Carleman linearization approach for chemical kinetics integration toward quantum computation

Abstract

Abstract The Harrow, Hassidim, Lloyd (HHL) algorithm, known as the pioneering algorithm for solving linear equations in quantum computers, is expected to accelerate solving large-scale linear ordinary differential equations (ODEs). To efficiently combine classical and quantum computers for high-cost chemical problems, non-linear ODEs (e.g., chemical reactions) must be linearized to the highest possible accuracy. However, the linearization approach has not been fully established yet. In this study, Carleman linearization was examined to transform nonlinear first-order ODEs of chemical reactions into linear ODEs. Although this linearization theoretically requires the generation of an infinite matrix, the original nonlinear equations can be reconstructed. For the practical use, the linearized system should be truncated with finite size and the extent of the truncation determines analysis precision. Matrix should be sufficiently large so that the precision is satisfied because quantum computers can treat such huge matrix. Our method was applied to a one-variable nonlinear $$\dot{y}=-{y}^{2}$$ y ˙ = - y 2 system to investigate the effect of truncation orders and time step sizes on the computational error. Subsequently, two zero-dimensional homogeneous ignition problems for H2–air and CH4–air gas mixtures were solved. The results revealed that the proposed method could accurately reproduce reference data. Furthermore, an increase in the truncation order improved accuracy with large time-step sizes. Thus, our approach can provide accurate numerical simulations rapidly for complex combustion systems.