Abstract
Many-body quantum systems are subjected to the Curse of Dimensionality: The dimension of the Hilbert space H, where these systems live in, grows exponentially with number of their components. However, with some systems, it is possible to escape the curse by using a low-rank tensor approximation 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 scheme, which allows keeping the length of the state descriptions fixed, it is in principle possible to reach the asymptotic states. We propose and realize a cluster implementation of this idea. The implementation on four nodes allowed us to resolve the asymptotic states of the model systems with N = 128 spins (total dimension of the Hilbert space dimH = 2128 ≈ 1039).
Highlights
Many-body systems are at the focus of the current research in theoretical and experimental quantum physics
The evolution of an open quantum system towards its steady states is usually modeled with a Markovian master equation, which describes the dynamics of the system density operator (t), ̇(t) = L (t) [3]
In the case of open quantum systems, the complexity squares: to describe a density operator one needs L(N ) ∼ d2N real-valued parameters. This is a famous problem in modern data science – manipulations with data tensors becomes impossible when the data are sorted in high-dimensional spaces
Summary
Many-body systems are at the focus of the current research in theoretical and experimental quantum physics. In the case of open quantum systems, the complexity squares: to describe a density operator one needs L(N ) ∼ d2N real-valued parameters. This is a famous problem in modern data science – manipulations (or even storing) with data tensors becomes impossible when the data are sorted in high-dimensional spaces. The MPS/MPO representation allows for effective propagation of quantum many-body systems in time by using the so-called Time-Evolving Block Decimation (TEBD) scheme [11] This is a procedure to reduce the description of the state, obtained after every propagation step, to a given fixed length Lcut. We report the results of our studies in these directions
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