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RZ Gate (Z-Rotation)

The RZ gate performs a rotation around the Z-axis of the Bloch sphere. It applies a phase shift to the |1⟩ component while leaving the |0⟩ component unchanged, making it essential for phase manipulation in quantum algorithms.

Mathematical Definition

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

\[R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}\]

Alternatively, up to a global phase, it can be written as:

\[R_z(\theta) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\theta} \end{pmatrix}\]

Action on Basis States

  • \(R_z(\theta)|0\rangle = e^{-i\theta/2}|0\rangle\)
  • \(R_z(\theta)|1\rangle = e^{i\theta/2}|1\rangle\)

For a general state \(\alpha|0\rangle + \beta|1\rangle\):

\[R_z(\theta)(\alpha|0\rangle + \beta|1\rangle) = \alpha e^{-i\theta/2}|0\rangle + \beta e^{i\theta/2}|1\rangle\]

Geometric Interpretation

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

  • θ = 0: Identity operation
  • θ = π/2: S gate (phase gate)
  • θ = π: Z gate (phase flip)
  • θ = π/4: T gate (π/8 gate)

Usage in Quantum Simulator

from quantum_simulator import QuantumSimulator, QuantumCircuit
from quantum_simulator.gates import RZ, H_GATE
import numpy as np

# Create RZ gate with specific angle
theta = np.pi/4  # π/4 rotation (T gate)
rz_gate = RZ(theta)

# Apply to superposition state to see phase effect
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)

# Create superposition first
circuit.add_gate(H_GATE, [0])  # |+⟩ = (|0⟩ + |1⟩)/√2

# Apply phase rotation
circuit.add_gate(rz_gate, [0])  # Adds phase to |1⟩ component

circuit.execute(sim)
print(f"State after H then RZ(π/4): {sim.get_state_vector()}")

Special Cases

Common Phase Gates

from quantum_simulator.gates import RZ
import numpy as np

# S gate (Phase gate): RZ(π/2)
s_gate = RZ(np.pi/2)

# T gate (π/8 gate): RZ(π/4)  
t_gate = RZ(np.pi/4)

# Z gate (Phase flip): RZ(π)
z_gate = RZ(np.pi)

# Custom phase rotation
custom_phase = RZ(np.pi/6)  # 30-degree phase rotation

Common Applications

  1. Phase Manipulation: Adding relative phases between |0⟩ and |1⟩
  2. Quantum Fourier Transform: Essential for QFT implementations
  3. Quantum Phase Estimation: Phase kickback and controlled rotations
  4. Variational Algorithms: Parameter optimization in quantum circuits
  5. Error Correction: Phase error correction protocols
  6. Quantum Chemistry: Molecular simulation and phase evolution

Properties

  • Diagonal: Only affects phases, not populations
  • Unitary: \(R_z(\theta)^\dagger R_z(\theta) = I\)
  • Commutative: \(R_z(\theta_1) R_z(\theta_2) = R_z(\theta_1 + \theta_2)\)
  • Inverse: \(R_z(\theta)^{-1} = R_z(-\theta)\)
  • Global Phase: Often written with global phase \(e^{i\theta/2}\) factored out

Controlled RZ Gates

The RZ gate is commonly used in controlled operations:

# Controlled-RZ gates are used in QFT and other algorithms
# CRZ applies RZ to target only when control is |1⟩

# In quantum algorithms, controlled phases are crucial:
# |00⟩ → |00⟩
# |01⟩ → |01⟩  
# |10⟩ → |10⟩
# |11⟩ → e^{iθ}|11⟩

Relationship to Other Gates

  • Z Gate: \(Z = R_z(\pi)\)
  • S Gate: \(S = R_z(\pi/2)\)
  • T Gate: \(T = R_z(\pi/4)\)
  • Phase Gate: \(P(\phi) = R_z(\phi)\) (up to global phase)

Circuit Symbol

|ψ⟩ ──RZ(θ)── |ψ'⟩

Or for specific cases: 

|ψ⟩ ──S── |ψ'⟩    (θ = π/2)
|ψ⟩ ──T── |ψ'⟩    (θ = π/4)  
|ψ⟩ ──Z── |ψ'⟩    (θ = π)

The RZ gate is fundamental for phase control and appears in virtually all quantum algorithms requiring precise phase manipulation, making it one of the most important single-qubit gates in quantum computing.