Propagating large open quantum systems towards their steady states: cluster implementation of the time-evolving block decimation scheme

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 number of their components ('bodies'). %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 keeping 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 (total dimension of the Hilbert space $\mathrm{dim}\mathcal{H} = 2^{128} \approx 10^{39}$).