Quantum Simulation of Nonlinear Dynamical Systems Using Repeated Measurement

TL;DR

This paper introduces a quantum algorithm that uses repeated measurements to simulate nonlinear dynamical systems, enabling analysis of complex behaviors like chaos with quantum computing techniques.

Contribution

The paper proposes a novel quantum algorithm that maps nonlinear ODEs to Hamiltonian form and uses repeated measurements for simulation, incorporating stochasticity and enabling analysis of chaotic systems.

Findings

Solution accuracy depends on sampling rate and system nature.

The method successfully simulates logistic and Lorenz systems.

Applicable to integrable and chaotic regimes.

Abstract

We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations (ODEs), which may be generated from partial differential equations in plasma physics. We map a dynamical system to a Hamiltonian form, where the Hamiltonian matrix is a function of dynamical variables. To advance in time, we measure expectation values from the previous time step, and evaluate the Hamiltonian function classically, which introduces stochasticity into the dynamics. We then perform standard quantum Hamiltonian simulation over a short time, using the evaluated constant Hamiltonian matrix. This approach requires evolving an ensemble of quantum states, which are consumed each step to measure required observables. We apply this approach to the classic logistic and Lorenz systems, in both integrable and chaotic regimes. Out analysis shows that solutions' accuracy is influenced by both the stochastic sampling rate and the nature of the dynamical system.