Sublinear scaling in non-Markovian open quantum systems simulations

While several numerical techniques are available for predicting the dynamics of non-Markovian open quantum systems, most struggle with simulations for very long memory and propagation times, e.g., due to superlinear scaling with the number of time steps n. Here, we introduce a numerically exact algorithm to calculate process tensors - compact representations of environmental influences - which provides a scaling advantage over previous algorithms by leveraging self-similarity of the tensor networks that represent the environment. It is applicable to environments with Gaussian statistics, such as for spin-boson-type open quantum systems. Based on a divide-and-conquer strategy, our approach requires only O(n log n) singular value decompositions for environments with infinite memory. Where the memory can be truncated after n_c time steps, a nominal scaling O(n_c log n_c) is found, which is independent of n. This improved scaling is enabled by identifying process tensors with repeatable blocks. To demonstrate the power and utility of our approach, we provide three examples. (1) We calculate the fluorescence spectra of a quantum dot under both strong driving and strong dot-phonon couplings, a task requiring simulations over millions of time steps, which we are able to perform in minutes. (2) We efficiently find process tensors describing superradiance of multiple emitters. (3) We explore the limits of our algorithm by considering coherence decay with a very strongly coupled environment. The observed computation time is not necessarily proportional to the number of singular value decompositions because the matrix dimensions also depend on the number of time steps. Nevertheless, quasilinear and sublinear scaling of computation time is found in practice for a wide range of parameters. While there are instances where existing methods can achieve comparable nominal scaling by precalculating effective propagators for time-independent or periodic system Hamiltonians, process tensors contain all the information needed to extract arbitrary multitime correlation functions of the system when driven by arbitrary time-dependent system Hamiltonians. The algorithm we present here not only significantly extends the scope of numerically exact techniques to open quantum systems with long memory times, but it also has fundamental implications for the simulation complexity of tensor network approaches.