Learning Parallel Computations with ParaLab

In this paper, we present the ParaLab teachware system, which can be used for learning the parallel computation methods. ParaLab provides the tools for simulating the multiprocessor computational systems with various network topologies, for carrying out the computational experiments in the simulation mode, and for evaluating the efficiency of the parallel computation methods. The visual presentation of the parallel computations taking place in the computational experiments is the key feature of the system. ParaLab can be used for the laboratory training within various teaching courses in the field of parallel, distributed, and supercomputer computations

Localization in periodically modulated speckle potentials

Disorder in a 1D quantum lattice induces Anderson localization of the eigenstates and drastically alters transport properties of the lattice. In the original Anderson model, the addition of a periodic driving increases, in a certain range of the driving's frequency and amplitude, localization length of the appearing Floquet eigenstates. We go beyond the uncorrelated disorder case and address the experimentally relevant situation when spatial correlations are present in the lattice potential. Their presence induces the creation of an effective mobility edge in the energy spectrum of the system. We find that a slow driving leads to resonant hybridization of the Floquet states, by increasing both the participation numbers and effective widths of the states in the strongly localized band and decreasing values of these characteristics for the states in the quasi-extended band. Strong driving homogenizes the bands, so that the Floquet states loose compactness and tend to be spatially smeared. In the basis of the stationary Hamiltonian, these states retain localization in terms of participation number but become de-localized and spectrum-wide in term of their effective widths. Signatures of thermalization are also observed.Comment: 6 pages, 3 figure

Unfolding quantum master equation into a system of real-valued equations: computationally effective expansion over the basis of SU(N)SU(N) generators

Dynamics of an open $N$-state quantum system is typically modeled with a Markovian master equation describing the evolution of the system's density operator. By using generators of $SU(N)$ group as a basis, the density operator can be transformed into a real-valued 'Bloch vector'. The Lindbladian, a super-operator which serves a generator of the evolution, %in the master equation, can be expanded over the same basis and recast in the form of a real matrix. Together, these expansions result is a non-homogeneous system of $N^2-1$ real-valued linear differential equations for the Bloch vector. Now one can, e.g., implement a high-performance parallel simplex algorithm to find a solution of this system which guarantees exact preservation of the norm and Hermiticity of the density matrix. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity $\mathcal{O}(N^{10})$. The complexity can be reduced when the number of dissipative operators is independent of $N$, which is often the case for physically meaningful models. Here we present an algorithm to transform quantum master equation into a system of real-valued differential equations and propagate it forward in time. By using a scalable model, we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with $N = 10^3$ states on a single node of a computer cluster

Transforming Lindblad Equations Into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm

With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell–Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per nod