Carleman linearization approach for chemical kinetics integration toward quantum computation

The Harrow, Hassidim, Lloyd (HHL) algorithm is a quantum algorithm expected to accelerate solving large-scale linear ordinary differential equations (ODEs). To apply the HHL to non-linear problems such as chemical reactions, the system must be linearized. In this study, Carleman linearization was utilized 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 analysis precision can be determined by the extent of the truncation. Matrix should be sufficiently large so that the precision is satisfied because quantum computers can treat. Our method was applied to a one-variable nonlinear dy/dt = -y^2 system to investigate the effect of truncation orders in Carleman linearization and time step size on the absolute 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 in Carleman linearization improved accuracy even with a large time-step size. Thus, our approach can provide accurate numerical simulations rapidly for complex combustion systems.