Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

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.1 Here, by 'solve', we mean to output a quantum state that efficiently encodes the classical solution vector. Thus approaches based on classical encodings such as in Ref.[27] are not covered here. We note that this state will, necessarily, need to be sampled many times to extract the desired information.

Koopman-Based Spectral Clustering of Directed and Time-Evolving Graphs

“…A few exemplary applications are illustrated in Figure 1. More details regarding the specific application areas can be found in the following publications: (a) molecular dynamics [1,2,3,4,5,6], (b) fluid dynamics [7,8,9,10,11,12], (c) climate science [13,14,15,16,17,18], (d) quantum physics [19,20,21,22,23,24], (e) chaotic dynamical systems [25,26,27,28,29], (f) system identification [30,31,32,33,34], (g) control theory [35,36,37,38,39,40], and (h) graphs and networks [41,42,43,44,45,46]. Koopman-based methods have also been applied to video data, EEG recordings, traffic flow data, stock prices, and various other data sets.…”

Operational classical mechanics: holonomic systems

“…Another active area of research involving the KvN theory is the development of consistent quantum-classical hybrid theories [9][10][11][12][13][14][15]. A third and more recent line of research deals with quantum simulations of non-linear systems [16][17][18]. However, apart from any connection with quantum physics, the operational methods had produced results entirely within the classical domain [19][20][21][22][23][24][25][26][27][28].…”

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

“…It is similar to moment closure technique in kinetic theory, in which one attempts to close the moment system with finite number of moments but often ends up with a closed system that has mathematical stability problems [7,27] or physical realizibility issues [39]. This is similarly true for methods in [40,50], see [43].…”