Quantum simulation of nonlinear dynamical systems using repeated measurement

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 the required observables. We apply this approach to the classic logistic and Lorenz systems, in both integrable and chaotic regimes. Our analysis shows that the solutions’ accuracy is influenced by both the stochastic sampling rate and the nature of the dynamical system.