The simulation of gas flows for reentry problems can require to resolve a larger range of spatial scales in order to accurately model the physics of the flow field. In hypersonic flows, the modeling additionally requires to cover a large range in the velocity subspace of the phase space. Scale resolving methods like the Direct Simulation Monte Carlo (DSMC) method have been developed but can become computationally very expensive and unfeasible. Therefore, approximation methods allowing coarser grids are used which, however, may not be sufficiently accurate. To enable more efficient simulation methods, first approaches of a quantum lattice-gas model have been developed but these model do not scale efficiently with the grid resolution. Building on these quantum algorithms, promising modeling approaches using a quantum lattice Boltzmann method (QLBM) with amplitude encoding have shown to efficiently deal with very large grids and fine discretizations. Fine resolution of position and velocity space in the context of lattice Boltzmann methods (LBM) translate to many grid cells and a large set of velocity vectors in the LBM velocity discretization stencil, which can be encoded to scale efficiently in a qubit representation using the QLBM. For the investigation of collisionless flows a method based on Todorova et al. has been adopted and extended (cf. Fig. 1) to build upon our previous publication. The streaming and boundary interaction are modeled as two independent modular blocks, where the streaming block is adapted to be performed independently of the boundary. Additionally, our quantum algorithm includes the summation of the velocity distribution functions to calculate the mass density distribution directly in the quantum simulation. In contrast to a summation by classical post processing, the quantum summation block enables further development to extract reduced data from the quantum computer. This is a key point to not having to extract the entire high dimensional quantum state vector. In this study, a hypersonic collisionless flow around a flat plate with specular reflection is simulated using a quantum lattice Boltzmann method and compared with results using the established DSMC method for modeling rarefied flows. The quantum simulation is performed using a quantum algorithm simulator. Further approaches on how to extend the specular reflection model to a diffusive reflecting model are sketched out.
The simulation of gas flows for reentry problems can require to resolve a larger range of spatial scales in order to accurately model the physics of the flow field. In hypersonic flows, the modeling additionally requires to cover a large range in the velocity subspace of the phase space. Scale resolving methods like the Direct Simulation Monte Carlo (DSMC) method have been developed but can become computationally very expensive and unfeasible. Therefore, approximation methods allowing coarser grids are used which, however, may not be sufficiently accurate. To enable more efficient simulation methods, first approaches of a quantum lattice-gas model have been developed but these model do not scale efficiently with the grid resolution. Building on these quantum algorithms, promising modeling approaches using a quantum lattice Boltzmann method (QLBM) with amplitude encoding have shown to efficiently deal with very large grids and fine discretizations. Fine resolution of position and velocity space in the context of lattice Boltzmann methods (LBM) translate to many grid cells and a large set of velocity vectors in the LBM velocity discretization stencil, which can be encoded to scale efficiently in a qubit representation using the QLBM. For the investigation of collisionless flows a method based on Todorova et al. has been adopted and extended (cf. Fig. 1) to build upon our previous publication. The streaming and boundary interaction are modeled as two independent modular blocks, where the streaming block is adapted to be performed independently of the boundary. Additionally, our quantum algorithm includes the summation of the velocity distribution functions to calculate the mass density distribution directly in the quantum simulation. In contrast to a summation by classical post processing, the quantum summation block enables further development to extract reduced data from the quantum computer. This is a key point to not having to extract the entire high dimensional quantum state vector. In this study, a hypersonic collisionless flow around a flat plate with specular reflection is simulated using a quantum lattice Boltzmann method and compared with results using the established DSMC method for modeling rarefied flows. The quantum simulation is performed using a quantum algorithm simulator. Further approaches on how to extend the specular reflection model to a diffusive reflecting model are sketched out.
The computational power to resolve large complex aerospace flow problems has increased over the past decades but is still yet not able to simulate these problems sufficiently accurate to a scale resolved level. At the same time, the interest in quantum computing has recently increased rapidly due to the massive improvements of the quantum computer hardware technology. Many fields of research explore the possible advantages that quantum computing can offer in their respective fields. For fluid flow problems, the exponentially increasing storage capacity of flow properties by quantum amplitude encoding offers an approach to efficiently deal with highly resolved flow fields. Within the DLR-QCI project ToQuaFlics, we adjust and extend the quantum lattice Boltzmann method (QLBM) from the literature for single time steps and present our extension to multiple time steps. The structure of our algorithm is shown in Fig. 1. Our extended QLBM can perform all time steps of a simulation without having to extract the quantum state or parts of it at any time. Therefore, with our algorithm there is no need to measure any qubit during or at the end of the simulation. This saves the computationally expensive read out and reinitialization after every time step and enables an efficient time stepping scheme. We verify our algorithm on 1D and 2D test cases simulating a linear advection-diffusion process with periodic boundary conditions. The algorithm is simulated using the IBM simulator package Qiskit with the shot method. For verification reasons, we measure the entire flow field at the end of the simulation to demonstrate the agreement with the reference solution given by the classic lattice Boltzmann solution. The results show excellent agreement and arbitrary convergence towards the reference. We further discuss the inherent amplitude decay of a QLBM time step and how that effects our solution in terms of signal loss of the quantum state. An outlook to include a body with a surface interaction model and to capture nonlinearities is given and briefly sketched.
We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.
The Quantum Lattice Boltzmann Method (QLBM) has emerged as one of the most promising quantum computing approaches for the numerical simulation of problems in computational fluid dynamics (CFD). The dynamics is formulated in terms of mesoscopic particle distribution functions governed by a discrete Boltzmann transport equation, comprising local streaming and collision operations. In this work, the resulting macroscopic behavior corresponds to the advection-diffusion equation, which we adopt as a canonical model problem for transport phenomena. Building upon recent progress in QLBM implementations, we advance towards more realistic problem settings that better reflect conventional CFD requirements. We address, for the first time, transport under the action of non uniform velocity fields on quantum hardware. We implement our demonstration using IonQ's trapped-ion systems including Forte generation systems and a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We identify the density readout and subsequent reloading of the fluid density as a potential bottleneck of the current algorithm and discuss several approaches to mitigate this bottleneck. We identify the use of MPS shadow tomography as a promising method to efficiently scale the readout to large system with complex density distributions. Lastly, we introduce and simulate a novel method to implement wall boundaries for advection-diffusion in QLBM, and discuss the prospects of scaling to higher-complexity problems.
To simulate highly-resolved flow fields, we extend the Quantum Lattice Boltzmann Method (QLBM) to be able to simulate multiple time steps without state extraction or reinitialization. We adjust and extend given QLBM approaches from the literature to completely remove the need to measure or reinitialize the flow field in between the simulation time steps. Therefore, our algorithm does not require to sample the entire flow field at any time. We solve the linear advection-diffusion problem with periodic boundary conditions and derive all necessary equations and build the corresponding quantum circuit diagrams, including details on the QLBM blocks and explicitly drawing the circuit gates. We discuss the general decay of a QLBM step and how that effects our algorithm. The new algorithm is verified on 1D and 2D test cases using the shot method of IBMs Qiskit package. We show excellent agreement and convergence between our QLBM and the classical Lattice Boltzmann method. The conclusion section includes a discussion on the advantages of our algorithm as well as limitations and to what extent it is more efficient.
The Quantum Lattice Boltzmann Method (QLBM) is one of the most promising approaches for realizing the potential of quantum computing in simulating computational fluid dynamics. Many recent works mostly focus on classical simulation, and rely on full state tomography. Several key algorithmic issues like observable readout, data encoding, and impractical circuit depth remain unsolved. As a result, these are not directly realizable on any quantum hardware. We present a series of novel algorithmic advances which allow us to implement the QLBM algorithm, for the first time, on a quantum computer. Hardware results for the time evolution of a 2D Gaussian initial density distribution subject to a uniform advection-diffusion field are presented. Furthermore, 3D simulation results are presented for particular non-uniform advection fields, devised so as to avoid the problem of diminishing probability of success due to repeated post-selection operations required for multiple timesteps. We demonstrate the evolution of an initial quantum state governed by the advection-diffusion equation, accounting for the iterative nature of the explicit QLBM algorithm. A tensor network encoding scheme is used to represent the initial condition supplied to the advection-diffusion equation, significantly reducing the two-qubit gate count affording a shorter circuit depth. Further reductions are made in the collision and streaming operators. Collectively, these advances give a path to realizing more practical, 2D and 3D QLBM applications with non-trivial velocity fields on quantum hardware.
RÉSUMÉ: Les phénomènes de transport dans les milieux poreux, tels que l'écoulement des fluides, le transfert de chaleur et la diffusion de masse, sont essentiels dans les systèmes naturels et techniques. La modélisation de ces processus dans des géométries complexes ou multi échelles nécessite des méthodes de calcul avancées. La méthode de Boltzmann sur réseau (LBM), avec son cadre mésoscopique et son traitement discret de l'espace, du temps et des vitesses, capture le comportement macroscopique par le biais d'interactions locales entre les particules. À mesure que les exigences en matière de vitesse et de précision augmentent, l'intérêt s'est porté sur l'informatique quantique, qui exploite les propriétés de la mécanique quantique pour offrir un parallélisme dépassant les capacités classiques. Cette thèse présente une formulation quantique de la méthode Lattice Boltzmann (QLBM) pour simuler des écoulements de Stokes incompressibles bidimensionnels dans des domaines poreux. Une méthode basée sur la vitesse et la pression a été développée et mise en œuvre en utilisant des circuits quantiques. Cette méthode propose un support complet pour les lattices de type D2Q9, incluant la propagation diagonale de la population pour permettre la récupération complète du tenseur de contrainte. Des sous-programmes quantiques ont également été conçus pour calculer les moments d'ordre zéro et un, ce qui permet d'extraire directement des quantités acroscopiques telles que la densité et la vitesse à partir des états quantiques. Cette méthode incorpore des conditions limites périodiques sans glissement intégrées dans la logique quantique pour fonctionner indépendamment à chaque nœud de la lattice. Une vérification et une validation approfondies par rapport aux solveurs LBM classiques ont montré un excellent accord entre nos résultats quantiques et ceux obtenus par un solveur classique, avec des erreurs absolues moyennes relatives (RMAE) inférieures à 0,001 % pour la densité et inférieures à 0,1 % pour les champs de vitesse, et ce sur l'ensemble des résolutions de grille considérées. Tous les sous-programmes ont été vérifiés et intégrés dans une pipeline modulaire, ce qui permet de simuler des milieux poreux complexes dont la taille et la densité des obstacles varient. Une analyse détaillée des ressources quantiques a été réalisée pour étudier le comportement de mise à l'échelle en fonction de la taille du domaine et de la complexité géométrique, en quantifiant l'impact sur la profondeur du circuit, le nombre de qubits et les opérations de porte. Les simulations ont été effectuées en utilisant des vecteurs d'état idéaux (sans bruit) en raison des limitations matérielles actuelles telles que la décohérence, le bruit de grille et la disponibilité restreinte des qubits. Néanmoins, l'algorithme est conçu pour exploiter le parallélisme quantique et peut être adapté pour être exécuté sur des dispositifs NISQ au fur et à mesure des progrès du matériel. Ce travail établit un solveur QLBM fonctionnel et extensible qui fait le lien entre la dynamique des fluides numérique classique et l'informatique quantique. ABSTRACT: Transport phenomena in porous media, such as fluid flow, heat transfer, and mass diffusion are essential in both natural and engineered systems. Modeling these processes in complex or multiscale geometries requires advanced computational methods. The Lattice Boltzmann Method (LBM), with its mesoscopic framework and discrete treatment of space, time, and velocities, captures macroscopic behavior through local particle interactions. As speed and accuracy demands grow, interest has shifted toward quantum computing, which exploits quantum mechanic properties to offer parallelism beyond classical capabilities. This thesis presents a quantum formulation of the Lattice Boltzmann Method (QLBM) for simulating two-dimensional incompressible Stokes flows in porous domains. A velocity–pressure based framework was developed and implemented using quantum circuits, with complete support for D2Q9 lattices including diagonal population propagation to enable full stress tensor recovery. Quantum subroutines were designed for computing zero- and first-order moments, allowing direct extraction of macroscopic quantities such as density and velocity from the quantum states. The framework incorporates no-slip and periodic boundary conditions, embedded within the quantum logic to operate independently at each lattice node. Thorough verification and validation against classical LBM solvers showed excellent agreement, with relative mean absolute errors (RMAE) below 0.001% for density and below 0.1% for velocity fields across grid resolutions. All subroutines were verified and integrated into a modular pipeline, enabling simulation of complex porous media with varying obstacle size and density. A detailed quantum resource analysis was performed to study the scaling behavior with domain size and geometric complexity, quantifying the impact on circuit depth, qubit count, and gate operations. Simulations were carried out using ideal statevector backends due to current hardware limitations such as decoherence, gate noise, and restricted qubit availability. Nevertheless, the algorithm is designed to harness quantum parallelism and can be adapted for execution on NISQ devices as hardware advances. This work establishes a functional and extensible QLBM solver that bridges classical numerical fluid dynamics and quantum computation.