Quantum computational supremacy in the sampling of bosonic random walkers on a one-dimensional lattice

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.