Efficient classical simulation of time dynamics in Fermi-Hubbard models with imaginary interactions

Abstract

Using a map between the Lindbladian evolution of dephasing in free fermions and the time evolution of imaginary-interaction Fermi-Hubbard models in bipartite lattices, we present an efficient classical algorithm to solve the Schrödinger equation in these interacting systems. This algorithm leverages the recently discovered algorithm for simulating Lindbladian evolution by sampling mixed unitary channels (Wang et al arXiv:2601.06298). We comment on the expected classical complexity of the problem for general complex values of the parameters and discuss some applications.

Figures (2)

• Figure 1: 2-particle wavefunction $\Psi_{1,1}$ with $n_\uparrow=n_\downarrow=1$. Here $\Psi_{1,1}$ is computed by direct solution of the (complexified) Schrödinger equation for $J=1$, $U=1$. $\mathbb{E}(\Psi_{1,1})$ is the sample mean obtained with the classical protocol Alg. \ref{['alg:class_sim']}. This results are obtained with 200 samples per data point, $t/R=0.01$ and $R=1000$.
• Figure 2: Classical complexity landscape of time evolution in (complexified) Fermi-Hubbard models (Complexity doughnut). The yellow star ${\rm arg}(J)={\rm arg}(U)=0$ represents the time evolution of usual Fermi-Hubbard models. Time dynamics simulation of these is believed to be difficult classically for generic values of the parameters. The surface of the torus $s=1\rightarrow U=0$ on the contrary is efficiently simulable due to the algebraic structure of free fermion Hamiltonians. The results of this work shows that on the disk ${\rm arg}(U)=\frac{\pi}{2}$ the line ${\rm arg}(J)=0$ is also classically simulable. The classically accessible region with the method of this work extends over $\Im(J)=O((Lt)^{-1})$ where $L$ is the size of the system and $t$ is the simulation time.