  We study the sampling complexity of a probability distribution associated
with an ensemble ofidentical noninteracting bosons undergoing a quantum random
walk on a one-dimensional lattice.With uniform nearest-neighbor hopping we show
that one can efficiently sample the distribution fortimes logarithmic in the
size of the system, while for longer times there is no known efficient
samplingalgorithm. With time-dependent hopping and optimal control, we design
the time evolution toapproximate an arbitrary Haar-random unitary map analogous
to that designed for photons in alinear optical network. This approach
highlights a route to generating quantum complexity byoptimal control only of a
single-body unitary matrix. We study this in the context of two
potentialexperimental realizations: a spinor optical lattice of ultracold atoms
and a quantum gas microscope.
