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X Gate (Pauli-X)

The X gate, also known as the Pauli-X gate or NOT gate, is a fundamental single-qubit quantum gate that performs a bit-flip operation.

Matrix Representation

\[X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]

Action on Basis States

The X gate flips computational basis states:

\[X|0\rangle = |1\rangle\]
\[X|1\rangle = |0\rangle\]

General Action

For an arbitrary qubit state \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\):

\[X|\psi\rangle = X(\alpha|0\rangle + \beta|1\rangle) = \alpha|1\rangle + \beta|0\rangle\]

The amplitudes are swapped between the \(|0\rangle\) and \(|1\rangle\) components.

Bloch Sphere Representation

The X gate corresponds to a 180° rotation around the X-axis of the Bloch sphere.

  • Maps north pole (\(|0\rangle\)) to south pole (\(|1\rangle\))
  • Maps south pole (\(|1\rangle\)) to north pole (\(|0\rangle\))
  • Maps points on the X-axis to themselves
  • Inverts points on the Y and Z axes

Properties

Involutory

\[X^2 = I\]

The X gate is its own inverse - applying it twice returns to the original state.

Hermitian

\[X^\dagger = X\]

The X gate is Hermitian, making it both unitary and self-adjoint.

Eigenvalues and Eigenvectors

Eigenvalues: \(\lambda_1 = +1\), \(\lambda_2 = -1\)

Eigenvectors:

\[|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle), \quad X|+\rangle = +|+\rangle\]
\[|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle), \quad X|-\rangle = -|-\rangle\]

Pauli Group

The X gate belongs to the Pauli group along with Y, Z, and I:

\[\{I, X, Y, Z\}\]

Commutation Relations

\[XY = iZ, \quad YZ = iX, \quad ZX = iY\]
\[[X,Y] = XY - YX = 2iZ\]

Anti-commutation

\[\{X,Y\} = XY + YX = 0\]
\[\{X,Z\} = XZ + ZX = 0\]

Circuit Symbol

|ψ⟩ ──X── X|ψ⟩

Or with the traditional NOT gate symbol: 

|ψ⟩ ──⊕── X|ψ⟩

Applications

Classical NOT Operation

For computational basis states, X gate performs classical NOT:

  • \(|0\rangle \rightarrow |1\rangle\) (0 → 1)
  • \(|1\rangle \rightarrow |0\rangle\) (1 → 0)

State Preparation

Prepare \(|1\rangle\) state from initialized \(|0\rangle\):

\[X|0\rangle = |1\rangle\]

Conditional Operations

X gate is used as the target in controlled operations like CNOT.

Implementation Examples

Using the Simulator

from quantum_simulator import QuantumSimulator, QuantumCircuit
from quantum_simulator.gates import X_GATE

# Apply X gate to |0⟩
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)
circuit.add_gate(X_GATE, [0])
circuit.execute(sim)

print(sim.get_state_vector())  # [0, 1] = |1⟩

Multiple X Gates

# Apply X gate twice (should return to original state)
circuit = QuantumCircuit(1)
circuit.add_gate(X_GATE, [0])  # |0⟩ → |1⟩
circuit.add_gate(X_GATE, [0])  # |1⟩ → |0⟩
circuit.execute(sim)

print(sim.get_state_vector())  # [1, 0] = |0⟩

X Gate on Superposition

from quantum_simulator.gates import H_GATE

# Create superposition then apply X
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)
circuit.add_gate(H_GATE, [0])  # |0⟩ → |+⟩
circuit.add_gate(X_GATE, [0])  # |+⟩ → |+⟩ (eigenstate)
circuit.execute(sim)

print(sim.get_state_vector())  # ≈ [0.707, 0.707] = |+⟩

Controlled-X (CNOT)

The X gate serves as the target in the controlled-X (CNOT) gate:

\[\text{CNOT} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes X\]

Rotation Relationship

The X gate can be expressed as a rotation:

\[X = e^{-i\pi\sigma_x/2} = \cos(\pi/2)I - i\sin(\pi/2)\sigma_x\]

Where \(\sigma_x = X\) is the Pauli-X operator.

Relationship to Hadamard

\[HXH = Z\]

The X gate conjugated by Hadamard gates becomes a Z gate.

Physical Implementations

In physical quantum systems, the X gate can be implemented by:

  • Microwave pulses in superconducting qubits
  • Laser pulses in trapped ions
  • Magnetic field pulses in NMR systems
  • Optical pulses in photonic systems

The exact implementation depends on the physical platform and how qubits are encoded.