All around us is mathematics in motion: the shift of global economic figures in the news, the rush of wind over an airplane’s wings, or the flocks of tourists descending on post-pandemic towns. Complex scenarios like these can be modelled with partial differential equations (PDEs), mathematical rules that help us model and predict how systems with multiple variables change over time.
“PDEs are fundamental to solving engineering problems ranging from transport phenomena and electromagnetics to finance and social sciences,” said Fong Yew Leong, a Scientist at A*STAR’s Institute of High Performance Computing (IHPC).
Solving PDEs can require power-hungry simulations that take hours, or even days, to run on classical computers. For example, consider how many variables go into recreating the physics of airflow over a wing: the wing’s shape, drag, air resistance, angles of attack, and so on. In real life, some of those variables change from moment to moment, and PDEs trying to simulate that dynamic evolution can get extra complicated.
To deal with that complexity, one solution researchers are turning to is quantum computing. In theory, quantum algorithms could help speed things up with processing capabilities and methods that surpass classical systems. However, today’s quantum computing hardware—held back by limited bandwidth and high error rates—aren’t quite ready to run practical PDE simulations in full.
To sidestep this problem, Leong and colleagues examined a hybrid classical-quantum approach, known as variational quantum algorithms (VQAs), where classical computers help take on some of the processing grunt work, allowing quantum hardware to focus on solving the PDEs themselves.
Based on that approach, they designed the Variational Quantum Evolution Equation Solver: an algorithm that breaks down a complex PDE into simpler quasi-steady equations through time-stepping. The algorithm then feeds them into quantum hardware which executes the simulation. Over repeated cycles, the classical hardware provides real-time feedback to its quantum counterpart, gradually guiding the simulation towards accurate solutions that align with real-world models.
To showcase their algorithm’s feasibility, the team successfully solved several different problems ranging from heat transfer to fluid dynamic model PDEs. “Our proposed algorithm inherits the advantages of VQAs—which are great for solving problems that don’t change with time—yet can also tackle a wide range of dynamic problems,” said Leong.
Leong added that commercialising and scaling their algorithm will take time; several quantum hardware engineering kinks, such as prohibitive costs and noisy quantum bits, first need to be ironed out before they can be routinely used for everyday simulations. Nonetheless, the team continues to explore new possibilities in quantum algorithms, and is currently working on variational quantum simulations that help reduce the costs of quantum encoding.
The A*STAR-affiliated researchers contributing to this research are from the Institute of High Performance Computing (IHPC).