Linearized quantum lattice-Boltzmann method for the advection-diffusion equation using dynamic circuits (2025)
Supplementary code for the publication: "Linearized quantum lattice-Boltzmann method for the advection-diffusion equation using dynamic circuits". https://doi.org/10.1016/j.cpc.2025.109856
We propose a quantum algorithm for the linear advection-diffusion equation (ADE) Lattice-Boltzmann method (LBM) that leverages dynamic circuits. Dynamic quantum circuits allow for an optimized quantum collision operator algorithm by incorporating partial measurements as an integral step. The quantum circuit efficiently adapts during execution based on digital information obtained via mid-circuit measurements. The proposed new collision algorithm is implemented as a fully unitary operator, which facilitates the computation of multiple time steps without state reinitialization. Unlike previous quantum collision operators that rely on linear combinations of unitaries, the proposed algorithm does not exhibit a probabilistic failure rate. Our proposed algorithm embeds no more than two distribution functions simultaneously within the quantum state, irrespective of the velocity set. Compared to previous quantum algorithms, this approach reduces both the qubit overhead and circuit complexity required to execute the collision operator and encode the distributions. The quantum collision algorithm is validated against classical LBM simulations in 1D and 2D, showing excellent agreement. Performance analysis over multiple time steps highlights advantages of the proposed method compared to previous methods. As an additional variant, a hybrid quantum-digital approach is proposed, which reduces the number of mid-circuit measurements, thus improving the efficiency of the quantum collision algorithm.
David Wawrzyniak
Correspondence via mail.
@article{WAWRZYNIAK2025109856,
title = {Linearized quantum lattice-Boltzmann method for the advection-diffusion equation using dynamic circuits},
journal = {Computer Physics Communications},
volume = {317},
pages = {109856},
year = {2025},
issn = {0010-4655},
doi = {https://doi.org/10.1016/j.cpc.2025.109856},
url = {https://www.sciencedirect.com/science/article/pii/S0010465525003583},
author = {David Wawrzyniak and Josef Winter and Steffen Schmidt and Thomas Indinger and Christian F. Janßen and Uwe Schramm and Nikolaus A. Adams}
Supplementary code for the publication: "A Quantum Algorithm for the Advection-Diffusion Equation in the Lattice-Boltzmann Method". https://doi.org/10.1016/j.cpc.2024.109373
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.
David Wawrzyniak
Correspondence via mail.
@article{WAWRZYNIAK2025109373,
title = {A quantum algorithm for the lattice-Boltzmann method advection-diffusion equation},
journal = {Computer Physics Communications},
volume = {306},
pages = {109373},
year = {2025},
issn = {0010-4655},
doi = {https://doi.org/10.1016/j.cpc.2024.109373},
url = {https://www.sciencedirect.com/science/article/pii/S0010465524002960},
author = {David Wawrzyniak and Josef Winter and Steffen Schmidt and Thomas Indinger and Christian F. Janßen and Uwe Schramm and Nikolaus A. Adams},
}