|Résumé||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.