Over the past three decades, a fruitful approach for analysis and data-driven modeling of dynamical systems has been to consider the action of (nonlinear) dynamics in state space on linear spaces of observables. These methods leverage the linearity of the associated evolution operators, namely the Koopman and transfer operators, to carry out tasks such as mode decomposition, forecasting, and uncertainty quantification using linear operator techniques. Mathematically, the operator-theoretic approach has close connections with representations of nonlinear transformations (the state space dynamics) into spaces of functions (the observables) with a commutative algebraic structure.

In this talk we discuss generalizations of this framework to the setting of non-commutative algebras of operators using ideas from quantum information science. Central to our approach is a representation of observables and probability densities through multiplication operators and density operators (“quantum states”), respectively. Using these objects, and the dynamical operators governing their evolution, we formulate two common problems in dynamical systems modeling, namely data assimilation and dynamical closure, in an operator-theoretic language. We discuss how the operator-theoretic approach leads to structure-preserving computational schemes (e.g., positivity-preserving function approximation) which are also amenable to data-driven implementation using kernel methods for operator approximation. We present applications to data assimilation of the El Nino Southern Oscillation of the climate system and subgrid-scale modeling in multiscale systems. In the second part of the talk, we discuss implementations of these methods on quantum computers.