Efficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation
Year: 2022
Type: preprint
Abstract: Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition $R < 1$, where $R$ measures the ratio of nonlinearity to dissipation using the $\ell_2$ norm. Here we develop an efficient quantum algorithm based on [1] for reaction-diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in [1] to obtain a faster convergence rate under the condition $R_D < 1$, where $R_D$ measures the ratio of nonlinearity to dissipation using the $\ell_{\infty}$ norm. Since $R_D$ is independent of the number of spatial grid points $n$ while $R$ increases with $n$, the criterion $R_D<1$ is significantly milder than $R<1$ for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.
Source: arXiv (Cornell University)
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Towards Quantum Speedup for Solving High-Dimensional Partial Differential Equations, QLCI-CI: NSF Quantum Leap Challenge Institute for Present and Future Quantum Computing, G3894017315, G4020565512, CAREER: Turbo-Charging Hybrid Functional Electronic Structure Calculations via Adaptive Compression Methods, G4800547303, Convergence QL: Workshop Series: Cross- Sector Connections in Quantum Leap, G5129568324, G5298095028, GREEN'S FUNCTION METHODS FOR MULTIPHYSICS SIMULATIONS, On the Dynamics of Nonlinear Systems in Applied Sciences: From Theory, Computations, and Experiments to Insights, GREEN'S FUNCTION METHODS FOR MULTIPHYSICS SIMULATIONS, AF: Small: Toward Applications and Verification of Early Quantum Computers
OpenAlex ID: W4229066487