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Phase Gate (Single-Qubit Phase Shift)

The Phase gate (also known as the P gate or phase shift gate) is a fundamental single-qubit gate that applies a phase rotation to the \(|1\rangle\) state while leaving the \(|0\rangle\) state unchanged. It's essential for creating quantum interference effects and controlling relative phases.

Mathematical Definition

The Phase gate with phase angle φ is defined by the matrix:

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

Action on Basis States

  • \(P(\phi)|0\rangle = |0\rangle\) (unchanged)
  • \(P(\phi)|1\rangle = e^{i\phi}|1\rangle\) (acquires phase \(e^{i\phi}\))

Action on Superposition States

For a general single-qubit state \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\):

\[P(\phi)|\psi\rangle = \alpha|0\rangle + \beta e^{i\phi}|1\rangle\]

The measurement probabilities remain unchanged, but the relative phase between \(|0\rangle\) and \(|1\rangle\) is modified by φ.

Geometric Interpretation

On the Bloch sphere, the Phase gate corresponds to a rotation around the Z-axis by angle φ:

  • The north pole (\(|0\rangle\)) remains fixed
  • The south pole (\(|1\rangle\)) acquires phase \(e^{i\phi}\)
  • Points on the equator (superposition states) rotate around the Z-axis

Usage in Quantum Simulator

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

# Create phase gate with specific angle
phi = np.pi/4
p_gate = phase_gate(phi)

# Apply to superposition state
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)
circuit.add_gate(H_GATE, [0])     # Create superposition
circuit.add_gate(p_gate, [0])     # Apply phase shift

sim.execute_circuit(circuit)
print(f"State after Phase(π/4): {sim.get_state_vector()}")

Special Cases and Named Gates

Common Phase Gates

  • φ = 0: Identity gate \(I\)
  • φ = π/4: T gate (π/8 gate)
  • φ = π/2: S gate (phase gate)
  • φ = π: Z gate (Pauli-Z)
  • φ = 2π: Identity (full rotation)

S Gate (φ = π/2)

\[S = P(\pi/2) = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}\]

T Gate (φ = π/4)

\[T = P(\pi/4) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1+i}{\sqrt{2}} \end{pmatrix}\]

Key Properties

Unitarity

Phase gates are unitary: \(P(\phi)^\dagger P(\phi) = I\)

Commutativity

Phase gates commute with each other and with Z gate:

\[P(\phi_1) P(\phi_2) = P(\phi_2) P(\phi_1) = P(\phi_1 + \phi_2)\]

Inverse

\[P(\phi)^{-1} = P(-\phi) = P(\phi)^\dagger\]

Diagonal Property

Phase gates are diagonal in the computational basis, making them easy to implement and analyze.

Physical Significance

Quantum Interference

Phase gates create the relative phases necessary for quantum interference effects in algorithms like:

  • Quantum Fourier Transform (QFT)
  • Grover's Algorithm
  • Phase Estimation
  • Variational Quantum Algorithms

No Effect on Measurement

Phase gates don't change measurement probabilities when measuring in the computational basis, but they're crucial for:

  • Interference patterns in superposition
  • Entanglement creation when combined with other gates
  • Quantum algorithm implementation

Relationship to Other Gates

Connection to Z-Rotation

The Phase gate is equivalent to a Z-rotation:

\[P(\phi) = R_Z(\phi) = e^{-i\phi/2} \begin{pmatrix} e^{i\phi/2} & 0 \\ 0 & e^{-i\phi/2} \end{pmatrix}\]

(up to a global phase factor \(e^{-i\phi/2}\))

Pauli-Z Relationship

\[Z = P(\pi) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]

X-Gate Conjugation

\[X \cdot P(\phi) \cdot X = P(-\phi)^*\]

This swaps which basis state gets the phase.

Circuit Representation

In quantum circuit diagrams, phase gates are represented as:

q: ──P(φ)──

Or for specific cases: 

q: ──S──    (for φ = π/2)
q: ──T──    (for φ = π/4) 
q: ──Z──    (for φ = π)

Applications

1. Quantum Fourier Transform

Phase gates implement the controlled phase rotations: 

# QFT uses phase gates of the form P(2π/2^k)
for k in range(1, n+1):
    phase_k = phase_gate(2*np.pi / (2**k))
    circuit.add_gate(phase_k, [target_qubit])

2. Quantum Phase Estimation

Accumulates phases through controlled-phase operations.

3. Variational Quantum Circuits

Phase gates provide continuous parameterization for optimization: 

def parameterized_layer(theta):
    circuit = QuantumCircuit(n_qubits)
    for i in range(n_qubits):
        circuit.add_gate(phase_gate(theta[i]), [i])
    return circuit

4. Quantum Error Correction

Phase gates help implement stabilizer codes and error syndromes.

Example: Creating Quantum Interference

# Demonstrate phase-dependent interference
sim = QuantumSimulator(1)

# Create superposition
circuit = QuantumCircuit(1)
circuit.add_gate(H_GATE, [0])
circuit.add_gate(phase_gate(np.pi), [0])  # Apply π phase
circuit.add_gate(H_GATE, [0])  # Second Hadamard

sim.execute_circuit(circuit)
# Result: |1⟩ (destructive interference for |0⟩)

The phase gate is fundamental to quantum computing, enabling the phase relationships that make quantum algorithms more powerful than their classical counterparts.