Tensor network approximation of Koopman operators

TL;DR

The paper develops a quantum-inspired, tensor-network framework to approximate the time evolution of observables under measure-preserving ergodic dynamics via a spectrally convergent generator $W_\tau$ acting on a reproducing kernel Hilbert algebra $\mathcal{H}_\tau$. It lifts the regularized Koopman evolution to a unitary group on the Fock space $F(\mathcal{H}_\tau)$, enabling a multiplicative, tensor-network representation (tree TTN) that captures high-dimensional information from a small set of eigenfunctions. The method preserves positivity, provides a convergence analysis, and demonstrates effective predictions on 2-torus dynamics (torus rotation with pure point spectrum and Stepanoff flow with weak mixing) compared with classical and purely quantum approaches. The framework offers a scalable, structure-preserving alternative for Koopman analysis with potential for quantum-inspired computation and efficient TTN implementations on classical hardware, and it motivates future work toward quantum hardware realizations and broader dynamical regimes.

Abstract

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 $W_τ$ that acts on a reproducing kernel Hilbert space $\mathcal H_τ$ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators $e^{t W_τ}$ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space $F(\mathcal H_τ)$ generated by $\mathcal H_τ$ 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 $\mathcal H_τ^{\otimes n} \subset F(\mathcal H_τ)$ 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 $W_τ$. 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.

Figures (11)

• Figure 1: Evolution of quantum states (Schrödinger picture) and observables (Heisenberg picture) under the tensor network approximation framework described in this paper. Given a classical probability density $p \in S_*(\mathfrak A)$, our scheme approximates the evolution of a vector state $\rho = \langle p^{1/2}, \cdot \rangle_H p^{1/2} \in S_*(\mathfrak B)$ under the induced transfer operator $\mathcal{P}^t$ by a vector state $\rho_\tau = \langle \xi_\tau, \cdot \rangle_{\mathcal{H}_\tau} \xi_\tau$ on an RKHA $\mathcal{H}_\tau$ under an evolution operator $\mathcal{P}^t_\tau$ induced from a regularized Koopman operator $U^t_\tau$ on $\mathcal{H}_\tau$ with pure point spectrum. Similarly, the evolution of quantum observables $A \in \mathfrak B$ under the induced Koopman operator $\mathcal{U}^t$ is approximated by a regularized operator $\mathcal{U}^t_\tau$ induced from $U^t_\tau$ acting on smoothed quantum observables $A_\tau = K_\tau A K_\tau^*$, where $K_\tau\colon H \to \mathcal{H}_\tau$ is a kernel integral operator associated with $\mathcal{H}_\tau$. Our scheme then dilates the regularized Koopman dynamics on $\mathcal{H}_\tau$ to unitary dynamics $\tilde{U}^t \colon F(\mathcal{H}_\tau) \to F(\mathcal{H}_\tau)$ on the Fock space generated by $\mathcal{H}_\tau$, such that $\tilde{U}^t$ acts multiplicatively (tensorially) on the tensor algebra $T(\mathcal{H}_\tau) \subset F(\mathcal{H}_\tau)$. Furthermore, the quantum observables $A_\tau$ are amplified using the coalgebra structure of $\mathcal{H}_\tau$ to quantum observables $A_{\tau, n}$ that act on any grading $\mathcal{H}_\tau^{\otimes (n+1)} \subset F(\mathcal{H}_\tau)$. The state vector $\xi_\tau$ is also dilated to a vector $\eta_\tau \in F(\mathcal{H}_\tau)$ that projects non-trivially on $\mathcal{H}^{\otimes n}_\tau$ for every $n \in \mathbb N$, with an associated quantum state $\nu_\tau = \langle \eta_\tau, \cdot\rangle_{F(\mathcal{H}_\tau)} \eta_\tau$. The expectation of $A_{\tau, n}$ with respect to the time-evolved quantum state $\tilde{\rho}_\tau$ under the transfer operator $\tilde{\mathcal{P}}^t_\tau$ induced by $\tilde{U}^t_\tau$ is then used to approximate the corresponding evolution under the true dynamics on $\mathfrak B$ using a tree tensor network construction (see \ref{['fig:network']}).
• Figure 2: Time evolution of the von Mises density $p_{\mu, \kappa}$ with $\mu=\pi$ and $\kappa = 6$ under the circle rotation with frequency $\alpha=1$ ($f^{(t)}_\text{true}$; blue), the classical approximation ($f^{(t)}_\text{cl}$; orange), the quantum mechanical approximation ($f^{(t)}_\text{qm}$; green), and the tensor network/ Fock space approximation ($f^{(t)}_\text{Fock}$; red). The top and bottom rows show the evolution times $t= 0$ and $t=5$, respectively. The panels in the right-hand column show the pointwise approximation error $f^{(t)}_\text{approx} - f^{(t)}_\text{true}$ for each evolution time and approximation method.
• Figure 3: Simple examples of tensor networks. From left to right: A scalar, $s$; a rank-1 tensor (vector), $\bm v$; a rank-2 tensor (matrix), $\bm A$; the contraction between two rank-2 tensors, $\bm M$ and $\bm N$.
• Figure 4: Tree tensor network used to compute the Fock space evolution $f^{(t)}_{\sigma, \tau, n, d}$ in \ref{['eq:ft_fock_d']}. Structurally, the comultiplication operators form a tree surrounded by smoothing operators $\tilde{G}_{\tau, \sigma}$.
• Figure 5: Quiver plots of the dynamical vector fields $\vec{V}$ for the ergodic torus rotation (left) and Stepanoff flow (right) examples.

Theorems & Definitions (24)

• Example : Circle rotation
• Definition 1: RKHA
• Definition 2
• Definition 3
• Example : Circle rotation
• Lemma 4
• proof
• Definition 5
• Lemma 6
• proof