In many body quantum mechanics, the non-equilibrium Green's function gives us knowledge of physical quantities such as electron density, current, and available energy states. The integro-differential equation that governs the dynamics of the Green's function, known as the Kadanoff-Baym equation, poses significant computational challenges as the cost scales cubically with the final time. Furthermore, storage of the full Green's function becomes costly as it scales quadratically with the final time, with some long-time calculations requiring terabytes of data. In practice, people typically make approximations to these equations, which lowers the scaling of both cost and memory, however these approximations are uncontrolled and do not have parameters that can be tuned in order to converge to the exact solution. Recently, it has been shown that these Green's function objects are highly compressible and have a hierarchical off-diagonal low-rank (hodlr) structure. This compressibility allows for a construction of an algorithm that scales as T 2 log(T) instead of cubically, and requires much less memory to represent the Green's function. We have developed a high-order BDF integration scheme that solves the Kadanoff-Baym equations, while updating the hodlr representation on the fly. We apply our method to several interesting physical systems, including a molecule driven by a laser pulse and a superconductor with a temperature quench. We show that our method performs orders of magnitude better than the existing quantum Green's function solver codes and is thus able to probe much longer time scales.
