Rigorous free fermion entanglement renormalization from wavelet theory

TL;DR

This paper develops entanglement renormalization schemes for free fermion ground states using wavelet transforms, providing hierarchical quantum circuits that accurately approximate these states without variational optimization.

Contribution

It introduces a novel wavelet-based entanglement renormalization method for non-interacting fermions, capturing complex entanglement structures efficiently.

Findings

Hierarchical quantum circuits constructed from wavelet transforms.

Approximate ground states with controlled error without variational methods.

Effective handling of Fermi surface entanglement in 2D models.

Abstract

We construct entanglement renormalization schemes which provably approximate the ground states of non-interacting fermion nearest-neighbor hopping Hamiltonians on the one-dimensional discrete line and the two-dimensional square lattice. These schemes give hierarchical quantum circuits which build up the states from unentangled degrees of freedom. The circuits are based on pairs of discrete wavelet transforms which are approximately related by a "half-shift": translation by half a unit cell. The presence of the Fermi surface in the two-dimensional model requires a special kind of circuit architecture to properly capture the entanglement in the ground state. We show how the error in the approximation can be controlled without ever performing a variational optimization.