  Many-body quantum systems are subjected to the Curse of Dimensionality: The
dimension of the Hilbert space $\mathcal{H}$, where these systems live in,
grows exponentially with systems' 'size' (number of their components,
"bodies"). It means that, in order to specify a state of a quantum system, we
need a description whose length grows exponentially with the system size.
However, with some systems it is possible to escape the curse by using low-rank
tensor approximations known as `matrix-product state/operator (MPS/O)
representation' in the quantum community and `tensor-train decomposition' among
applied mathematicians. Motivated by recent advances in computational quantum
physics, we consider chains of $N$ spins coupled by nearest-neighbor
interactions. The spins are subjected to an action coming from the environment.
Spatially disordered interaction and environment-induced decoherence drive
systems into non-trivial asymptotic states. The dissipative evolution is
modeled with a Markovian master equation in the Lindblad form. By implementing
the MPO technique and propagating system states with the time-evolving block
decimation (TEBD) scheme (which allows to keep the length of the state
descriptions fixed), it is in principle possible to reach the corresponding
steady states. We propose and realize a cluster implementation of this idea.
The implementation on four nodes allowed us to resolve steady states of the
model systems with $N = 128$ spins.
