Talking Across Fields: A Physicist’s Presentation of some Mathematical Aspects of Quantum Monte Carlo Methods

Caffarel, Michel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 949-972.

Dans cet article nous discutons quelques aspects mathématiques des méthodes Monte Carlo quantique du point de vue du physicien. Une liste (non-exhaustive) de techniques probabilistes utilisées et développées en physique et de problèmes ouverts est présentée. Afin d’illuster l’approche des physiciens, nous décrivons en détail une des variantes des méthodes Monte Carlo quantique basée sur la formule de Feynman-Kac. Le problème de l’efficacité numérique au cœur des applications physiques où l’espace de configuration est en général de très grande dimension est présenté, ainsi que la solution adoptée. Finalement, nous explicitons les contraintes spécifiques associées à la simulation des systèmes de fermions et présentons le fameux “problème du signe” considéré comme un des problèmes les plus importants à résoudre en physique numérique.

This paper discusses some mathematical aspects related to the use of probabilistic techniques in quantum Monte Carlo (QMC) methods from a physicist’s point of view. A selected list of problems and techniques employed in computational physics and of interest to the applied probability community is presented. One of the variants of QMC approaches based on the Feynman-Kac formula is described in some detail. The problem of numerical efficiency at the heart of physical applications defined in (very) high-dimensional space is discussed and the commonly used solution through importance sampling is presented. Finally, the specific constraints related to fermion systems in QMC are presented and the celebrated “fermion sign problem” (considered as one of the most important open problem in computational physics) is discussed.

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