  The multiscale entanglement renormalization ansatz (MERA) provides a
constructive algorithm for realizing wavefunctions that are inherently scale
invariant. Unlike conformally invariant partition functions however, the finite
bond dimension $\chi$ of the MERA provides a cut-off in the fields that can be
realized. In this letter, we demonstrate that this cut-off is equivalent to the
one obtained when approximating a thermal state of a critical Hamiltonian with
a matrix product operator (MPO) of finite bond dimension $\chi$. This is
achieved by constructing an explicit mapping between the isometries of a MERA
and the local tensors of the MPO.
