Linearized quantum lattice-Boltzmann method for the advection-diffusion equation using dynamic circuits

Abstract

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.