# Quantum Circuit-Width Reduction through Parameterisation and Specialisation
## 1 Introduction
Another type of transformations, closer related to the present work, involves circuit transformation aimed at facilitating classical simulation on large-scale distributed memory computing facilities. Graph-based transformation techniques have been introduced by a range of researchers to achieve a reduction in memory requirements. [PERSON] et al. [7], [PERSON] et al. [8], and [PERSON] et al. [9] employ a tensor network-based approach, while [PERSON] et al. [10] use an undirected graph model. Further transformation methods presented in the literature include the circuit partitioning scheme described by [PERSON] et al. [11] and the implicit decomposition technique described by [PERSON] et al. [12], transforming the original circuit into multiple, smaller transformed circuits so that communication in parallel simulations on classical computers could be reduced. The quantum circuit transformations introduced in this work are mainly aimed at facilitating simulating quantum circuits on classical computers. Additionally, the introduced transformation can be used in quantum hardware realisations, where the qubit count reduction can be used to facilitate realisation of circuits that would not fit without the transformations introduced here. It is important to note that the output of the proposed transformation involves gate operations with multiple control qubits (as detailed later). Therefore, a subsequent transpilation step accounting for available native gates and qubit connectivity of target quantum hardware is required for quantum hardware realisations. For quantum circuit simulation on classical hardware, a similar transpilation step is often required for applications where the impact of (modelled) quantum gate errors is investigated. In the transpilation step, a well-considered trade-off between quantum circuit width and quantum circuit depth is required to avoid undoing the circuit width reduction achieved by the quantum circuit transformations introduced here. If IBM quantum hardware is used in realisation, the transformation step in IBM Qiskit [13] would be performed after the transformations proposed here. Beyond IBM's Qiskit, several other tools exist that were designed to facilitate the specification and transformation of quantum circuits, e.g., Quipper [14], OpenQASM 3 [15; 16], Xanadu's Strawberry Fields [17], Penny-Lane [18], and Microsoft Q# [19]. Xanadu's tools primarily focus on Quantum Machine Learning and compilation to photonic quantum computers; OpenQASM is an effort to standardise quantum circuit specification. While the circuit specification and manipulation techniques provided by Qiskit, Quipper, and Microsoft Q# are similar to those provided by our toolchain, these tools were developed for more general circuit transformations and do not provide a similar functionality in terms of static analysis to achieve circuit width reduction of parameterised quantum arithmetic circuits. The transformations introduced in this work rely on the concept of identifying qubits in the circuit that can be _specialised_ to a constant value, i.e., \(|0\rangle\) or \(|1\rangle\), as explained later. If the identified qubit only acts as _control_ qubit in the quantum gate operations in the circuit, then two independent circuits (with one fewer qubit) can be defined for both choices of the specialised qubit. This approach works independent of the application domain for which the considered quantum circuit was developed, and was used in a different context in the works on circuit partitioning [11] and circuit decomposition [12], as mentioned previously. The second concept on which the transformations introduced in this work rely involves exploiting application domain knowledge to facilitate the identification of qubits suitable for _specialisation_. As explained later in this section, a wide range of practically important quantum algorithms exist where quantum arithmetic occurs as part of the larger computation. This includes the famous Shor algorithm for integer factorization [20].