  We introduce a new technique for the simulation of dissipative quantum
systems. This method is composed of an approximate decomposition of the
Lindblad equation into a Kraus map, from which one can define an ensemble of
wavefunctions. Using principal component analysis, this ensemble can be
truncated to a manageable size without sacrificing numerical accuracy. We term
this method \emph{Ensemble Rank Truncation} (ERT), and find that in the regime
of weak coupling, this method is able to outperform existing wavefunction
Monte-Carlo methods by an order of magnitude in both accuracy and speed. We
also explore the possibility of combining ERT with approximate techniques for
simulating large systems (such as Matrix Product States (MPS)), and show that
in many cases this approach will be more efficient than directly expressing the
density matrix in its MPS form. We expect the ERT technique to be of practical
interest when simulating dissipative systems for quantum information, metrology
and thermodynamics.

Quantum Physics