What is the Pauli-Z gate?

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The Pauli-Z gate is one of the three fundamental single-qubit quantum gates (Pauli-X, Pauli-Y, Pauli-Z) in quantum computing. It is also called the phase-flip gate because, unlike the X gate (which flips states), the Z gate flips the phase of the qubit state.

🔑 Definition

The Pauli-Z gate is represented by the matrix:

$$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

🧩 Action on Basis States

• $Z\|0⟩ = \|0⟩$ (unchanged)

• $Z\|1⟩ = -\|1⟩$ (adds a negative phase)

For a superposition:

$$Z(\alpha\|0⟩ + \beta\|1⟩) = \alpha\|0⟩ - \beta\|1⟩$$

Thus, it flips the phase of the |1⟩ component, but leaves |0⟩ unchanged.

⚡ Properties

1. Unitary → Preserves probability.

2. Hermitian & Self-inverse → $Z^\dagger = Z$, $Z^2 = I$.

3. Bloch Sphere → Represents a 180° rotation around the Z-axis.

📌 Applications

• Phase-flip operations in quantum error correction.

• Used in Hadamard-Z-Hadamard combinations to implement X gates (showing symmetry).

• Plays a role in controlled gates like the Controlled-Z (CZ) gate.

• Forms building blocks for quantum algorithms and universal gate sets.

👉 In summary:
The Pauli-Z gate is a phase-flip gate that changes the sign of the |1⟩ state while leaving the |0⟩ state unchanged. On the Bloch sphere, it corresponds to a 180° rotation around the Z-axis, making it essential for controlling phase and building complex quantum circuits.

Read More :

Explain the Hadamard (H) gate.

What is the Pauli-X gate?

What is the Pauli-Y gate?

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