Fast-forwardability of Jordan-Wigner-transformed Fermion models based on Cartan decomposition

TL;DR

This paper analyzes the algebraic structure of Jordan-Wigner-transformed fermion models and demonstrates that their Hamiltonian algebra's exponential growth limits the efficiency of Cartan-based fast-forwarding simulation methods.

Contribution

It establishes a lower bound on the Hamiltonian algebra's dimension for certain fermion models and shows this leads to exponential scaling in simulation complexity.

Findings

Hamiltonian algebra dimension grows exponentially with system size.

Cartan-based fast-forwarding circuit depth also scales exponentially.

Fermion models like the Hubbard model are not efficiently simulatable with this method.

Abstract

We study the Hamiltonian algebra of Jordan-Wigner-transformed interacting fermion models and its fast-forwardability. We prove that the dimension of the Hamiltonian algebra of the fermion model with single-site Coulomb interaction is bounded from below by the exponential function of the number of sites, and the circuit depth of the Cartan-based fast-forwarding method for such model also exhibits the same scaling. We apply this proposition to the Anderson impurity model and the Hubbard model and show that the dimension of the Hamiltonian algebra of these models scales exponentially with the number of sites. These behaviors of the Hamiltonian algebras imply that the qubit models obtained by the Jordan-Wigner transformation of these fermion models cannot be efficiently simulated using the Cartan-based fast-forwarding method.