# Quantum Circuit Implementation of Multi-Dimensional Non-Linear Lattice Models
###### Abstract  
The application of Quantum Computing (QC) to fluid dynamics simulation has developed into a dynamic research topic in recent years. With many flow problems of scientific and engineering interest requiring large computational resources, the potential of QC to speed-up simulations and facilitate more detailed modeling forms the main motivation for this growing research interest. Despite notable progress, many important challenges to creating quantum algorithms for fluid modeling remain. The key challenge of non-linearity of the governing equations in fluid modeling is investigated here in the context of lattice-based modeling of fluids. Quantum circuits for the DIQ3 (one-dimensional, three discrete velocities) Lattice Boltzmann model are detailed along with design trade-offs involving circuit width and depth. Then, the design is extended to a one-dimensional lattice model for the non-linear Burgers equation. To facilitate the evaluation of non-linear terms, the presented quantum circuits employ quantum computational basis encoding. The second part of this work introduces a novel, modular quantum-circuit implementation for non-linear terms in multi-dimensional lattice models. In particular, the evaluation of kinetic energy in two-dimensional models is detailed as the first step toward quantum circuits for the collision term of two- and three-dimensional Lattice Boltzmann methods. The quantum circuit analysis shows that with \(O(100)\) fault-tolerant qubits, meaningful proof-of-concept experiments could be performed in the near future. q 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 2022 20222 2022 202  
Thus far, the overhead associated with repeated quantum measurement and quantum-state re-initialization has played an important part in preventing quantum speed-up for fluid dynamics simulations. Recently the available number of qubits on available hardware has grown to \(O(100)\). These qubits are still 'noisy', but with the predicted further growth to \(O(10^{4})\) or \(O(10^{5})\) physical qubits, the creation of \(O(100)\) 'functional qubits' (as termed by IBM, i.e., with a degree of fault tolerance enabling much deeper quantum circuits than on NISQ-era machines) within the next 5-10 years appears realistic. It is this context that motivated the current work. The development of quantum algorithms for effective three-dimensional flow simulations based on the Lattice Boltzmann method (LBM) with limited exchange of information between the quantum and classical domains in a hybrid quantum/classical approach forms the main aim. The aspect of minimizing the quantum-classical information exchange is driven by the expectation that even for next-generation quantum hardware, the cost of frequent re-initialization and quantum measurement will remain a bottleneck in achieving computational efficiency. Recently, [PERSON] [11] presented an interesting quantum algorithm for two-dimensional flow simulation based on the LBM. However, its formulation uses a tight coupling of the quantum and classical domains of the simulations introducing significant overhead for each time step performed. In a previous work, [PERSON] [14] showed a quantum implementation for lattice-based modeling of the linear advection-diffusion equation, where due to the linearity of the problem, some of the expensive re-initialization steps needed for the non-linear LBM could be avoided. An interesting approach to incorporating non-linearity in quantum algorithms based on Carleman linearization of the LBM was presented by [PERSON] and [PERSON] [10]. However, since this approach is based on linearization, it appears that this approach is mainly suited to weakly non-linear low Reynolds number flows. The current approach differs from these existing works in a number of key aspects. The approach followed here uses encoding of data in the quantum computational basis along with a quantum floating-point format previously introduced by the author [7; 12].