Nonlinear quantum dynamics

János A. Bergou, Mark S. Hillery, Mark E. Saffman

Baoyang Zhang, Dong An, Zhaoyuan Meng +4 more

Nonlinear dynamics is ubiquitous in nature, ranging from chemical pattern formation to ocean circulation, yet its simulation on quantum computers is fundamentally limited by the unitary nature of quantum evolution. We propose the quantum Koopman method, a data-driven framework that embeds nonlinear dynamics into a learned linear representation and implements the resulting evolution using shallow quantum circuits. This method learns Koopman observables from trajectory data, projects the lifted dynamics onto a finite-dimensional subspace, and decomposes the corresponding non-unitary propagator into parallel spectral channels. We utilize the Koopman method on a superconducting processor to simulate three distinct nonlinear systems, comprising reaction-diffusion dynamics, fluid motion on a sphere, and satellite-derived observations of Gulf Stream currents, employing up to 32 parallel circuits of 10 qubits. These quantum simulations capture the dominant multiscale patterns and statistical signatures of the underlying dynamics, and reveal a transition from performance limited by hardware noise in weakly nonlinear systems to performance limited by finite-dimensional Koopman representations as nonlinear scale interactions increase. This transition identifies a practical boundary for quantum-amenable nonlinear dynamics, establishing a hardware-validated route for simulating moderately nonlinear dynamics on near-term quantum hardware.

S. Gheorghiu-Svirschevski

We outline an approach that streamlines considerably the construction and analysis of well-behaved nonlinear quantum dynamics, with completely positive extensions to entangled systems. A few notes are added on the issue of quantum measurements under a nonlinear dynamics.

János A. Bergou, Mark S. Hillery

J. R. Croca, João L. Cordovil

Jincheng Zhang

This paper presents a novel approach to modeling dynamic quantum control scenarios involving nonlinear quantum systems. The core claim is the establishment of a model capable of accurately describing the behavior of such systems under dynamic control. This model is built upon the introduction of a nonlinear Schrödinger equation, coupled with principles from dynamic control theory. The methodology allows for detailed analysis and manipulation of nonlinear quantum systems, expanding the theoretical foundation for quantum control and manipulation research. We explore the challenges and potential solutions in designing control strategies for systems exhibiting complex, nonlinear interactions. The presented framework offers a foundation for further research into advanced quantum control techniques. The key mathematical components are detailed within the subsequent sections, including the nonlinear Schrödinger equation, control vector formulation, and analysis of system dynamics. The focus is on providing a rigorous mathematical framework rather than a specific experimental implementation.