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Proceedings of the Combustion Institute

Efficient evaluation of Arrhenius rates for quantum computing applications in reactive flow problems using Carleman linearization

Takaki Akiba, Youhi Morii, Minhyeok Lee, Kaoru Maruta, Yuji Suzuki

Published Jan 1, 2025• 0 citations
Publisher

Abstract

This research introduces a novel approach toward integrating quantum computing into reactive flow analysis in combustion science, addressing the challenges posed by the nonlinear behavior of Arrhenius reaction rates. Combustion systems are described by nonlinear equations. Quantum computers, however, work based on linear algebra and are good at solving linear systems, necessitating a method to bridge this disparity. The study utilizes Carleman linearization, a technique that transforms nonlinear equations into linear ones, making them suitable for quantum algorithms. To address the exponential temperature dependence of Arrhenius rate constants, Taylor series expansions are employed. The method is validated through numerical simulations of ignition in H 2 /air and CH 4 /air mixtures. By examining the impact of truncation order in Carleman linearization and fitting order in the Taylor expansion, the study balances computational accuracy and cost. Results reveal that higher-order approximations improve accuracy but can introduce numerical stiffness, particularly in systems with high activation energies, such as CH 4 /air in this study. The findings highlight the trade-offs between computational feasibility and solution precision, emphasizing the importance of optimizing fitting and truncation parameters. This work establishes a critical foundation for applying quantum computing to combustion research.

DOI
10.1016/j.proci.2025.105918

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This research introduces a novel approach toward integrating quantum computing into reactive flow analysis in combustion science, addressing the challenges posed by the nonlinear behavior of Arrhenius reaction rates. Combustion systems are described by nonlinear equations. Quantum computers, however, work based on linear algebra and are good at solving linear systems, necessitating a method to bridge this disparity. The study utilizes Carleman linearization, a technique that transforms nonlinear equations into linear ones, making them suitable for quantum algorithms. To address the exponential temperature dependence of Arrhenius rate constants, Taylor series expansions are employed. The method is validated through numerical simulations of ignition in H 2 /air and CH 4 /air mixtures. By examining the impact of truncation order in Carleman linearization and fitting order in the Taylor expansion, the study balances computational accuracy and cost. Results reveal that higher-order approximations improve accuracy but can introduce numerical stiffness, particularly in systems with high activation energies, such as CH 4 /air in this study. The findings highlight the trade-offs between computational feasibility and solution precision, emphasizing the importance of optimizing fitting and truncation parameters. This work establishes a critical foundation for applying quantum computing to combustion research.
Reacting flows are governed by the conservation equations of mass, momentum, energy, and chemical species, and exhibit strong nonlinearity due to convection and chemical reactions. As quantum computers are inherently limited to linear operations, a bridging method is required to enable their application to such nonlinear systems. This study proposes an approach using the Carleman linearization to integrate quantum computing into reacting flow analysis, with a focus on the nonlinear temperature dependence described by the Arrhenius equation. To handle the exponential form of the reaction rate, a Taylor series expansion is employed. The present method is validated through numerical simulations of fuel ignition. The impacts of the truncation order in the Carleman linearization and the polynomial order in the Taylor expansion are systematically examined. The results demonstrate that higher-order approximations enhance the accuracy but may also introduce numerical stiffness. Overall, the findings confirm the validity and potential of the proposed framework for quantum-assisted reacting flow computations.
Abstract 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 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.
Quantum computation of fluid dynamics has attracted growing attention as a key application of fault-tolerant quantum computers anticipated in the coming decade, with lattice Boltzmann methods emerging as a particularly promising approach. Explicit and efficient elementary-gate-level circuit simulations, however, have so far been demonstrated only in the linear case. Here we include the leading nonlinearity through second-order Carleman linearization of the one-dimensional Boltzmann equation, and demonstrate, via explicit quantum-circuit simulation, the preparation of the final-time state using a Taylor-expansion-based ODE solver based on the quantum singular value transformation. With this construction, we analyze the gate and qubit complexities, which scale logarithmically with the grid size, the nonlinearity captured by the higher-order Carleman linearization, and the practical utility of higher-order expansions in the Taylor ODE solver. The construction provides a concrete baseline for computational cost reduction and further developments such as extensions to higher dimensions, complex geometries, and the extraction of physical quantities, towards industrially useful quantum CFD.
Quantum computing offers the promise of speedups for scientific computations, but its application to reacting flows is hindered by nonlinear source terms, the challenges of time-dependent simulations, and the difficulty of extracting meaningful physical quantities from quantum states. We employ a probability density function (PDF) formulation to transform the nonlinear reacting-flow governing equations into high-dimensional linear ones. The entire temporal evolution is then solved as a single large linear system using the history state method, which avoids the measurement bottleneck of conventional time-marching schemes and fully leverages the advantages of quantum linear system algorithms. To extract the quantity of interest from the resulting quantum state, we develop an efficient algorithm to measure the statistical moments of the PDF, bypassing the need for costly full-state tomography. A computational complexity analysis shows that the measurement algorithm achieves a complexity polynomial in the logarithm of the system size using low-order polynomial approximations, compared to the exponential cost of the exact operator, thereby retaining the quantum advantage gained from solving the linear system. We validate the framework in two stages: an a priori test confirms the accuracy of the measurement algorithm on beta distributions with known analytical moments, and a perfectly stirred reactor simulation demonstrates the capability to capture the PDF evolution and statistics of a nonlinear reactive system. This work establishes a pathway for applying quantum computing to nonlinear reacting flows.
The solution of large systems of nonlinear differential equations is essential for many applications in science and engineering. We present three improvements to existing quantum algorithms based on the Carleman linearisation technique. First, we use a high-precision method for solving the linearised system that yields logarithmic dependence on the error and near-linear dependence on time. Second, we introduce a rescaling strategy that significantly reduces the cost, which would otherwise scale exponentially with the Carleman order, thus limiting quantum speedups for PDEs. Third, we derive tighter error bounds for Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We also show that enforcing a stability criterion independent of the discretisation can conflict with rescaling due to the mismatch between the max-norm and the 2-norm. Nonetheless, efficient quantum solutions remain possible when the number of discretisation points is constrained, as enabled by higher-order schemes.
We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. By shifting the dynamics by a pivot state prior to Carleman lifting, and combining this with a Lyapunov transform and rescaling, we enlarge the class of nonlinear systems that can be efficiently simulated on quantum computers. For systems that exhibit stability in the shifted coordinates, we establish long time convergence of the truncated Carleman embedding. We prove that the truncation order scales only logarithmically with the simulation time and target precision, and we derive end-to-end quantum query complexity bounds for preparing a state proportional to the final solution. By introducing a modified nonlinearity condition, this framework entirely removes the conventional lower bound requirement on the initial condition. For more general systems that remain unstable after shifting, we provide short time convergence guarantees that are similarly free from the initial condition constraints. Numerical experiments on the logistic and the Lotka-Volterra equations demonstrate that an appropriate pivot choice improves stability and accuracy, and yields exponential error decay with truncation order. These results show that pivot shifting provides a practical and theoretically justified route for extending Carleman-based quantum algorithms to a broader class of nonlinear dynamical systems.