Simulation of nonlinear partial differential equations with quantum algorithms: A Fokker-Planck approach*

As manufacturing of classical computing chips has reached the scale of a few nanometers, i.e. a few dozen atoms, the progress of enhancements of classical computing resources is slowing down and, possibly, coming to an end. Quantum Algorithms represent a new paradigm of information processing that have theoretically been shown to outperform their classical counterparts on a number of specific tasks such as sparse matrix inversion, Fourier transformation and Singular Value Decomposition to only name a few. All these tasks being linear, the important question arises whether Quantum Computers can be successfully deployed to integrate non-linear dynamics. After two decades of research, a number of conceptually different algorithms have been proposed recently. Unfortunately, these algorithms have either no proven Quantum Advantage or cannot yet be run on currently available quantum hardware. 

In this talk, an ansatz to integrate non-linear dynamics on Quantum Computers will be presented. We consider systems with a non-linear flow field that are subject to white Gaussian noise. This allows us to describe the evolution of the probability distribution function by means of the Fokker-Planck equation which is a linear Partial Differential Equation. We then discuss the application of Quantum Algorithms to solve the Fokker-Planck equation. Special care is given to incorporating the boundary conditions. Key steps of our new Quantum solver involve the discretisation of the space domain and a direct exponentiation of the discretised Fokker-Planck operator. In addition to theoretical performance bounds of our new algorithm, we demonstrate its application to a set of prototypical model problems. This work opens opportunities for solving nonlinear partial differential equations with hybrid algorithms.