Order $O(1)$ algorithm for first-principles transient current through open quantum systems

First principles transient current through molecular devices is known to be extremely time consuming with typical computational complexity $T^3 N^3$ where $N$ and T are the dimension of the scattering system and the number of time steps respectively. Various algorithms have been developed which eventually brings the complexity down to $c T N^3$, a linear scaling in $T$, where $c$ is a large coefficient comparable to $N$. Here we provide an order $O(1)$ algorithm that reduces it further to $c_1 N^3+c_2 T N^2$ where $c_1$ and $c_2$ are $\sim$50 and 0.1 respectively. Hence for $TN$. Benchmark calculation has been done on graphene nanoribbons using Tight-binding (TB) Hamiltonian with a huge speed up factor of $100T$, confirmed the $O(1)$ scaling.