Quantum Simulation of Nonlinear Dynamical Systems Using Repeated Measurement

We present a novel quantum algorithm based on repeated measurement to solve initial-value problems for non-linear, polynomial, real ordinary differential equations (ODEs), which may be generated from partial differential equations in plasma physics and related fields. We present an algorithm to map a nonlinear, dynamical system to a Hamiltonian form, where the Hamiltonian matrix is a function of dynamical variables and the weighted sum of associated sub-Hamiltonians. The solution is stored as the amplitudes of the evolving quantum state. To advance in time, we measure the expectation value of various observables, using the previous time step’s quantum state as the distribution. We then use these expectation values as the weights in the summation to evaluate the Hamiltonian function classically. This measurement process introduces stochasticity into the system dynamics. We then perform standard quantum Hamiltonian simulation over a short time, using the evaluated Hamiltonian matrix as a constant for the entire period. This approach requires initializing and evolving an ensemble of identical quantum states, with some of this ensemble being consumed each step to measure the required observables. We apply this approach to the classic logistic system, associated with population growth, and the Lorenz system, in both integrable and chaotic regimes. Our analysis shows that the stochastic, quantuminspired solutions’ accuracy, measured by trace distance, is influenced by both the stochastic sampling rate and the nature of the dynamical system. We find a link between the trace distance error and the von Neumann entropy, which we link to stochastic spread in the solutions.