Quantum algorithm for statistical characterization of chaotic systems

A central challenge in the quantum simulation of chaotic systems is the development of methods for efficiently evolving nonlinear dynamics. Much of the prior work has focused on hybrid classical-quantum integration, multiple state copies, or linear approximations. However, quantum-classical latency, exponential resource scaling, and the inability of linear frameworks to efficiently encode nonlinearity limit the applicability of existing techniques to high-dimensional, strongly nonlinear systems. Our objective is to develop a fully quantum algorithm capable of efficiently obtaining the statistical description of chaotic systems, with potential applications to turbulent flows. To this end, we transform the underlying partial differential equations into algebraic update laws using explicit time and spatial discretization. These update laws are then mapped onto efficient quantum circuits that directly evaluate and evolve the nonlinearity from efficiently prepared arbitrary initial states. To mitigate exponential temporal resource scaling, we employ approximate state cloning and quantum bit reuse techniques. Statistical quantities of interest are obtained via quantum ensemble operators. In this talk, we discuss the accuracy and computational cost of the quantum algorithm and identify the conditions under which it offers a speedup over the best available classical techniques.