Operator theory, quantum information science, and statistical modeling of climate dynamics 

We survey aspects of operator theory and quantum information science for statistical modeling of climate dynamics, focusing on two core problems: data assimilation and dynamical closure. We discuss how techniques from operator theory lead to classical (Bayesian) and quantum formulations of these problems through a common set of axioms, implemented for appropriately chosen spaces of observables. This leads in turn to natural ways of consistently embedding the dynamical evolution of classical observables into a quantum evolution, wherein the Koopman and transfer operators play a key role. We describe how projections of the resulting quantum system onto finite-dimensional Hilbert spaces yield quantum-inspired numerical algorithms with favorable structure preservation properties (e.g., positivity preservation), asymptotic convergence guarantees, and amenability to data-driven approximation from time series data. We illustrate these methods with applications to prediction of the El Nino Southern Oscillation and coarse-graining of cloud-resolving atmospheric models. We also discuss prospects and challenges of implementing these methods on quantum computers.