Carleman linearization approach for chemical kinetics integration toward quantum computation

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y}=-{y}^{2}$$\end{document}y˙=-y2 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.

Carleman linearization is based on the Taylor expansion and was proposed by Carleman 7 .The Jacobian matrix is another widely considered linearization approach.In the Jacobian approach, only the first derivative of the original system is considered.Therefore, the order considered in the linearization process is limited.Carleman linearization theoretically provides a linearized system with infinite orders, which has the same information as the original nonlinear system, and allows the determination of the degree of orders to which the analysis is considered.Thus, the degree of accuracy considered in the Carleman

FORMULATION
Chemical kinetics are typically first-order ODEs.
Therefore, the governing equations for ODEs are summarized as follows: where  is the system variable vector and () is a nonlinear function.
In this section, the widely used Jacobian matrix linearization is explained.Next, we explain Carleman linearization.Finally, the differences between the Jacobian and Carleman linearization were explained.

Linearization using the Jacobian matrix
The governing equations (equation ( 1)) can be rewritten with linearization using the Jacobian matrix as follows: where  = ' ' is the Jacobian matrix, and  is the time indicator for the solution vector.By discretizing (2) over time, the differential equation can be transformed as follows: where  and ∆ represent the time step and the time step size, respectively.We compare equation ( 3) with other methods.

Linearization using the Carleman matrix
Next, we briefly explain the formulation of Carleman linearization.Here, () of equation ( 1) can be transformed as follows: where superscripts of "⊗" represent the Kronecker power, which is expressed as follows: Equation ( 1) can be written down with the expression of equation ( 5) in linear expression by using Carleman linearization as follows: and where for  ≥ 1.From definition  / , the matrix has infinite rows and columns, and such an infinite matrix cannot be considered in the simulations.
Thus, the matrix should be truncated in the order of  1 , and we can summarize the matrix as follows: where nt is termed as the truncation order hereafter.
The number of elements in the matrix is ) " ,+ . The final target of this study is the chemical reaction problem.Most elementary reactions involve three chemical species at most.The problem is reduced to an upper triangle matrix as follows: When linearization is completed, the system is discretized for time using an explicit approach as follows: Discretization can also be performed using the implicit approach.An  / can be assumed as a constant through the time between  and  + 1 with a small time step size as follows: By comparing the matrices whose inverse matrix is required for Eqs. ( 3) and ( 12), both the Jacobian  The Kronecker power of solution  is prepared as follows:

RESULTS AND DISCUSSION
The problem is expressed as follows: where  Furthermore, the absolute maximum errors were almost the same regardless of the truncation order.
These results implied that the truncation order is sufficient with the same order as that of the problem.

First application to chemical reactive systems -H2-air combustion
This study is the first to attempt the H2-air where  -( =  − ) is the chemical species involved and can be evaluated as follows:   Although obvious evidence was not obtained, the stiffness of the system influenced the limitation of the timesteps.).An increase in the truncation order had a limited effect on the computational costs for matrix preparation.

Second application -CH4-air combustion
The  ) in the formulation section around Eq. (12)., which will be investigated in the future.Another possible cause of this delay is the insufficient truncation order of Carleman linearization.However, the order cannot be increased because of machine limitations.This discrepancy will be investigated in the future with the temperature-dependent reaction systems.

CONCLUSION
In order to utilize the quantum computation resources for large-cost chemical reaction analysis, the nonlinear nature of chemical reactions needs to be linearized for the application of the HHL algorithm, the powerful quantum algorithm for large-scale equations.In this study, the Carleman linearization was applied as a linearization method.
The linearization method was evaluated using three simple problems.The results showed the validity and reliability of the proposed approach and implied the potential of using the proposed method in quantum computation of chemical kinetics.In the future, we will focus on evaluating more practical combustion problems.
should be controlled under extreme conditions, such as high pressure, high temperature, and lean-fuel conditions.However, analyzing combustion phenomena under such extreme conditions by using an experimental approach is difficult because of the short characteristic time and nonlinear nature of the phenomena.Therefore, numerical approaches have been proposed for detailed analyses.Numerical analyses of combustion systems under extreme conditions require high-precision methods with detailed information related to chemical reactions.However, detailed analyses of reactive flow systems results in high computational costs because of numerous variables and stiffness of the phenomena.The combustion reaction generally involves 10-1000 chemical species.Furthermore, the dimensional parameters increases the scale of the problems because the thickness of the reaction zones is approximately 10 −5 -10 −4 m, whereas the scale of practical combustion systems is 10 −1 -10 1 m.Because of the thin reaction zone, the required mesh size for reactive flow problems is approximately 30 3 times finer than that for nonreactive flow problems 1 .The characteristic times of fluid dynamics, molar transport, and chemical reactions differ by approximately 10 0 -10 −2 , 10 −2 -10 −5 , and 10 −6 -10 −12 s for the fluid dynamics time, molecular-transport time, and chemical reaction time scale, respectively 2 .A possible approach for overcoming these problems is to develop an efficient method or algorithm to evaluate chemical reaction problems without precision loss.As the other approach, powerful machine resources can be used to solve high-cost chemical problems.Rapid developments have been achieved in both quantum computing hardware and software.IBM has shared its roadmap of the scale of quantum computers and will launch a quantum computer with a capacity of more than 1000 qubits in 2023 3 .Although utilization of hardware development is limited, quantum machine resources have been used for high-cost problems.The Harrow, Hassidim, Lloyd (HHL) algorithm is widely known as a powerful solver of large linear equations and expected to be utilized for solving large-scale ordinary differential equations ODEs 4 .The HHL algorithm can be used to solve a largescale linear problem with K variables within O(poly(log(K))) compared with O(K) time required for the best classical algorithm.Studies have focused on specialized reactive flow problems, such as the application to Burgers equations, and primarily pure fluid problems 5,6 .Carleman linearization was used because quantum computers can only be used for linear problems, whereas fluid problems involve nonlinear properties.
method and Carleman linearization produce similar expressions: ( − Δ) and ( − Δ / ) ) .The differences in the expression is the solution vectors ( and ) and matrices used ( and  / ) .The definitions of  / and  clearly reveal that Carleman linearization involves higher-order elements in terms of the original solution vector  to be obtained.An implicit approach was used in this study because of the heavy stiffness caused by the chemical reactions mentioned in the following sections.As mentioned, the system size can be estimated (  ), and becomes large with a slight increase in the number of variables  or the order of truncation  .The sparse matrix solver in the SciPy library in Python was implemented to solve the implicit approach.The solution vector was obtained by using the direct method because we will utilize the HHL method in a quantum computer in the future, and iterative operations should be eliminated, which could potentially increase the communication between classical and quantum computers.SageMath 11 and its library 12 were used for the Carleman linearization procedure using Python interfaces.Python was used to solve the discretized problem.

3. 1 .
Nonlinear sample system A simple description of the problem is presented first.The target problem is expressed as follows: This problem was selected because of the simplicity of the formulation with the smallest nonlinearity and the consideration of applying the approach to chemically reactive problems.Only single alpha value was applied in this paper because the change of alpha just affects the duration of decay.The initial condition was chosen to simulate the unreacted state of reactants.The solution to this problem is displayed in Fig. 1.In the first step, Carleman linearization was applied to the problem.

Figure 1 .
Figure 1.Effect of truncation orders and time step sizes on the maximum or minimum absolute value.The representative absolute error is defined as the maximum or minimum absolute error, which has maximum absolute value.

Figure 4 .
Figure 4. Comparison of the results by Carleman linearization and reference for nondimensional isothermic transient reactor system of H2-air combustion.
figure, nine species, including N2 as an inert gas, were selected for the H2-O2 reactions.The equivalence ratio, initial temperature, and pressure were 0.8, 2000 K, and 1 atm, respectively.These conditions are consistent throughout the following discussions, unless otherwise mentioned.Fig.4displays the successful evaluation of the chemically reactive system because the overall transition of chemical species and the reaction timing agree.Similar to the analysis in the previous problem, the effect of the size of the time steps on the simulation error was investigated.

Fig. 5
Fig. 5 displays the convergence of the time step sizes for two truncation orders.As references for the slopes, lines 102 and 103 are plotted as dashed and dotted lines, respectively.As expected, the decrease in the time step size improved accuracy.With an increase in the truncation order, the convergence of time step sizes improved considerably, particularly for large time steps.The calculation diverged when the time step increased to more than 2.0 × 108.

3 Figure 5 .
Figure 5.Effect of truncation order and time step sizes on the convergence.

Fig. 6
Fig. 6 displays the effect of the truncation order on the computational precision and cost.The matrix size is calculated as (  & !*+ − 1)/(  − 1), where   and  1 are the number of variables and the

Figure 6 .Figure 7 .
Figure 6.Estimation of the effect of truncation order on computational costs.