The multiscale entanglement renormalization ansatz describes quantum many-body states
by a hierarchical entanglement structure organized by length scale. Numerically, it has been
demonstrated to capture critical lattice models and the data of the corresponding conformal
field theories with high accuracy. However, a rigorous understanding of its success and precise
relation to the continuum is still lacking. To address this challenge, we provide an explicit
construction of entanglement-renormalization quantum circuits that rigorously approximate
correlation functions of the massless Dirac conformal field theory. We directly target the
continuum theory: discreteness is introduced by our choice of how to probe the system, not
by any underlying short-distance lattice regulator. To achieve this, we use multiresolution
analysis from wavelet theory to obtain an approximation scheme and to implement entanglement
renormalization in a natural way. This could be a starting point for constructing quantum circuit
approximations for more general conformal field theories