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

## TL;DR

This paper presents a cluster implementation of the TEBD scheme to efficiently simulate large open quantum systems and determine their steady states using tensor network methods.

## Contribution

It introduces a cluster-based approach to extend TEBD for large-scale open quantum systems, enabling the analysis of systems with up to 128 spins.

## Key findings

- Successfully resolved steady states for 128-spin systems.
- Demonstrated the effectiveness of cluster implementation in large quantum simulations.
- Extended the applicability of tensor network methods to larger open quantum systems.

## Abstract

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.