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Studies in computational intelligence

An Efficient Adaptive Monte Carlo Approach for Multidimensional Quantum Mechanics

Venelin Todorov, Ivan Dimov

Published Jan 1, 2022• 0 citations
Publisher
DOI
10.1007/978-3-031-06839-3_18

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Multidimensional numerical integration is a central ingredient of theoretical predictions in high-energy physics, where multiloop Feynman diagrams and phase-space integrals are computationally demanding due to divergences and complex mathematical structures. Established Adaptive Importance Sampling methods for numerical integration, such as VEGAS, iteratively refine a grid in a separable way, dimension by dimension. This keeps the algorithm scalable but reduces performance when strong inter-variable correlations are present. In this work, we introduce a hybrid quantum-classical algorithm that performs Quantum Adaptive Importance Sampling (QAIS) for multidimensional Monte Carlo integration. Our approach uses a Parametrized Quantum Circuit to encode a non-separable Probability Density Function on a multidimensional grid and allocate samples efficiently in the integration domain. We apply the method to a sharply peaked loop Feynman integral and to multi-modal benchmark integrals. Our results show that QAIS provides an efficient route for high-precision evaluation of multidimensional integrals. Theoretical calculations in particle physics rely on multidimensional integrals that become difficult to evaluate in the presence of correlations. Here, the authors show that a hybrid quantum-classical algorithm implementing Quantum Adaptive Importance Sampling can learn correlated structures efficiently and deliver precise results for nontrivial integrals.
This chapter presents development of a Bohmian trajectory-based approach for computing the quantum mechanical structure, energetics, and thermodynamics of multiatom systems. It reviews the salient features of the Bohmian approach, focusing upon how one might use it to develop new computational approaches for many-body systems. The chapter presents a variational approach that finds the quantum ground state for N-atom rare clusters using a statistical modeling approach for determining a best estimate of the quantum potential for a multidimensional system. It explores an efficient strategy for determining the quantum density associated with a statistical ensemble of space-time trajectories. The thermodynamics of small mesoscale systems is of considerable interest since what are typically extensive variables that scale monotonically with system size can exhibit anomalous behavior as the system size becomes small. The “node problem” has been a bugbear in the development of time-dependent quantum trajectory approaches and plagues Monte Carlo approaches.
Quantum optimization, particularly utilizing Monte Carlo methods, offers a promising pathway to accelerate the design of novel materials. However, classical Monte Carlo simulations often suffer from slow convergence and high computational cost. This paper proposes an adaptive quantum Monte Carlo algorithm that dynamically adjusts simulation parameters during the design process, aiming to significantly enhance the efficiency of material design. We introduce a novel Bayesian optimization framework incorporating gradient descent to optimize simulation parameters, leading to improved convergence rates and reduced computational time. The algorithm's adaptability is achieved through a feedback loop that continuously evaluates the quality of the simulation and adjusts the parameters accordingly. This results in a more efficient and robust approach to material exploration and design. The study demonstrates the effectiveness of this adaptive quantum Monte Carlo method in generating promising material candidates, highlighting its potential to revolutionize the field of materials science.