2012/02/26 by Nihal Yapage, Yapage, Nihal
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Complex Systems and Time Series Analysis #FOS: Physical sciences #Quantum Physics (quant-ph) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy #cond-mat.stat-mech #quant-ph
paper · pdf · doi:10.48550/arxiv.1202.5726
10 pages
arxiv created 2012/02/26 · openalex publication_date 2012/02/26 · arxiv updated 2012/02/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This work is a simple extension of \citeNNjpa. We apply the concepts of information geometry to study the mean-field approximation for a general class of quantum statistical models namely the higher-order quantum Boltzmann machines (QBMs). The states we consider are assumed to have at most third-order interactions with deterministic coupling coefficients. Such states, taken together, can be shown to form a quantum exponential family and thus can be viewed as a smooth manifold. In our work, we explicitly obtain naive mean-field equations for the third-order classical and quantum Boltzmann machines and demonstrate how some information geometrical concepts, particularly, exponential and mixture projections used to study the naive mean-field approximation in \citeNNjpa can be extended to a more general case. Though our results do not differ much from those in \citeNNjpa, we emphasize the validity and the importance of information geometrical point of view for higher dimensional classical and quantum statistical models.