# Quantum Circuit Implementation of Multi-Dimensional Non-Linear Lattice Models
###### Abstract
Non-linear terms are then evaluated using arithmetic for the quantum floating-point format. This avoids restriction to weakly non-linear low-Reynolds numbers flows. Further, based on the used data encoding and floating-point format, the current work aims to perform multiple time steps in a fluid dynamics simulation on the quantum processor, in contrast to approaches where measurement and re-initialization take place at the end of each step. For the chosen approach, it is expected that with a greatly reduced number of bits representing the mantissa as compared to IEEE-754 single precision [15], e.g., 4-8 qubits are suggested here, the required number of qubits in the derived circuits is \(O(100)\). This is before 'transpilation' (as termed by IBM) into quantum circuits expressed in terms of native gates on the considered quantum hardware. This clearly creates a requirement for fault tolerance and coherency well beyond the current NISQ-type hardware. However, considering current predictions, quantum hardware capable of executing the derived circuits can be expected this decade. The main contributions of the present work can be summarized as follows:  
* Detailed analysis of quantum-circuit design for the evaluation of non-linear equilibrium distribution function for the D1Q3 Lattice Boltzmann model as well as for the one-dimensional lattice model for the Burgers equation. The modular design for both applications facilitates large-scale re-use of the different components of the circuits;
* A complexity analysis for the one-dimensional models is presented, highlighting the dependency of quantum-circuit width on the number of mantissa qubits in floating-point representation;
* Detailed description of the design process for quantum circuit implementations of non-linear terms in two- and three-dimensional lattice models, including a complexity analysis showing the dependency on the number of mantissa qubits used;
* Demonstration of how a modified shift-and-add-based multiplier can be used to create efficient, modular circuits for kinetic energy evaluation in two-dimensional models;Detailed quantum-circuit description for the proposed modular kinetic energy evaluation circuits based on quantum floating-point arithmetic;  
For a previously developed quantum-circuit implementation of the D1Q3 lattice model with fewer qubits but larger circuit depth, the analysis and verification using quantum computing simulation techniques were addressed recently by the author and co-workers in [PERSON] et al. [12]. The systematic evaluation of the quantum circuits presented in this work forms part of ongoing work by the author and co-workers. The present work is structured as follows. First, non-linear lattice models used in fluid dynamics are reviewed in Section 2. This section also considers the non-linear convection terms in the Navier-Stokes equations to provide context. Key concepts and principles of QC relevant to this work are reviewed briefly in Section 3. Quantum circuits for the evaluation of the non-linear distribution functions of the D1Q3 Lattice Boltzmann model are detailed in Section 4. For a lattice model for the one-dimensional Burgers equation, similar circuits are presented in Section 5. The extension of quantum-circuit design to two- and three-dimensional lattice models are described in Section 6. For two-dimensional kinetic energy evaluation, an efficient, modular quantum circuit is described in detail in Sections 7 and 8. Section 9 summarizes a step-by-step methodology for the evaluation of algorithms. Finally, Section 10 presents the conclusions and directions for future research work.