Quantum topological data analysis algorithm for dynamical systems

Abstract

Dynamical systems appear in nearly every aspect of the physical world. As such, understanding the properties of dynamical systems is of great importance. Typically, a dynamical system is described by a system of ordinary differential equations (ODE). Most ODEs do not admit analytical solutions, which makes dynamical systems challenging to understand. In this work, we introduce a quantum framework for determining certain properties of dynamical systems. We combine many recent advances in quantum algorithms, particularly quantum ODE solver and quantum topological data analysis. Leveraging the prior results regarding the quantum ODE solvers, we use the output of these quantum algorithms as a means to build the graph associated with the trajectory of the dynamical system in the phase space. This graph information is then fed into existing quantum TDA algorithms, respectively, to obtain the (normalized) Betti numbers, which are the topological signatures. We then discuss how such signatures can be linked to the properties of a given ODE, and thus, they can reveal insight towards the original dynamical systems. As a by-product, our work provides an affirmative answer to the applicability of quantum ODE solvers, showing that the quantum state as output can be valuable for useful purposes.