What is the Pauli-Y gate?

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The Pauli-Y gate is a fundamental single-qubit quantum gate in quantum computing, belonging to the Pauli family (X, Y, Z). It performs a specific type of quantum state transformation and is closely related to both the Pauli-X (bit-flip) and Pauli-Z (phase-flip) gates.

🔑 Definition

• The Pauli-Y gate corresponds to the Pauli-Y matrix:

$$Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$$

🧩 Action on Basis States

• $Y\|0⟩ = i\|1⟩$

• $Y\|1⟩ = -i\|0⟩$

So unlike X (which flips states) or Z (which changes phase), the Y gate flips the qubit state with a phase factor of ±i.

⚡ Properties

1. Unitary → Preserves quantum probabilities.

2. Hermitian → Its own inverse ($Y = Y^\dagger$).

3. Self-inverse → Applying twice gives identity ($Y^2 = I$).

4. Bloch Sphere → Represents a 80° rotation around the Y-axis.

📌 Applications1

• Provides bit-flip + phase-flip operation.

• Useful in error correction codes and quantum cryptography.

• Appears in constructing universal gate sets and entangled states.

• Helps implement rotations: $R_y(\theta) = e^{-i \theta Y / 2}$.

👉 In summary:
The Pauli-Y gate is a quantum gate that flips a qubit state like the X gate but also adds a complex phase factor. On the Bloch sphere, it corresponds to a 180° rotation around the Y-axis, making it essential for full 3D quantum state manipulation.

Read More :

Explain the Hadamard (H) gate.

Explain the no-cloning theorem.

What is the Pauli-X gate?

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