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 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

Case Study for Running Memory-Bound Kernels on RISC-V CPUs

The emergence of a new, open, and free instruction set architecture, RISC-V, has heralded a new era in microprocessor architectures. Starting with low-power, low-performance prototypes, the RISC-V community has a good chance of moving towards fully functional high-end microprocessors suitable for high-performance computing. Achieving progress in this direction requires comprehensive development of the software environment, namely operating systems, compilers, mathematical libraries, and approaches to performance analysis and optimization. In this paper, we analyze the performance of two available RISC-V devices when executing three memory-bound applications: a widely used STREAM benchmark, an in-place dense matrix transposition algorithm, and a Gaussian Blur algorithm. We show that, compared to x86 and ARM CPUs, RISC-V devices are still expected to be inferior in terms of computation time but are very good in resource utilization. We also demonstrate that well-developed memory optimization techniques for x86 CPUs improve the performance on RISC-V CPUs. Overall, the paper shows the potential of RISC-V as an alternative architecture for high-performance computing

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 node

On Solving the Problem of Finding Kinetic Parameters of Catalytic Isomerization of the Pentane-Hexane Fraction Using a Parallel Global Search Algorithm

This article is devoted to the problem of developing a kinetic model of a complex chemical reaction using a parallel optimization method. The design of the kinetic model consists of finding the kinetic parameters of the reaction, which cannot be calculated analytically, and since the chemical reaction involves many stages, the optimization problem is multiextremal. As a chemical reaction, the process of catalytic isomerization of the pentane-hexane fraction is considered, which is now important due to the switch of the oil refining industry to the production of gasoline corresponding to the Euro-5 standard. On the basis of known industrial data on the concentrations of reaction components and the temperature at the outlet of the third reactor, the activation energies and pre-exponential factors of each reaction stage were calculated. To solve the optimization problem, the authors developed a parallel global search algorithm and a program based on Lipschitz optimization. The kinetic parameters found made it possible to develop a mathematical model of the process, which is in good agreement with industrial data. The developed mathematical model in future works will make it possible to study the dynamics of the gas–liquid flow in the reactor unit, taking into account diffusion and heat exchange processes through the catalyst layer