  We propose a tensor network framework for approximating the evolution of
observables of measure-preserving ergodic systems. Our approach is based on a
spectrally-convergent approximation of the skew-adjoint Koopman generator by a
diagonalizable, skew-adjoint operator $W_\tau$ that acts on a reproducing
kernel Hilbert space $\mathcal H_\tau$ with coalgebra structure and Banach
algebra structure under the pointwise product of functions. Leveraging this
structure, we lift the unitary evolution operators $e^{t W_\tau}$ (which can be
thought of as regularized Koopman operators) to a unitary evolution group on
the Fock space $F(\mathcal H_\tau)$ generated by $\mathcal H_\tau$ that acts
multiplicatively with respect to the tensor product. Our scheme also employs a
representation of classical observables ($L^\infty$ functions of the state) by
quantum observables (self-adjoint operators) acting on the Fock space, and a
representation of probability densities in $L^1$ by quantum states. Combining
these constructions leads to an approximation of the Koopman evolution of
observables that is representable as evaluation of a tree tensor network built
on a tensor product subspace $\mathcal H_\tau^{\otimes n} \subset F(\mathcal
H_\tau)$ of arbitrarily high grading $n \in \mathbb N$. A key feature of this
quantum-inspired approximation is that it captures information from a tensor
product space of dimension $(2d+1)^n$, generated from a collection of $2d + 1$
eigenfunctions of $W_\tau$. Furthermore, the approximation is positivity
preserving. The paper contains a theoretical convergence analysis of the method
and numerical applications to two dynamical systems on the 2-torus: an ergodic
torus rotation as an example with pure point Koopman spectrum and a Stepanoff
flow as an example with topological weak mixing.
