A high-level Quantum Lattice Boltzmann Solver for pure advection with local measurement

Our quantum solver for the linear advection equation exhibits two key features. One of them is that we use for the implementation of the full QLBM advection cycle  a high-level functional quantum programming language called Qrisp [4]. This programming language has been developed by the Fraunhofer Institute FOKUS to facilitate the implementation of quantum algorithms based on concepts from classical programming languages. Two concepts that have been adapted from classical programming are, e.g., the use of variables as well as automated uncomputation of local variables in a function [5]. By means of principles from high-level programming, our implementation is less prone to errors compared to creating complex circuits consisting of a huge amount of qubits and gates. Moreover, we obtain a clear and modular mapping of the streaming step onto quantum registers. Unlike other algorithms that assume infinite domains, our approach incorporates physically relevant boundary conditions, including periodic, simple bounce-back, and slip setups, realized directly through high-level functional constructs. As a second key feature, we introduce a local measurement protocol based on algorithms for quantum amplitude estimation and amplification [6] to extract the macroscopic particle densities in selected sub-regions of the computational domain. Both algorithms have already been implemented in Qrisp. To be able to apply them in terms of our issue special oracles have to be designed, which represent the QLBM. In addition, the initial particle distribution encoded in a quantum state has to be provided. By focusing on localized areas of interest and using amplitude amplification, this strategy has the potential to avoid the exponential cost of extracting the different particle densities by measuring the final quantum state resulting from the QLBM.

To illustrate the performance of our quantum solver, we use the simulators provided by Qrisp to show that it produces correct simulation results. This means that we study whether a cluster of particles moves correctly in a given velocity field and whether the boundary conditions have the intended influence on the particle densities. Our simulations of advection problems demonstrate that the solver preserves sharp profiles without numerical diffusion that can occur in terms of classical discretization methods with a low approximation order. In addition to that our solver avoids the formation of unphysical oscillations, which are caused by numerical instabilities. Finally, the computation of local densities is tested and its correctness is evaluated by comparing our results to analytical solutions.  

Besides the numerical simulations, we use the tools of Qrisp to perform a resource estimation, i.e., we show how the number of qubits, gates and the depth of the underlying circuits scale with the dimensions of the grid and the number of time steps. In addition to that, we determine how many measurements are required to compute local particle densities with sufficient accuracy. It can be shown that the number of required measurements can be decreased in a significant way. Our future work will be concentrated on extending our quantum solver so that not only a pure advection process can be simulated but also the whole Boltzmann equation can be handled. Thereby, references such as [7] are taken into account. A solver for the entire Boltzmann equation with efficient local measurement would enable us to determine flow fields in sub-regions of interest. In terms of blood flow simulations within vessels having an aneurysm, a sub-region of interest would be the aneurysm, since one is primary interested in a detailed description of the flow field within the aneurysm. Using the solver for the pure advection problem, the propagation of contaminants e.g. in water flow can be studied at the end of some simulation period and within a sub-region of a computational domain.