Algebraic fast-forwarding of Jordan-Wigner transformed fermion models

The algebraic fast-forwarding method is a highly efficient approach for implementing time evolution operators in quantum systems. The circuit depth required for this method is determined by the dimension of the Hamiltonian algebra associated with the system's Hamiltonian. Despite its significance, the algebraic structure of interacting fermion models remains largely unexplored. In this presentation, we give a proof that the dimension of the Hamiltonian algebra of Jordan-Wigner transformed free-fermion models scales quadratically with the number of qubits. For a class of interacting fermion models with spin interactions, including the Anderson and Hubbard models, it scales exponentially with the qubit number. Furthermore, we point out that the dimension of the Hamiltonian algebra depends on the choice of the fermion basis, and we discuss how it can sometimes vary between polynomial and exponential order.