Quantum Simulation of Nonlinear Dynamics through Repeated Measurement

To simulate nonlinear dynamics on quantum computers, we present an approach to map nonlinear dynamics to linear evolution through repeated measurements. The dynamics is casted to a Hamiltonian form, where the Hamiltonian matrix is a functional of dynamical variables. To advance in time, we measure expectation values from the previous time step, and evaluate the Hamiltonian functional 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 an ensemble of quantum states, where each step consumes a subset of quantum states, which are used for measurements and are discarded from further time advance. We apply this approach to the classic logistic and Lorenz systems, in both chaotic and non-chaotic regimes. Analysis of solutions' accuracy and entropy is provided as influenced by the model's stochastic strength and system chaos. This approach may be applied to high-dimension systems, including dissipative plasma dynamics and the Vlasov equation.