Quantum computing offers the promise of speedups for scientific computations, but its application to reacting flows is hindered by nonlinear source terms and the challenges of time-dependent simulations. We present a quantum framework to address these issues. 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 indicates the potential for a near-exponential speedup over classical algorithms. We validate the framework by simulating a perfectly stirred reactor, demonstrating its 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.
Quantum computing offers the promise of speedups for scientific computations, but its application to reacting flows is hindered by nonlinear source terms and the challenges of time-dependent simulations. We present a quantum framework to address these issues. 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 indicates the potential for a near-exponential speedup over classical algorithms. We validate the framework by simulating a perfectly stirred reactor, demonstrating its 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.
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 to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, nonlinear PDEs. We embed the PDEs in a truncated, high-dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods with a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of the time-marching system size, simulation time, accuracy parameters, and post-selection parameters. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts, and query complexity. A physically motivated measure of nonlinearity is connected to a parameter similar to the flow Reynolds number ReH, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of Burgers, Fisher--KPP, damped Kuramoto--Sivashinsky, and real Ginzburg--Landau/Allen--Cahn equations. Together, these examples encompass cubic nonlinearity and saturation, quadratic transport, and fourth-order dissipative stiffness. This work shows the potential of hybrid quantum algorithms for simulating nonlinear problems on near-term and fault-tolerant devices.
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.
Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this work, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. The results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems.
Nonlinear terms present a fundamental challenge for quantum computational fluid dynamics, as their implementation on inherently linear quantum hardware typically requires resource-intensive workarounds that limit scalability to large-scale simulations. We present a hybrid quantum-classical tensor network algorithm that addresses this bottleneck by combining variational time-stepping with quantum tensor programming to efficiently compile operators and time-dependent fields into quantum circuits. Within a probabilistic framework, we replace prior state-based nonlinear implementations with tensor-based block encodings, stabilizing success probabilities that otherwise decay exponentially with system size. Benchmarking on turbulent flow fields demonstrates that the algorithm maintains high success probabilities and moderate measurement overhead across increasing Reynolds numbers and grid resolutions. Compared to fully classical tensor network solvers, our hybrid approach yields substantial reductions in both memory footprint and computational cost, establishing a scalable pathway toward practical quantum advantage in scale-resolving CFD simulations.
Fault-tolerant quantum computing is a promising technology to solve linear partial differential equations that are classically demanding to integrate. It is still challenging to solve non-linear equations in fluid dynamics, such as the Burgers equation, using quantum computers. We propose a novel quantum algorithm to solve the Burgers equation. With the Cole-Hopf transformation that maps the fluid velocity field u to a new field ψ, we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state ∣ψ⟩ that encodes the solution ψ. We also propose an efficient way to extract stochastic properties of u, namely the multi-point functions of u, from the quantum state of ∣ψ⟩. Our algorithm offers an exponential advantage over the classical finite difference method in terms of the number of spatial grids when a perturbativity condition in the information-extracting step is met.