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Understanding the emergent system-bath correlations in non-Markovian andnon-perturbative open systems is a theoretical challenge that has benefitedgreatly from the application of Matrix Product State (MPS) methods. Here, wepropose an autonmously adapative variant of the one-siteTime-Dependent-Variational-Principle (1TDVP) method for many-body MPSwave-functions in which the local bond-dimensions can evolve to capture growing  entanglement 'on the fly'. We achieve this by efficiently examining theeffect of increasing each MPS bond-dimension in advance of each dynamictimestep, resulting in an MPS that can dynamically and inhomogeneouslyrestructure itself as the complexity of the dynamics grows across time andspace. This naturally leads to more efficient simulations, oviates the need formultiple convergence runs, and, as we demonstrate, is ideally suited to thetypical, finite-temperature 'impurity' problems that describe open quantumsystem connected to multiple environments.

Dynamically Evolving Bond-Dimensions within the one-site  Time-Dependent-Variational-Principle method for Matrix Product States:  Towards efficient simulation of non-equilibrium open quantum dynamics