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.

翻译：近期多项研究提出，线性表示方法适用于用量子计算机求解非线性动力系统，因为量子计算机本质上是在希尔伯特空间中对波函数进行线性操作。诸如Koopman表示和Koopman-von Neumann力学等线性表示方法，已重新引起动力系统研究领域的关注。本文旨在建立一个目前文献中尚缺的统一理论框架，用以比较和关联现有方法、其概念基础及其表示形式。同时我们试图证明：尽管基于此类线性表示可能实现非线性经典系统的量子模拟，但实际计算中向有限维可行空间的必要投影，最终将不可避免地引入数值伪影，这些伪影往往难以消除甚至难以控制。因此，如何建立实用、可靠且精确的量子计算方法以求解一般非线性动力系统，仍然是一个悬而未决的问题。