Unitary Matrices
Quantum gates are mathematically represented by unitary matrices. A unitary matrix is one that is invertible, meaning that when multiplied by its own Hermitian conjugate, it results in the identity matrix.
Invertibility
All quantum gates must be invertible, meaning that the result of an operation can be reversed to return to the original state.
Operation on Qubits
Quantum gates can operate on one or more qubits. For example, the Hadamard gate operates on a single qubit, while the CNOT gate operates on two qubits.
1.1. Quantum Gates and Their Properties
Since this section will be the starting point of a long study, some explanations may be necessary here. Firstly, as mentioned in our previous work, quantum gates, like classical computer gates, can operate on either a single bit or multiple bits. To express this, we will first list quantum gates that work on single qubits and multiple qubits separately, and later, we will discuss these gates in detail.
| Quantum Gate Name | Number of Qubits | Description |
|---|---|---|
| Pauli – X | Single Qubit | Similar to the classical NOT gate, it flips the bit of the input. |
| Pauli – Y | Single Qubit | Rotates the quantum bit’s state by 180° around the Y-axis. |
| Pauli – Z | Single Qubit | Rotates the quantum bit’s state by 180° around the Z-axis. |
| Hadamard (H Gate) | Single Qubit | Brings the qubit into a superposition state. |
| S Gate | Single Qubit | Also known as the phase gate, it shifts the qubit’s phase by 90°. |
| T Gate | Single Qubit | A phase gate that shifts the phase by 45°. |
| I Gate | Single Qubit | Leaves the input qubit unchanged. |
| RX, RY, RZ Gates | Single Qubit | Enables the qubit to rotate around specific axes by certain angles. |
| Quantum Gate Name | Number of Qubits | Description |
|---|---|---|
| CNOT Gate | Two Qubits | Changes the state of the target qubit depending on the control qubit. |
| Toffoli Gate | Three Qubits | A three-input gate, the quantum version of the classical AND gate. |
| Swap Gate | Two Qubits | Exchanges the states of two qubits. |
| Fredkin Gate | Three Qubits | A controlled SWAP gate that swaps two other bits if the control bit is true. |
| CU (Controlled U) | Two Qubits | Applies a specific U operation to the target qubit when the control bit is verified. |
| Controlled Z Gate | Two Qubits | Applies a Z operation to the target qubit when the control bit is verified. |
Now, let’s briefly discuss the usage scenarios and justifications for quantum gates. These will be addressed in more detail and effectively in the later sections of the study.
- Pauli X Gate: This is the quantum counterpart of the classical NOT gate. It is used when it is necessary to invert the state of a qubit. For example, it plays a crucial role in error correction algorithms and basic qubit manipulations.
- Hadamard Gate (H): It is used to create superposition, which is fundamental to quantum algorithms. For example, in the Grover search algorithm or the Deutsch-Jozsa algorithm, it creates the initial state to run all possible states simultaneously.
- CNOT Gate: This gate is used to create quantum entanglement. Protocols like quantum teleportation and the Bell state use the CNOT gate to ensure qubits behave together.
- Toffoli Gate: As the quantum version of the commonly used AND gate in classical computing, it is used in quantum error correction codes and for quantum analogs of classical computations.
- SWAP Gate: It is used when there is a need to exchange the positions of physical qubits, particularly when transferring information between qubits. This is important for optimizing quantum circuits and adjusting qubit positions in physical hardware.
- Controlled-Z and Controlled-U Gates: These gates are used when phase shifts or specific rotational operations are required under control. For instance, they are used in quantum Fourier transforms and phase estimation algorithms to apply specific phases.
Here, we have briefly discussed how quantum computer gates work from both a physical and engineering perspective, and in which situations they are used. Now, we can examine the details of these gates.