Quantum embedding approaches for materials simulations on quantum computers

Quantum embedding approaches for materials simulations, such as the dynamical mean-field theory (DMFT), provide corrections to first-principles calculations for strongly correlated electrons, which are poorly described at lower levels of theory. These embedding methods are computationally demanding on classical computing architectures and hence remain restricted to small systems, limiting the scope of their applicability.

Here, we present an algorithm to compute Green's functions on quantum computers for interacting electron systems. It uses a continued fraction representation based on the Lanczos method, where the wave functions are expanded as linear combinations of basis states within a quantum subspace. While on conventional computers the cost of the computation grows exponentially with system size, limiting the method to small systems, by representing the basis states on a quantum computer one may overcome this exponential scaling barrier. We propose a two-level multigrid Trotter time evolution for an efficient preparation of the basis states in a quantum circuit, which takes advantage of the robustness of the subspace expansion against Trotter errors. Using a quantum emulator we demonstrate the algorithm for the Hubbard model on a Bethe lattice with infinite coordination, which we map to a 16 qubit Anderson impurity model within the dynamical mean-field theory. Our algorithm computes the Green's function accurately for both the metallic and Mott insulating regimes, with a circuit depth several orders of magnitude below what has been proposed using time evolution. The two-level multigrid time evolution reduces the number of Trotter steps required to compute the Green's function to about four to six. We, therefore, expect that the method can be used on near-term quantum computers for moderate system sizes, while allowing for scalability to larger circuit depths and qubit numbers on future fault-tolerant quantum computers.