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.
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.
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.
Classical computational fluid dynamics (CFD) methods encounter significant computational bottlenecks in large systems and at high Reynolds number flows due to the rapid increase in degrees of freedom, even on flagship supercomputers. As quantum computers emerge, the Quantum Lattice Boltzmann Method (QLBM) has been proposed as a quantum-native alternative for solving fluid dynamics problems. However, until recently, nearly all studies have relied on classical quantum simulators rather than real quantum hardware. This study presents a benchmark of a one-dimensional QLBM implementation executed on three different IBM Quantum hardware. We design and test a quantum circuit for the advection-diffusion equation using amplitude encoding, a Bhatnagar-Gross-Krook (BGK)-type collision operator, and a streaming step with periodic boundary conditions. Our numerical experiments evaluate transpiled two-qubit gate count, total circuit depth, runtime, and accuracy metrics including maximum absolute error (L∞ norm) and root-mean-square error (RMSE). We find that the present collision and streaming operators dominate the overall cost and become the primary scalability bottleneck on current hardware. To mitigate the prohibitive cost of full quantum state tomography on hardware-based benchmarking exercises, we introduce a sampler-based measurement strategy that reconstructs macroscopic density fields directly from computational-basis measurements. This approach substantially reduces measurement overhead while preserving the expected advection-diffusion dynamics on hardware. These results provide a quantitative assessment of QLBM performance in the noisy intermediate-scale quantum (NISQ) regime and identify the principal algorithmic and experimental limitations that must be addressed for QLBM to become viable for fluid dynamics.
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 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.
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.
We present a quantum algorithm for computational fluid dynamics based on the Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a modified collision operator, assuming full relaxation to the local equilibrium within a single time step. Our quantum algorithm enables the computation of multiple time steps in the linearized case, specifically for solving the advection-diffusion equation, before necessitating a full state measurement. Moreover, our formulation can be extended to compute the non-linear equilibrium distribution function for a single time step prior to measurement, utilizing the measurement as an essential algorithmic step. However, in the non-linear case, a classical postprocessing step is necessary for computing the moments of the distribution function. We validate our algorithm by solving the one dimensional advection-diffusion of a Gaussian hill. Our results demonstrate that our quantum algorithm captures non-linearity.