## Solving nonlinear equations with quantum algorithms

**What is a Quantum Non-linear solver?**

Since their invention, classical computers have seen rapid developments in terms of clock time and memory capacity. Over the past ten years, this progress has slowed down and state-of-the art chip manufacturing is approaching the atomic scale, thereby reaching a fundamental limitation to further enhancements very soon. Quantum computing presents an entirely new paradigm of computing and information processing that makes use of the laws of quantum mechanics governing microscopic systems. Quantum computers have theoretically been proven to (exponentially) outperform their classical counterparts on a number of tasks, such as matrix inversion, SVD, Fourier transformations and others. These tasks are linear operations and therefore the question arises whether quantum computers can be used to integrate non-linear dynamical systems. In our research we are interested in exploring the suitability of quantum computers for future computational fluid dynamic simulations. Therefore, we aim to develop a Quantum Non-linear solver that hopefully soon will add quantum computers to the repertoire of fluid dynamic researchers. Arguably, the benefit of designing quantum algorithms to be run on emerging future quantum hardware not only lies in providing a sense for their applicability to real-world problems but also helps to inform the specific design of the future quantum hardware.

**What is Quantum Non-linear solver, a bit more technically?**

After three decades of research, a number of conceptually different approaches to integrating non-linear dynamics on quantum computers have been proposed in the past three years, based on, for instance, mean-field non-linearities, Carlemann-linearisation and others. In our research, we use the Fokker-Planck equation associated with a non-linear system to simulate non-linear dynamics on quantum computers. The Fokker-Planck equation is a

*linear*partial differential equation for the probability density distribution of a dynamical system's state, that encodes the coupling of macroscopic dynamical variables to (fast evolving) microscopic variables by including a diffusion term. Due to its linearity, quantum computers provide a powerful tool for the integration of the Fokker-Planck equation. However, a number of challanges must be addressed related to boundary conditions, finite sizes of mesh grids and non-normality of the arising discretised operators. Having successfully solved these problems we now work on implementing our new non-linear quantum solver to specific problems and to run simulations on cloud quantum computers.**An example of practical application:**

**Nonlinear ordinary differential systems**

To be displayed when the pre-preprint of the paper becomes available. Stay tuned.

**Material, activities, and people**

Felix Tennie, Luca Magri

Article written by Felix Tennie and Luca Magri.