Complexity of quantum impurity problems

TL;DR

This paper presents a quasi-polynomial time classical algorithm for estimating ground state energies of quantum impurity models, enabling efficient low-energy state computation and analysis of their properties.

Contribution

The authors introduce a novel classical algorithm for impurity models that approximates ground energies and constructs low-energy states with superpositions of Gaussian states, advancing simulation capabilities.

Findings

Algorithm achieves ground energy approximation with additive error 2^{-b} in time n^3 exp(O(b^3))

Eigenvalues of ground state covariance matrices decay exponentially, depending mildly on spectral gap

Benchmarking on the single impurity Anderson model demonstrates practical effectiveness

Abstract

We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of $n$ fermionic modes and has a Hamiltonian $H=H_0+H_{imp}$, where $H_0$ is quadratic in creation-annihilation operators and $H_{imp}$ is an arbitrary Hamiltonian acting on a subset of $O(1)$ modes. We show that the ground energy of $H$ can be approximated with an additive error $2^{-b}$ in time $n^3 \exp{[O(b^3)]}$. Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of $\exp{[O(b^3)]}$ fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of $H_0$. A key ingredient of our proof is Zolotarev's rational approximation to the $\sqrt{x}$ function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model.