Tree-Based Implementation of the Small Matrix Path Integral for System-Bath Dynamics (Communications in Computational Physics, 2024)
The small matrix path integral (SMatPI) method is an efficient numerical approach to simulating the evolution of a quantum system coupled to a harmonic bath.The method relies on a sequence of kernel matrices that defines the non-Markoviandynamics of the quantum system. In the original SMatPI method, these kernels arecomputed indirectly through the QuAPI method. Instead, we focus on the definitionof the kernel matrices and reveal a recurrence relation in these matrices. Using sucha relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to bemuch faster than straightforward computation of the kernel matrices based on theirdefinitions. This algorithm bypasses the step to compute the SMatPI matrices by otherpath integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numericalexperiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPIand SMatPI. In spite of this, our method may indicate some new properties of openquantum systems, and has the potential to be generalized to higher-order numericalschemes.