使用线性表示进行非线性动力系统量子计算的挑战
Challenges for quantum computation of nonlinear dynamical systems using linear representations
AI总结：
本文针对量子计算求解非线性动力系统的线性表示方法，构建统一理论框架以关联现有方法，并指出有限维空间投影会产生难控数值伪影，该领域实用可靠的求解方法仍待突破。
摘要：
近期多项研究提出，线性表示适用于借助量子计算机求解非线性动力系统，而量子计算机本质上是对希尔伯特空间中的波函数进行线性作用。Koopman表示（库普曼表示）与Koopman von Neumann力学（库普曼-冯·诺依曼力学）等线性表示已重新获得动力系统研究领域的关注。本文旨在提出目前文献中缺失的统一理论框架，借此可对现有方法、其概念基础及表示形式进行比较与关联。本文还旨在表明，尽管借助此类线性表示或可实现非线性经典系统的量子模拟，但向可行的有限维空间进行的必要投影，在实际中最终会产生难以消除甚至难以控制的数值伪影。因此，用量子计算求解一般非线性动力系统的实用、可靠且准确的方法，仍是一个未解决的问题。
英文摘要：
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.
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