# Quantum Circuit-Width Reduction through Parameterisation and Specialisation
## 1 Introduction
### Context of Present Work  
In recent years, significant research efforts have focused on developing quantum algorithms for computational engineering applications, including fluid dynamics, with the aim of achieving significant quantum speed-up. A promising line of investigation in the context of computational fluid dynamics focuses on developing quantum circuit implementations of lattice-based models of a range of problems [25; 26; 27; 28; 29; 30; 31]. A quantum algorithm for the collisionless Boltzmann equation and its application to rarefied supersonic flows was presented by [PERSON] and [PERSON] [25; 26]. [PERSON] [27] introduced a quantum algorithm for the linear advection-diffusion equation based on the Lattice Boltzmann method. In this work and in the work of [PERSON] and [PERSON] [25], particle distribution functions defined in each of the lattice sites on the considered regular lattice move to a neighbouring site during subsequent time steps. The quantum circuit implementation of this step shows strong similarity with quantum modulo adders. Therefore, it can be expected that the transformations introduced here can also be extended to this type of circuits. In the Lattice Boltzmann method, a collision term appears on the right-hand side of the equation and its non-linearity for the Navier-Stokes equations in fluid dynamics was considered by [PERSON] [29] using quantum floating-point arithmetic and using Carleman linearisation in the work of [PERSON] and [PERSON] [28]. In the context of the Lattice Boltzmann method for fluid dynamics, in [PERSON] et al. [30] the authors presented quantum circuit implementations of non-linear one-dimensional models. More recently, the quantum circuit implementation of multi-dimensional non-linear lattice models was presented by one of the authors [31]. Both these works employed an application-specific floating-point representation with reduced precision relative to the IEEE-754 single-precision standard [32]. A key feature of this ongoing work on quantum circuit implementation of lattice modelling is the reliance on a range of quantum arithmetic operations. An illustrative example is shown in Figure 2 where for a fixed-point definition of velocities \(u\) and \(v\) in a two-dimensional problem, the quantity \(u^{2}+v^{2}\) (i.e., directly related to kinetic energy) is evaluated using a shift-and-add based approach to sum contributions from \(u\) and \(v\) into the output register defining \(u^{2}+v^{2}\). The quantum circuit shown in Figure 2 is based on the more complex circuits shown in [PERSON] [31] that involve floating-point operations. For the current work, the modified 4-qubit full adders \(FA_{u}\) and \(FA_{v}\) are of particular interest. Specifically, for larger fixed-point data representations with an increased number of qubits, it is clear that the quantum circuit width often exceeds the size that can be routinely simulated using a full state vector Quantum Circuit Simulator (as discussed in more detail in Section 3). In the ongoing development of quantum algorithms for lattice-based modeling, but also many further quantum algorithms employing similar quantum arithmetic operations, the quantum circuit width is often a limiting factor. This means that for quantum algorithms involving quantum arithmetic operations, the introduction of circuit-width reduction steps on the circuit performing quantum arithmetic operations would facilitate the classical simulation of larger circuits, as well as a wider range of quantum algorithms.  
Figure 1: Illustration of simple two-qubit quantum circuits.