  We introduce a novel, model-independent method for the efficient simulation
of low-entropy systems, whose dynamics can be accurately described with a
limited number of states. Our method leverages the time-dependent variational
principle to efficiently integrate the Lindblad master equation, dynamically
identifying and modifying the low-rank basis over which we decompose the
system's evolution. By dynamically adapting the dimension of this basis, and
thus the rank of the density matrix, our method maintains optimal
representation of the system state, offering a substantial computational
advantage over existing adaptive low-rank schemes in terms of both
computational time and memory requirements. We demonstrate the efficacy of our
method through extensive benchmarks on a variety of model systems, with a
particular emphasis on multi-qubit bosonic codes, a promising candidate for
fault-tolerant quantum hardware. Our results highlight the method's versatility
and efficiency, making it applicable to a wide range of systems characterized
by arbitrary degrees of entanglement and moderate entropy throughout their
dynamics. We provide an implementation of the method as a Julia package, making
it readily available to use.
