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🎯 Pauli Gates (X, Y, Z)

Master the three fundamental gates that rotate qubits on the Bloch sphere

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Introduction to Quantum Gates

The Foundation of Quantum Computing

🎯 What Are Pauli Gates?

The Pauli gates—X, Y, and Z—are the fundamental single-qubit operations in quantum computing. Named after physicist Wolfgang Pauli, these gates perform rotations of π radians (180°) around the three axes of the Bloch sphere.

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Why They Matter

Pauli gates are the building blocks of quantum circuits. Any single-qubit operation can be decomposed into combinations of Pauli gates and rotations.

🔄 The Three Pauli Gates

X
Pauli-X Gate

Quantum NOT gate

|0⟩ ↔ |1⟩
Y
Pauli-Y Gate

Flip with phase

|0⟩ → i|1⟩
Z
Pauli-Z Gate

Phase flip only

|1⟩ → -|1⟩

🌀 Bloch Sphere Rotations

Each Pauli gate rotates the qubit state by 180° around a specific axis of the Bloch sphere:

X-axis Rotation

X gate rotates around x-axis

Flips |0⟩ ↔ |1⟩

Y-axis Rotation

Y gate rotates around y-axis

Flips with phase shift

Z-axis Rotation

Z gate rotates around z-axis

Changes phase only

📊 Matrix Representation

Pauli gates are represented as 2×2 matrices operating on qubit states:

X Gate
⎡0 1⎤
⎣1 0⎦
Y Gate
⎡0 -i⎤
⎣i 0⎦
Z Gate
⎡1 0⎤
⎣0 -1⎦

💡 Key Insight

Pauli gates are Hermitian (self-adjoint) and unitary, meaning they're reversible and preserve quantum information. Applying any Pauli gate twice returns the qubit to its original state: X² = Y² = Z² = I (identity).