**Overcoming the exponential tensor storage**

The discretized path integral expression for the reduced density matrix (RDM) of a system interacting with a dissipative harmonic bath is fully entangled because of influence functional terms that couple the variables at different time points. The QuAPI algorithm, which exploits the finite length of influence functional memory, involves a tensor propagator whose size grows exponentially with memory length. Recent work has shown that the path integral variables can be disentangled by recursively spreading the correlations over longer time intervals, while decreasing their contribution.

Eventually, the remaining entangled term becomes negligible, leading to an iterative propagation algorithm that involves simple multiplication of matrices whose size is equal to that of the bare system. The SMatPI algorithm is stable and extremely efficient, extending the applicability of numerically exact real-time path integral methods to multi-state systems.

The figure below shows the time evolution of populations in an 11-site system with nearest- and next-nearest interactions interacting with a dissipative harmonic bath.

Related Articles:

- Makri, “Small matrix path integral with extended memory”,
*J. Chem. Theory and Comput.***17**, 1-6 (2021). - N. Makri, “Small matrix path integral for system-bath dynamics”,
*J. Chem. Theory and Comput.***16**, 4038-4049 (2020). - N. Makri, “Small matrix disentanglement of the path integral: Overcoming the exponential tensor scaling with memory length”,
*J. Chem. Phys.*(Commun.) 152, 041104 (2020).