  We propose an algorithm to obtain Green's functions as a continued fraction
on quantum computers, which is based on the construction of the Krylov basis
using variational quantum algorithms, and included in a Lanczos iterative
scheme. This allows the integration of quantum algorithms with first principles
material science simulations, as we demonstrate within the dynamical mean-field
theory (DMFT) framework. DMFT enables quantitative predictions for strongly
correlated materials, and relies on the calculation of Green's functions. On
conventional computers the exponential growth of the Hilbert space with the
number of orbitals limits DMFT to small systems. Quantum computers open new
avenues and can lead to a significant speedup in the computation of expectation
values required to obtain the Green's function. We apply our Krylov variational
quantum algorithm combined with DMFT to the charge transfer insulator
La$_{2}$CuO$_4$ using a quantum computing emulator, and show that with 8 qubits
it predicts the correct insulating material properties for the paramagnetic
phase. We therefore expect that the method is ideally suited to perform
simulations for real materials on near term quantum hardware.
