  Sampling the distribution of bosons that have undergone a random unitary
evolution is strongly believed to be a computationally hard problem. Key to
outperforming classical simulations of this task is to increase both the number
of input photons and the size of the network. We propose driven boson sampling,
in which photons are input within the network itself, as a means to approach
this goal. When using heralded single-photon sources based on parametric
down-conversion, this approach offers an $\sim e$-fold enhancement in the input
state generation rate over scattershot boson sampling, reaching the scaling
limit for such sources. More significantly, this approach offers a dramatic
increase in the signal-to-noise ratio with respect to higher-order photon
generation from such probabilistic sources, which removes the need for photon
number resolution during the heralding process as the size of the system
increases.
