Challenges for quantum computation of nonlinear dynamical systems using linear representations

TL;DR

The paper analyzes the promise and limits of solving nonlinear dynamical systems on quantum computers using linear representations such as Koopman, KvN, and Liouville formalisms. It provides a unified framework tying forward Liouville/Perron–Frobenius and backward Koopman to KvN, and demonstrates that finite-dimensional projections of these infinite-dimensional linear systems introduce irreversible numerical artifacts that hinder reliable long-time predictions. The work shows Gibbs‑like phenomena in KvN discretization, instability in Carleman‑like closures, and the potential (but not guaranteed) advantages of Liouville/EDMD or direct linear solvers, while emphasizing the need for problem‑specific closures and sampling considerations. Overall, the results suggest that while QC methods can linearize nonlinear dynamics, practical deployment will require careful handling of discretization, closure, and measurement challenges, and remains an open, highly system‑dependent problem.

Abstract

A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear representations, such as the Koopman representation and Koopman von Neumann mechanics, have regained attention from the dynamical-systems research community. Here, we aim to present a unified theoretical framework, currently missing in the literature, with which one can compare and relate existing methods, their conceptual basis, and their representations. We also aim to show that, despite the fact that quantum simulation of nonlinear classical systems may be possible with such linear representations, a necessary projection into a feasible finite-dimensional space will in practice eventually induce numerical artifacts which can be hard to eliminate or even control. As a result, a practical, reliable and accurate way to use quantum computation for solving general nonlinear dynamical systems is still an open problem.

Figures (16)

• Figure 1: Schematic diagram of the relationship between the forward/Perron--Frobenius/Schrödinger representation and the backward/adjoint/Koopman/Heisenberg representation. In the forward picture, phase-space variables/distributions are evolving and the observables are fixed. In the backward picture, phase-space variables/distributions are fixed and the observables are evolving. (a) "Classical" systems given a single ($\delta$-distributed) initial condition; (b) "Classical statistical" systems given an initial distribution $\rho_0(x)$.
• Figure 2: Carleman linearization with truncation at the first hundred order and approximate Koopman projection to a subfunctional space spanned by only the first three order.
• Figure 3: The probability density predicted by KvN mechanics with 10-qubit discretization ($2^{10}$ points in $x\in[0,2.0]$.) Panel (a) shows the evolution of the probability density $\left\vert \psi\left(x,t\right) \right\vert^2$, visualized as a heatmap. $\left\vert\psi\right\vert^2 \ge 0.05$ is considered as saturated to increase the contrast of the image. The mode of the distribution evolves almost identically with the deterministic solution (cf. analytical solution in Figs. \ref{['fig:carleman']} and \ref{['fig:KvN-summaryStatistics']}). (b) We plot $\log_{10}\left\vert \psi\left(x,t\right) \right\vert^2$ as the heatmap to reveal the fine ripple structure resembling a Gibbs-like phenomenon due to the finite dimensional derivative operator.
• Figure 4: Summary statistics shows that the mode of the probability distribution follows the analytic solution over a short time horizon, $t\lesssim 5$, beyond which the Gibbs-like phenomenon is large enough to disturb the mode into almost random fluctuations. The mean, $\left\langle \psi(t) \vert \hat{X} \vert \psi(t) \right \rangle$, on the other hand, is not a useful predictor for all $t\ge 0$.
• Figure 5: The probability density predicted by the Chemical Master Equation with $2^{10}$ discretization points on a domain $x\in\left[0,2\right]$. Panel (a) shows the evolution of the joint probability distribution $\mathbf{p}(t)$, visualized as a heatmap. $p_i(t)\ge 0.05$, $i=1\ldots 1024$ is considered as saturated to increase the contrast of the image. We plot $\log_{10} \mathbf{p}(t)$ as the heatmap in Panel (b), for making comparison to KvN mechanics Fig. \ref{['fig:KvN']}(b).