Complexity of Quantum Impurity Problems
Abstract
We give a quasipolynomial 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 creationannihilation 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 {√{x}} function. We anticipate that our algorithms may be used in hybrid quantumclassical 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 December 2017
 DOI:
 10.1007/s0022001729769
 arXiv:
 arXiv:1609.00735
 Bibcode:
 2017CMaPh.356..451B
 Keywords:

 Quantum Physics;
 Condensed Matter  Strongly Correlated Electrons;
 Mathematical Physics
 EPrint:
 Comm. Math. Phys. Vol. 356(2), pp. 451500 (2017)