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Phase: The Z Gate and Relative Phase

The Z gate looked harmless when we first saw it. Now that you understand superposition, we can see its true power: the Z gate introduces relative phase, which is invisible in immediate measurements but crucial for interference.

Global Phase vs Relative Phase

This distinction is fundamental.

Global Phase

Multiplying the entire state by a complex number:

\[|\psi\rangle \rightarrow e^{i\phi}|\psi\rangle\]

This changes nothing physically. All measurement probabilities remain the same.

Relative Phase

Multiplying one amplitude by a complex number:

\[|0\rangle + |1\rangle \rightarrow |0\rangle - |1\rangle\]

This changes everything physically. The state is different, even if measurement statistics are the same.

The Z Gate and Relative Phase

The Z gate applies a phase flip to the \(|1\rangle\) component:

\[Z(\alpha|0\rangle + \beta|1\rangle) = \alpha|0\rangle - \beta|1\rangle\]

On basis states:

  • \(Z|0\rangle = |0\rangle\) (unchanged)
  • \(Z|1\rangle = -|1\rangle\) (phase flip)

Why Z Seems Useless on \(|0\rangle\)

When your qubit is definitely in \(|0\rangle\), Z does nothing visible:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(1)
qc.z(0)
print(Statevector(qc))

Output:

Statevector([1.+0.j, 0.+0.j],
            dims=(2,))

The state is still \(|0\rangle\). No change.

But watch what happens after a Hadamard:

# Without Z
qc1 = QuantumCircuit(1)
qc1.h(0)
print("H only:", Statevector(qc1))

# With Z
qc2 = QuantumCircuit(1)
qc2.h(0)
qc2.z(0)
print("H then Z:", Statevector(qc2))

Output:

H only: Statevector([0.70710678+0.j, 0.70710678+0.j], dims=(2,))
H then Z: Statevector([0.70710678+0.j, -0.70710678+0.j], dims=(2,))

The states are different! The Z gate introduces a relative phase of \(\pi\) (180°) between the two branches.

The Minus State

The state \(|-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}\) is created by H then Z:

qc = QuantumCircuit(1)
qc.h(0)
qc.z(0)
print(Statevector(qc))

Or equivalently by X then H:

qc = QuantumCircuit(1)
qc.x(0)
qc.h(0)
print(Statevector(qc))

Both give \(|-\rangle\). Verify this!

Visualizing Phase on the Bloch Sphere

On the Bloch sphere, phase corresponds to position around the equator:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_bloch_multivector
from math import pi

# |+⟩ - phase 0
qc = QuantumCircuit(1)
qc.h(0)
display(plot_bloch_multivector(Statevector(qc)))

# |−⟩ - phase π (180°)
qc = QuantumCircuit(1)
qc.x(0)
qc.h(0)
display(plot_bloch_multivector(Statevector(qc)))

# P(π/2) - phase π/2
qc = QuantumCircuit(1)
qc.p(pi/2, 0)
display(plot_bloch_multivector(Statevector(qc)))

# P(π/4) - phase π/4
qc = QuantumCircuit(1)
qc.p(pi/4, 0)
display(plot_bloch_multivector(Statevector(qc)))

           |+⟩ (phase 0)
            ●
           / \
          /   \
    |−⟩──●     ●──|+i⟩
         \   /
          \ /
           ●
          |0⟩
  • \(|+\rangle\) has phase 0
  • \(|-\rangle\) has phase π (180°)
  • \(|+i\rangle\) and \(|-i\rangle\) are eigenstates of Y (phase π/2 and 3π/2)

Phase in Multi-Qubit States

Relative phase becomes more interesting with multiple qubits:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# Create superposition on two qubits
qc = QuantumCircuit(2)
qc.h(0)
qc.h(1)

print("Equal superposition:", Statevector(qc))

# Now apply Z to qubit 0 only
qc2 = QuantumCircuit(2)
qc2.h(0)
qc2.h(1)
qc2.z(0)

print("With Z on q0:", Statevector(qc2))

The \(|1\rangle\) component on qubit 0 gets the phase flip, creating entanglement between the phase and which qubit is in state 1.

CZ Gate: Phase on Two Qubits

The controlled-Z (CZ) gate adds a phase flip when both qubits are \(|1\rangle\):

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# Create superposition on both qubits
qc = QuantumCircuit(2)
qc.h(0)
qc.h(1)

print("Before CZ:", Statevector(qc))

# Apply CZ
qc.cz(0, 1)

print("After CZ:", Statevector(qc))

Output shows a phase flip on the \(|11\rangle\) component.

Why Phase Matters

Phase is invisible in immediate measurements but matters because:

  1. Interference: Phases combine when states recombine
  2. Entanglement: Phase correlations create entangled states
  3. Quantum algorithms: Phase manipulation is the core of most algorithms

Example: Grover’s algorithm uses phase differences to amplify the probability of marked states.

Measuring Phase

You can’t directly “see” relative phase in a single measurement. But you can reveal it with basis changes:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.primitives import StatevectorSampler

# Create |−⟩
qc = QuantumCircuit(1)
qc.h(0)
qc.z(0)

# Measure in X basis (apply H then measure)
qc.h(0)  # Basis change: Z basis → X basis
qc.measure_all()

sampler = StatevectorSampler()
result = sampler.run([qc], shots=100).result()
print(result[0].data.meas.get_counts())

Output: Always 1 (never 0).

Why? Because \(|-\rangle\) is the \(|1\rangle\) state in the X basis. The phase is revealed by changing the measurement basis.

The Phase Gate (P)

Qiskit has a general phase gate p(theta) that adds an arbitrary phase:

\[P(\theta)|1\rangle = e^{i\theta}|1\rangle\]

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from math import pi

# P(π) should equal Z
qc = QuantumCircuit(1)
qc.p(pi, 0)
print("P(π):", Statevector(qc))

# P(π/2) adds 90° phase
qc2 = QuantumCircuit(1)
qc2.p(pi/2, 0)
print("P(π/2):", Statevector(qc2))

Checkpoint Exercises

Exercise 1

Show that \(Z = PHP(\pi)\) where H is the Hadamard gate in the X basis. (Hint: use Operator to compare)

Exercise 2

Create \(|-\rangle\) three different ways and verify they give identical states.

Exercise 3

Apply Z to a qubit in superposition. Show that the probabilities don’t change but the state vector does.

Exercise 4

Use the CZ gate to create phase entanglement between two qubits.

Exercise 5

Design a circuit that distinguishes \(|+\rangle\) from \(|-\rangle\) without destroying the state.

Summary

Key concepts:

  • Global phase (\(e^{i\phi}|\psi\rangle\)) is physically irrelevant
  • Relative phase between amplitudes is physically meaningful
  • Z gate introduces relative phase of π on \(|1\rangle\)
  • Phase is invisible in immediate measurement but visible in interference
  • Basis changes reveal hidden phase information

Understanding phase is essential for quantum computing. Many algorithms (Grover, QFT, phase estimation) are fundamentally about manipulating phases.

Official Documentation: For more on phase and phase gates, see IBM’s phase kickback tutorial which explains how phase is used in quantum algorithms.

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