Large-scale stochastic propagation method beyond the sequential approach

The $O(N)$ stochastic propagation method, which relies on the numerical solution of the time-dependent Schr\"odinger equation using random initial states, is widely used in large-scale first-principles calculations. In this work, we eliminate the conventional sequential computation of intermediate states by introducing a concurrent strategy that minimizes information redundancy. The new method, in its state-, moment-, and energy-based implementations, not only surpasses the time step constraint of sequential propagation but also maintains precision within the framework of the Nyquist-Shannon sampling theorem. Systematic benchmarking on one billion atoms within the tight-binding model demonstrates that our new concurrent method achieves up to an order-of-magnitude speedup, enabling the rapid computation of a wide range of electronic, optical, and transport properties. This performance breakthrough offers valuable insights for enhancing other time-propagation algorithms, including those employed in large-scale stochastic density functional theory.