Jin-Peng Liu of the University of Maryland presents “Efficient quantum algorithms for nonlinear ODEs and PDEs” at IPAM’s Quantum Numerical Linear Algebra Workshop.
Nonlinear dynamics play a prominent role in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, we gave the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in the evolution time. We also established a lower bound showing that nonlinear differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Furthermore, numerical results suggest that our algorithm may potentially address complex nonlinear phenomena even in regimes with weaker dissipation.
As a sequent work, we applied the quantum algorithm for nonlinear reaction-diffusion equations with an improved convergence rate. We post-processed the quantum encoding of the solutions of the Fisher-KPP equation and the Allen-Cahn equation for estimating thermodynamic free energies, laying the groundwork for practical applications of quantum computers for classical nonlinear systems.