Linearization-based quantum algorithms for the statistical description of chaotic systems

One of the primary challenges in quantum computing is developing algorithms that can efficiently encode the nonlinear terms of underlying equations. Much of the previous work has focused on linearization techniques combined with quantum linear systems algorithms (QLSA). However, these linearization-based algorithms are only efficient for flows with weak nonlinearity. Our objective is to create a fully quantum algorithm capable of efficiently obtaining the statistical description of chaotic systems, with potential applications to turbulent flows. To this end, we explore a linearization approach for chaotic dynamical systems that encodes solutions for multiple time steps within a single linear system. The success of this algorithm hinges on balancing two competing effects. On one hand, the efficiency of solving the linear systems via a QLSA-based quantum algorithm improves with the number of time steps encoded. On the other hand, the accuracy of the statistical descriptions derived from this method decreases as the size of the linear system increases. In this talk, we will discuss the conditions under which this quantum algorithm can accurately perform its task while providing a speedup over the best classical techniques.