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Bulletin of the American Physical Society

Quantum Monte Carlo simulations of complex Hamiltonians

V. G. Rousseau, Kalani Hettiarachchilage, Ka-Ming Tam, Juana Moreno, Mark Jarrell

Published Mar 22, 2013• 0 citations
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Quantum Monte Carlo (QMC) simulations for the recent studies on complex materials were confronted by new computational challenges. Traditional approach to accelerate the simulations by parallel Monte Carlo chains faces serious scalability problems since the speedup is reaching the limitation predicted by Amdahl's law. Fine-grained parallelization of matrix kernels is essential to achieve better performance. In this paper, we investigate the performance optimization techniques on GPU for the most time consuming computational kernel in the Determinant Quantum Monte Carlo (DQMC) simulation: multiplication of matrix exponentials. The matrix, derived from the kinetic Hamiltonian, is highly sparse, and its exponential is approximated by the block checkerboard method, which can represent a matrix exponential as a product of a sequence of sparse matrices. The matrix exponentials from 2D and 3D toruses are focused, and various optimization techniques, such as data streaming and concurrent kernels, are proposed. Experiments show that the proposed optimization techniques can improve the SpMM (Sparse Matrix Multiplication) function, modified from the CUDA SKD SpMV function, up to 16 times and 117 times for 2D and 3D problems respectively.
Monte Carlo simulations are crucial for examining the Ising model, especially when it's tough to find analytical solutions. However, traditional Monte Carlo methods, like the Metropolis algorithm, encounter significant hurdles, such as slowing down near phase transitions and issues related to finite sizes. This paper looks into both the advantages and limitations of these traditional Monte Carlo techniques. It also covers recent developments like Tensor Network Monte Carlo and Quantum Monte Carlo methods, which have shown promise in overcoming these challenges. Furthermore, the paper explores how machine learning techniques are being incorporated into Monte Carlo simulations to enhance their efficiency and accuracy. These advancements represent a major leap forward in simulating complex systems and have expanded the use of Monte Carlo methods into new areas, including quantum systems and models with disorder. This paper finds that traditional Monte Carlo methods, like the Metropolis algorithm, face issues such as critical slowing down and finite-size effects. New methods, such as Tensor Network Monte Carlo (TNMC) and Quantum Monte Carlo (QMC), effectively address these challenges, enhancing simulation accuracy and efficiency in complex systems. Additionally, integrating machine learning optimizes parameters and accelerates convergence, expanding Monte Carlo’s applicability to large-scale, high-dimensional systems across various scientific fields.
Although stoquastic Hamiltonians are known to be simulable via sign-problem-free quantum Monte Carlo (QMC) techniques, the nonstoquasticity of a Hamiltonian does not necessarily imply the existence of a QMC sign problem.We give a sufficient and necessary condition for the QMC-simulability of Hamiltonians in a given basis: We prove that a QMC simulation will be sign-problem-free if and only if all the overall total phases along the chordless cycles of the weighted graph whose adjacency matrix is the Hamiltonian are zero (modulo 2π ).We use our findings to provide a construction for nonstoquastic, yet sign-problem-free and hence QMC-simulable, quantum many-body models.We also demonstrate why the simulation of truly sign-problematic models using the QMC weights of the stoquasticized Hamiltonian is generally suboptimal.We offer a superior alternative.