  Understanding the emergent system-bath correlations in non-Markovian and
non-perturbative open systems is a theoretical challenge that has benefited
greatly from the application of Matrix Product State (MPS) methods. Here, we
propose an autonmously adapative variant of the one-site
Time-Dependent-Variational-Principle (1TDVP) method for many-body MPS
wave-functions in which the local bond-dimensions can evolve to capture growing
  entanglement 'on the fly'. We achieve this by efficiently examining the
effect of increasing each MPS bond-dimension in advance of each dynamic
timestep, resulting in an MPS that can dynamically and inhomogeneously
restructure itself as the complexity of the dynamics grows across time and
space. This naturally leads to more efficient simulations, oviates the need for
multiple convergence runs, and, as we demonstrate, is ideally suited to the
typical, finite-temperature 'impurity' problems that describe open quantum
system connected to multiple environments.
