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RX Gate (X-Rotation)

The RX gate performs a rotation around the X-axis of the Bloch sphere. It's one of the fundamental single-qubit rotation gates that, together with RY and RZ, forms a complete set for arbitrary single-qubit rotations.

Mathematical Definition

The RX gate with rotation angle θ is defined by the matrix:

\[R_x(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}\]

Action on Basis States

  • \(R_x(\theta)|0\rangle = \cos(\theta/2)|0\rangle - i\sin(\theta/2)|1\rangle\)
  • \(R_x(\theta)|1\rangle = -i\sin(\theta/2)|0\rangle + \cos(\theta/2)|1\rangle\)

Geometric Interpretation

The RX gate rotates the qubit state vector around the X-axis of the Bloch sphere by angle θ:

  • θ = 0: Identity operation (no rotation)
  • θ = π/2: Rotates around X-axis by 90°, equivalent to \(\frac{1}{\sqrt{2}}(X + I)\)
  • θ = π: Equivalent to X gate (bit flip)
  • θ = 2π: Full rotation, returns to original state (up to global phase)

Usage in Quantum Simulator

from quantum_simulator import QuantumSimulator, QuantumCircuit
from quantum_simulator.gates import RX
import numpy as np

# Create RX gate with specific angle
theta = np.pi/3  # 60-degree rotation
rx_gate = RX(theta)

# Apply to qubit
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)
circuit.add_gate(rx_gate, [0])
circuit.execute(sim)

print(f"State after RX(π/3): {sim.get_state_vector()}")

Relationship to Pauli-X Gate

The RX gate generalizes the Pauli-X gate:

from quantum_simulator.gates import RX, X_GATE
import numpy as np

# X gate is RX(π)
x_equivalent = RX(np.pi)

# These should produce the same results (up to global phase)
sim1 = QuantumSimulator(1)
sim2 = QuantumSimulator(1)

circuit1 = QuantumCircuit(1)
circuit1.add_gate(X_GATE, [0])
circuit1.execute(sim1)

circuit2 = QuantumCircuit(1)
circuit2.add_gate(x_equivalent, [0])
circuit2.execute(sim2)

Common Applications

  1. Arbitrary State Preparation: Combined with RY and RZ for universal single-qubit control
  2. Quantum Algorithms: Rotation sequences in optimization algorithms
  3. Error Correction: Correcting bit-flip errors with controlled rotations
  4. Quantum Machine Learning: Parameterized circuits and feature maps
  5. Adiabatic Evolution: Smooth state transitions in quantum annealing

Properties

  • Unitary: \(R_x(\theta)^\dagger R_x(\theta) = I\)
  • Hermitian: \(R_x(\theta)^\dagger = R_x(-\theta)\)
  • Periodic: \(R_x(\theta + 2\pi) = R_x(\theta)\) (up to global phase)
  • Composition: \(R_x(\theta_1)R_x(\theta_2) = R_x(\theta_1 + \theta_2)\)

Universal Single-Qubit Rotations

Any single-qubit unitary can be decomposed using RX, RY, and RZ gates:

\[U = e^{i\alpha} R_z(\gamma) R_y(\beta) R_z(\delta)\]

or alternatively:

\[U = e^{i\alpha} R_z(\gamma) R_x(\beta) R_z(\delta)\]
# Example: Arbitrary single-qubit rotation
from quantum_simulator.gates import RX, RY, RZ
import numpy as np

def arbitrary_rotation(alpha, beta, gamma):
    """Create circuit for arbitrary single-qubit rotation."""
    circuit = QuantumCircuit(1)
    circuit.add_gate(RZ(gamma), [0])
    circuit.add_gate(RY(beta), [0])
    circuit.add_gate(RZ(alpha), [0])
    return circuit

Bloch Sphere Visualization

On the Bloch sphere, RX rotations: - Keep states on the YZ plane fixed on their meridians - Rotate the +Z axis toward +Y axis for positive θ - Are particularly useful for creating and manipulating states in the XY plane

Special Angles

  • RX(π/2): Creates superposition when applied to |0⟩: \((|0\rangle - i|1\rangle)/\sqrt{2}\)
  • RX(π): Bit flip operation (equivalent to X gate)
  • RX(π/4): Common in quantum algorithms, creates \((|0\rangle - i|1\rangle/\sqrt{2})/\sqrt{2}\)

Circuit Symbol

|ψ⟩ ──RX(θ)── |ψ'⟩

The RX gate completes the set of fundamental rotation gates and is essential for achieving universal quantum computation when combined with RY and RZ gates.