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While several numerical techniques are available for predicting the dynamicsof non-Markovian open quantum systems, most struggle with simulations for verylong memory and propagation times, e.g., due to superlinear scaling with thenumber of time steps $n$. Here, we introduce a numerically exact algorithm tocalculate process tensors -- compact representations of environmentalinfluences -- which provides a scaling advantage over previous algorithms byleveraging self-similarity of the tensor networks that represent Gaussianenvironments. Based on a divide-and-conquer strategy, our approach requiresonly $\mathcal{O}(n\log n)$ singular value decompositions for environments withinfinite memory. Where the memory can be truncated after $n_c$ time steps, ascaling $\mathcal{O}(n_c\log n_c)$ is found, which is independent of $n$. Thisimproved scaling is enabled by identifying process tensors with repeatableblocks. To demonstrate the power and utility of our approach we provide threeexamples. (1) We calculate the fluorescence spectra of a quantum dot under bothstrong driving and strong dot-phonon couplings, a task requiring simulationsover millions of time steps, which we are able to perform in minutes. (2) Weefficiently find process tensors describing superradiance of multiple emitters.(3) We explore the limits of our algorithm by considering coherence decay witha very strongly coupled environment. The algorithm we present here not onlysignificantly extends the scope of numerically exact techniques to open quantumsystems with long memory times, but also has fundamental implications forsimulation complexity.

Sublinear scaling in non-Markovian open quantum systems simulations