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.
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.
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.
Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) dXt=(A(t)Xt+F(t)Xt⊗s)dt+B(t)dWt for an RN-valued random process Xt, aiming at quantum speed-up with respect to the dimension N. We introduce auxiliary processes Ytj that approximate Xt⊗j, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of Xt by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order J in CL, and we can estimate the target expectation with a polylog(N) number of queries to unitaries encoding coefficient matrices A, F, and B. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as J increases.
Nonlinear differential equations underlie many scientific fields, yet classical solvers scale poorly with increasing resolution. Quantum linear systems algorithms (QLSAs), such as Harrow-Hassidim-Lloyd (HHL), can offer asymptotic advantages for sparse, well-conditioned linear systems. However, they do not directly treat nonlinear dynamics. We study two linearization techniques—Carleman linearization and Liouville linearization—as preprocessors for prospective QLSA use. We assess the stability of the linearization step on two models with bounded phase spaces and single- or multi-attractor dynamics. Carleman linearization tracks single trajectories until truncation error dominates. The Liouville ensemble formulation, especially with narrow initial distributions, captures multi-basin splitting and trajectory statistics while providing built-in uncertainty quantification. These results suggest that Liouville may offer broader practical applicability as a QLSA preprocessor, particularly in regimes not covered by existing efficiency guarantees for Carleman linearization.
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.
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.
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as du/dt=G0+G1eiu, where u denotes a vector of n complex variables evolving with t, G0 is an n-dimensional complex vector, G1 is an n×n complex matrix and eiu denotes the vector with entries {eiuj}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of u, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.