# Quantum circuit approximations and entanglement renormalization for the Dirac field in 1+1 dimensions

## TL;DR

This paper constructs explicit quantum circuits using entanglement renormalization and wavelet analysis to accurately approximate correlation functions of the massless Dirac conformal field theory directly in the continuum.

## Contribution

It provides a rigorous, continuum-targeted construction of entanglement-renormalization circuits for the Dirac field, bridging the gap between lattice models and continuum quantum field theories.

## Key findings

- Successfully approximates Dirac CFT correlation functions
- Uses wavelet-based multiresolution analysis for circuit construction
- Offers a framework for quantum circuit approximations of other CFTs

## Abstract

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.