Single-qubit gates are fundamental operations in quantum computing. They manipulate individual qubits by rotating their states on the Bloch sphere, enabling state flipping, phase shifts, and superposition creation.

Pauli gates (X, Y, Z) perform specific rotations, while the Hadamard gate creates superposition. Phase gates (S, T) introduce phase shifts. These operations form the building blocks for more complex quantum algorithms.

Single-Qubit Gates

Operation of Pauli gates

Pauli gates perform rotations on the Bloch sphere representing single-qubit states
- Pauli-X gate (X gate or NOT gate) flips the qubit state from $|0\rangle$ $∣0 ⟩$ to $|1\rangle$ $∣1 ⟩$ and vice versa by rotating the state by $\pi$ $π$ radians about the x-axis
  - Matrix representation: $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
- Pauli-Y gate (Y gate) flips the qubit state and applies a phase shift of $i$ $i$ to $|1\rangle$ $∣1 ⟩$ by rotating the state by $\pi$ $π$ radians about the y-axis
  - Matrix representation: $Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
- Pauli-Z gate (Z gate) applies a phase shift of -1 to $|1\rangle$ $∣1 ⟩$ , leaving $|0\rangle$ $∣0 ⟩$ unchanged, by rotating the state by $\pi$ $π$ radians about the z-axis
  - Matrix representation: $Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

Function of Hadamard gate

The Hadamard gate (H gate) creates superposition by transforming a qubit from a basis state ( $|0\rangle$ $∣0 ⟩$ or $|1\rangle$ $∣1 ⟩$ ) to an equal superposition of both states
- Applies the following transformations:
  - $H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$
  - $H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$
- Applying the Hadamard gate twice in succession returns the qubit to its original state (self-inverse property)
- Matrix representation: $H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

Operation of Pauli gates, Single-qubit transform of a state vector - Algowiki

Application of phase gates

Phase gates introduce phase shifts to the $|1\rangle$ $∣1 ⟩$ state without changing the $|0\rangle$ $∣0 ⟩$ state
- S gate (Phase gate or $\sqrt{Z}$ $Z$ gate) applies a phase shift of $i$ $i$ to $|1\rangle$ $∣1 ⟩$ by rotating the state by $\frac{\pi}{2}$ $\frac{π}{2}$ radians about the z-axis
  - Matrix representation: $S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}$
- T gate ( $\sqrt{S}$ $S$ gate or $\sqrt[4]{Z}$ $4 Z$ gate) applies a phase shift of $e^{i\frac{\pi}{4}}$ $e^{i \frac{π}{4}}$ to $|1\rangle$ $∣1 ⟩$ by rotating the state by $\frac{\pi}{4}$ $\frac{π}{4}$ radians about the z-axis
  - Matrix representation: $T = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\frac{\pi}{4}} \end{bmatrix}$

Geometric representation on Bloch sphere

The Bloch sphere geometrically represents a qubit's state, with pure states on the surface and mixed states inside the sphere
Single-qubit gates can be visualized as rotations on the Bloch sphere:
1. Pauli-X gate: $\pi$ rotation about the x-axis (flips state)
2. Pauli-Y gate: $\pi$ rotation about the y-axis (flips state and applies phase shift)
3. Pauli-Z gate: $\pi$ rotation about the z-axis (applies phase shift)
4. Hadamard gate: $\pi$ rotation about the axis equally dividing the x and z axes (creates superposition)
5. S gate: $\frac{\pi}{2}$ rotation about the z-axis (applies phase shift)
6. T gate: $\frac{\pi}{4}$ rotation about the z-axis (applies phase shift)

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