The exact dynamics of many-particle quantum systems coupled to an environment is governed by the Kadanoff-Baym equations, which describe the time evolution of Green's functions containing both population and spectral information. While these equations can, in principle, be solved exactly for small systems, the numerical cost grows rapidly and becomes prohibitive for longer times or larger systems. Beyond many-body correlations, the primary computational challenge arises from memory effects, which cause the numerical complexity to scale cubically with physical time. In certain cases, this scaling can be reduced to linear by reconstructing the lesser and greater Green's function components from the electron density, however, at the expense of losing spectral information. It is demonstrated here that the Kadanoff-Baym equations for open systems can be solved numerically exactly with optimal quadratic time scaling, without introducing any approximations. This result significantly extends the feasibility of nonequilibrium simulations and provides a simple, exact benchmark for testing approximate methods.

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• Quantum transport
• Ultrafast phenomena
• Nonequilibrium systems
• Quantum many-body systems
• Nonequilibrium Green's function