# Quantum Circuit-Width Reduction through Parameterisation and Specialisation
## 2 Data Encoding in Quantum Information  
The quantum circuit transformation approaches introduced here were designed to reduce quantum circuits performing quantum arithmetic operations. In the intended application, the quantum circuit considered represents a larger, and more general quantum algorithm were the arithmetic represents a part of the computational work. Quantum arithmetic operations typically rely on a specific type of data encoding. To explain this further, the two main data encoding approaches used in quantum information processing are summarized here. In amplitude-based encoding, a vector of normalised complex data (of size up to \(N=2^{n}\)) is encoded into the amplitudes, \(C_{k}\) in Equation (3). This encoding technique is the most widely used encoding technique in quantum algorithms since it creates the most direct means of taking advantage of quantum parallelism (i.e., exponential growth of number of degrees of freedom with linear increase in the number of qubits). For this type of data encoding, the most widely used quantum circuit simulation approach on classical computers is **full state vector simulation**, where the \(2^{n}\) complex amplitudes are all stored and the gate operations in a considered circuit lead to step-by-step modifications of these amplitudes. In the present work, the focus is on the alternative approach termed **computational-basis encoding**. In this data encoding approach, the quantum algorithm is designed such that at initialisation only the complex amplitude of a single computational basis state has non-zero amplitude. After completion of the quantum algorithm the output is represented similarly by a single non-zero amplitude for one of the quantum basis states. For quantum algorithms employing the computational basis encoding a few important observations relevant to the present work can be made:  
* The motivation for this type of encoding is typically performing quantum arithmetic operations;
* To maintain the property that only a single computational basis state has a non-zero amplitude throughout the computation, the gates in the Quantum Circuit model are limited to quantum equivalents of logic gates (e.g., \(X\) as equivalent of \(NOT\) and Toffoli as doubly-controlled \(NOT\)). By doing so, the quantum circuit can efficiently be simulated on a classical computer using a **logic-based simulator**. In such a simulator, \(n\) classical bits suffice to represent the state of \(n\) qubits. Then, the controlled logic gate operations conditionally flip states between 0 and 1;
* If quantum arithmetic operations are implemented in the quantum circuit model based on the Quantum Fourier Transform [43; 44], then efficient simulation using classical logic-based simulation is not possible, since the QFT in the case of quantum arithmetic circuits temporarily moves the encoding approach to amplitude-based encoding, before finally returning an output in computational basis encoding. Since the circuit transformation approaches introduced here target quantum arithmetic operations, key aspects of computational basis encoding were used in creating these transformation steps. Specifically, identifying qubits that throughout a quantum computation are guaranteed to remain in either state \(|0\rangle\) or \(|1\rangle\) relies on this type of encoding throughout the computation or at least for the considered part of the algorithm performing arithmetic operations. For more general algorithms using amplitude encoding the transformation approaches introduced cannot be employed.