal in both the symmetric as well as the asymmetric setting in the sense
that it requires the least mean number of copies out of all other procedures,
including the ones that fix the number of copies ahead of time. For qubit
states we derive explicit expressions for the minimum average number of copies
and show that a sequential strategy based on fixed local measurements
outperforms the best collective measurement on a predetermined number of
copies. Whereas for general states the number of copies increases as $\log
1/\epsilon$, for pure states sequential strategies require a finite average
number of samples even in the case of perfect discrimination, i.e.,
$\epsilon=0$.
