Daniel Peterseim, Professor, Universität Augsburg, Germany

Wei Gong, Professor & Weiying Zheng, Professor

Quantum Algorithms for Computational PDEs

10:00-11:00 May 29(Thursday), N109

Numerical simulation plays a central role in understanding and predicting the behavior of complex processes in science and engineering modeled by partial differential equations (PDEs). However, solving high-dimensional, nonlinear, and multiscale PDEs often exceeds the capabilities of current computing environments. Quantum computing, with its near-logarithmic complexity in data processing for certain types of problems, offers the potential to address these limitations.\\

In this talk, we explore two classes of quantum algorithms for computational PDEs. First, we show how classical multilevel preconditioning techniques are adapted to design quantum-efficient solvers for prototypical linear elliptic PDEs with multiscale features and high dimensionality. Second, we introduce quantum algorithms for solving large systems of nonlinear equations that often arise in the context of nonlinear PDEs. In both approaches, we handle the inherent unitarity of quantum operations by block encodings and their careful amplification. We demonstrate theoretical quantum advantages through rigorous runtime bounds. Furthermore, through proof-of-concept numerical experiments conducted on both quantum simulators with controlled noise and actual noisy quantum hardware, we provide a practical assessment of the current capabilities and future opportunities of quantum computing in advancing the numerical simulation of PDEs.