  Tensor network states are powerful variational ans\"atze for many-body ground
states of quantum lattice models. The use of Monte Carlo sampling techniques in
tensor network approaches significantly reduces the cost of tensor
contractions, potentially leading to a substantial increase in computational
efficiency. Previous proposals are based on a Markov chain Monte Carlo scheme
generated by locally updating configurations and, as such, must deal with
equilibration and autocorrelation times, which result in a reduction of
efficiency. Here we propose a perfect sampling scheme, with vanishing
equilibration and autocorrelation times, for unitary tensor networks -- namely
tensor networks based on efficiently contractible, unitary quantum circuits,
such as unitary versions of the matrix product state (MPS) and tree tensor
network (TTN), and the multi-scale entanglement renormalization ansatz (MERA).
Configurations are directly sampled according to their probabilities in the
wavefunction, without resorting to a Markov chain process. We also describe a
partial sampling scheme that can result in a dramatic (basis-dependent)
reduction of sampling error.
