The multiscale entanglement renormalization ansatz describes quantum manybody 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.
Communications in Mathematical Physics

Witteveen, F, Scholz, V, Swingle, B, & Walter, M. (2021). Quantum circuit approximations and entanglement renormalization for the Dirac field in 1+1 dimensions. Communications in Mathematical Physics, 389, 75–120. doi:10.1007/s00220-021-04274-w