We will proceed with the graphical representation of quantum gates, which is highly significant when reading theoretical works and conducting literature reviews. Understanding the graphical representation of quantum gates is essential for comprehending how quantum states are processed and transformed. Similar to classical computing, quantum computing operations are fundamentally based on basic gates. Up to this point in the study, we have examined quantum states and measurement processes in succession. Here, we will explore how quantum states perform computational operations using gates.

Although quantum gates may resemble classical digital gates, there are specific quantum gates that have no equivalents in classical systems due to particular conditions and scenarios. These unique quantum gates will be specifically examined in the following section of this study.

Before delving into the details, let us discuss some key notations:

• We use the letter q to denote quantum bits (qubits).

• Horizontal lines are used to represent quantum gates and circuits, indicating the order of operations for the qubits. These lines do not represent physical wires but instead convey the sequence of operations.
• In classical computing, wires indicate both the order of operations and physical connections, making classical representations simpler in this regard. However, quantum computing can involve complex phenomena such as entanglement, which may require additional notations in diagrams to be accurately represented.

We will discuss all representations in detail in the subsequent parts of this work; for now, this overview should suffice.