  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.
