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Y Gate (Pauli-Y)

The Y gate, also known as the Pauli-Y gate, is a fundamental single-qubit quantum gate that performs a combined bit-flip and phase-flip operation.

Matrix Representation

\[Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\]

Action on Basis States

The Y gate transforms computational basis states with a phase factor:

\[Y|0\rangle = i|1\rangle\]
\[Y|1\rangle = -i|0\rangle\]

General Action

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

\[Y|\psi\rangle = Y(\alpha|0\rangle + \beta|1\rangle) = \alpha \cdot i|1\rangle + \beta \cdot (-i)|0\rangle\]
\[= -i\beta|0\rangle + i\alpha|1\rangle\]

The Y gate both swaps amplitudes (like X) and introduces phase factors.

Bloch Sphere Representation

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

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

Properties

Involutory

\[Y^2 = -I\]

Applying Y twice gives negative identity (global phase of -1).

Hermitian

\[Y^\dagger = Y\]

The Y gate is Hermitian.

Eigenvalues and Eigenvectors

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

Eigenvectors:

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

Pauli Group Relations

The Y gate is part of the Pauli group with commutation relations:

\[YZ = iX, \quad ZX = iY, \quad XY = iZ\]

Anti-commutation

\[\{Y,X\} = YX + XY = 0\]
\[\{Y,Z\} = YZ + ZY = 0\]

Circuit Symbol

|ψ⟩ ──Y── Y|ψ⟩

Decomposition

The Y gate can be decomposed using other gates:

\[Y = iXZ = -iZX\]

Or using rotations: $\(Y = e^{-i\pi\sigma_y/2}\)$

Applications

Combined Operations

Y gate performs both:

  1. Bit flip: \(|0\rangle \leftrightarrow |1\rangle\)
  2. Phase modification: Introduces \(\pm i\) factors

Quantum Algorithms

  • Used in quantum error correction
  • Component of arbitrary single-qubit rotations
  • Basis transformations between X and Z eigenstates

State Preparation

Prepare specific superposition states:

\[Y|0\rangle = i|1\rangle\]
\[Y|+\rangle = |-i\rangle\]

Measurement Basis

The Y gate eigenstates form the Y-measurement basis:

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

Measuring in this basis projects onto Y eigenstates.

Implementation Examples

Using the Simulator

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

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

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

Y Gate on |1⟩

from quantum_simulator.gates import X_GATE

# Apply Y to |1⟩ state
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)
circuit.add_gate(X_GATE, [0])  # Prepare |1⟩
circuit.add_gate(Y_GATE, [0])  # Apply Y
circuit.execute(sim)

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

Y on Superposition

from quantum_simulator.gates import H_GATE

# Apply Y to |+⟩ state
sim = QuantumSimulator(1)
circuit = QuantumCircuit(1)
circuit.add_gate(H_GATE, [0])  # Create |+⟩
circuit.add_gate(Y_GATE, [0])  # Apply Y
circuit.execute(sim)

# Result: Y|+⟩ = |-i⟩
print(sim.get_state_vector())  # ≈ [0.707, -0.707i]

Relationship to Other Gates

Conjugation Relations

\[HYH = -Y\]
\[XYX = -Y\]
\[ZYZ = -Y\]

Rotation Equivalence

The Y gate is equivalent to a π rotation around the Y-axis:

\[Y = R_y(\pi) = e^{-i\pi\sigma_y/2}\]

Sequential Applications

\[XY = iZ, \quad YZ = iX, \quad ZY = -iX\]

Physical Implementations

In quantum hardware:

  • Phase and amplitude control in superconducting qubits
  • Combined pulse sequences in trapped ions
  • Polarization rotations in photonic systems
  • Composite pulses in NMR systems

The Y gate often requires more complex control than X or Z gates due to its phase requirements.

Error Analysis

The Y gate can introduce errors through:

  • Amplitude errors: Incorrect rotation angle
  • Phase errors: Wrong phase accumulation
  • Decoherence: Loss of quantum coherence during gate operation

These errors can be mitigated through:

  • Calibrated pulse sequences
  • Dynamical decoupling
  • Quantum error correction