The Pauli-X gate in quantum mechanics functions similarly to the classical NOT gate and is represented by a 2×2 matrix:

| 0 1 |
| 1 0 |

his matrix flips the state of a qubit. That is, it converts the state ∣0⟩ to ∣1⟩ and the state ∣1⟩ ∣0⟩.

• In the Bloch sphere, the X gate corresponds to rotating the qubit’s state by 180° around the X axis. A qubit is generally expressed as:

∣ψ⟩=α∣0⟩+β∣1⟩

When the X gate is applied, this expression transforms into:

∣ψ⟩ = α∣1⟩ + β∣0⟩

This changes the qubit’s polar coordinates and ensures a new position in the quantum computer’s state space.

• The X gate is typically applied physically using a magnetic field or laser pulses. In physical qubit systems (e.g., superconducting quantum circuits or ion traps), the application of the X gate is typically achieved by sending microwave or radio frequency pulses to the qubit at a specific frequency for a certain duration.
  
• The X gate is a fundamental component in quantum algorithms. For example, it is used to alter the initial state in algorithms or to flip certain bits in error correction codes.
  
• In quantum error correction protocols, the use of the X gate is necessary to correct specific errors. For instance, this gate is used to correct a bit-flip error.

The common application of the X gate in quantum circuits is in algorithms that require the manipulation of qubits in their basic states.