  KSZ velocity reconstruction is a recently proposed method for mapping the
largest-scale modes of the universe, by applying a quadratic estimator
$\hat{v}_r$ to the small-scale CMB and a galaxy catalog. We implement kSZ
velocity reconstruction in an $N$-body simulation pipeline and explore its
properties. We find that the reconstruction noise can be larger than the
analytic prediction which is usually assumed. We revisit the analytic
prediction and find additional noise terms which explain the discrepancy. The
new terms are obtained from a six-point halo model calculation, and are
analogous to the $N^{(1)}$ and $N^{(3/2)}$ biases in CMB lensing. We implement
an MCMC pipeline which estimates $f_{NL}$ from $N$-body kSZ simulations, and
show that it recovers unbiased estimates of $f_{NL}$, with statistical errors
consistent with a Fisher matrix forecast. Overall, these results confirm that
kSZ velocity reconstruction will be a powerful probe of cosmology in the near
future, but new terms should be included in the noise power spectrum.
