  It is usually assumed that a quantum computation is performed by applying
gates in a specific order. One can relax this assumption by allowing a control
quantum system to switch the order in which the gates are applied. This
provides a more general kind of quantum computing, that allows transformations
on blackbox quantum gates that are impossible in a circuit with fixed order.
Here we show that this model of quantum computing is physically realizable, by
proposing an interferometric setup that can implement such a quantum control of
the order between the gates. We show that this new resource provides a
reduction in computational complexity: we propose a problem that can be solved
using $O(n)$ blackbox queries, whereas the best known quantum algorithm with
fixed order between the gates requires $O(n^2)$ queries. Furthermore, we
conjecture that solving this problem in a classical computer takes exponential
time, which may be of independent interest.
