Phase Kickback: Encoding in Phase

Phase kickback is one of the most elegant tricks in quantum computing. It lets you transfer information from control qubits to phases—enabling many quantum algorithms.

The Core Idea

When a controlled operation acts on an eigenstate of the target operation, the control qubit picks up the eigenvalue as a phase.

In simpler terms: a controlled operation can move phase information from the target to the control.

The Simplest Example

The \(|-\rangle\) state is an eigenstate of X:

\[X|-\rangle = -|-\rangle\]

This means:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# Prepare |-⟩ on qubit 1
qc = QuantumCircuit(2)
qc.x(1)      # |1⟩
qc.h(1)      # H|1⟩ = |−⟩

# Apply CNOT with |−⟩ as target
qc.cx(0, 1)

print(Statevector(qc))

The \(|1\rangle\) component on qubit 0 picks up a minus sign!

What Happened?

Before: \[|0\rangle \otimes |-\rangle + |1\rangle \otimes |-\rangle = (|0\rangle + |1\rangle) \otimes |-\rangle\]

After CNOT: \[|0\rangle \otimes |-\rangle + |1\rangle \otimes X|-\rangle = |0\rangle \otimes |-\rangle - |1\rangle \otimes |-\rangle = (|0\rangle - |1\rangle) \otimes |-\rangle\]

The control qubit now has the phase!

The Key Identity

Remember from the basis change chapter:

\[HZH = X\]

This means:

• \(|+\rangle\) is the \(+1\) eigenstate of \(Z\) (eigenvalue \(+1\))
• \(|-\rangle\) is the \(-1\) eigenstate of \(Z\) (eigenvalue \(-1\))

And:

• \(|0\rangle\) is the \(+1\) eigenstate of \(Z\) (eigenvalue \(+1\))
• \(|1\rangle\) is the \(-1\) eigenstate of \(Z\) (eigenvalue \(-1\))

Phase Kickback with Z

Using \(Z|1\rangle = -|1\rangle\):

# Kickback with Z
qc = QuantumCircuit(2)
qc.h(0)      # Put control in superposition
qc.x(1)      # Prepare |1⟩ (eigenstate of Z with eigenvalue -1)
qc.cz(0, 1) # Controlled-Z

print(Statevector(qc))

The \(|1\rangle\) branch on qubit 0 picks up a minus sign.

Why Is This Useful?

Phase kickback lets you:

1. Encode classical information in phases
2. Transfer phase information between qubits
3. Create controlled operations from simpler ones
4. Build oracles for Grover’s algorithm

A Concrete Application: Phase Oracle

A phase oracle marks a target state \(|w\rangle\) by flipping its phase:

\[O_w|x\rangle = \begin{cases} -|x\rangle & \text{if } x = w \ |x\rangle & \text{otherwise} \end{cases}\]

For \(|w\rangle = |11\rangle\):

# Phase oracle for |11⟩
qc = QuantumCircuit(2)
qc.h(0)
qc.h(1)  # Put both in superposition
qc.cz(0, 1)  # Mark |11⟩
qc.h(0)
qc.h(1)

print(Statevector(qc))

Try this and verify that \(|11\rangle\) gets the minus sign.

The Phase Kickback Pattern

The standard phase kickback pattern:

def phase_kickback(n_qubits, eigenstate_prep, controlled_op):
    """Generic phase kickback circuit."""
    qc = QuantumCircuit(n_qubits)
    
    # 1. Put control qubits in superposition
    for i in range(n_qubits - 1):
        qc.h(i)
    
    # 2. Prepare eigenstate on ancilla
    eigenstate_prep(qc)
    
    # 3. Apply controlled operation
    controlled_op(qc)
    
    return qc

Basis Changes Reveal Phase

After phase kickback, you can’t see the phase in computational basis measurement. Change to X basis:

# Create phase, then measure in X basis
qc = QuantumCircuit(2)
qc.h(0)
qc.x(1)
qc.cz(0, 1)

# Measure in X basis
qc.h(0)  # Change to X basis
qc.measure_all()

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

The phase becomes a bit flip in the X basis.

Checkpoint Exercises

Exercise 1

Show that \(HZH = X\) using phase kickback logic.

Exercise 2

Create a circuit where qubit 0 picks up phase \(\pi/2\) (not \(\pi\)).

Exercise 3

Build a phase oracle that marks \(|00\rangle\).

Exercise 4

Explain why \(|+\rangle\) and \(|-\rangle\) measure the same in Z basis but differently in X basis.

Exercise 5

Design a circuit that transfers phase from qubit A to qubit B.

Summary

Key concepts:

• Phase kickback: controlled operations on eigenstates create phase on controls
• \(|-\rangle\) is the \(-1\) eigenstate of \(X\)
• \(|1\rangle\) is the \(-1\) eigenstate of \(Z\)
• Phase kickback enables oracles and many algorithms

Phase kickback is fundamental to understanding Grover’s algorithm and quantum phase estimation.

Try These On QCoder

• QPC003 B2: Convert Bit-Flip into Phase-Flip I
• QPC003 Ex1: Convert Bit-Flip into Phase-Flip II
• QPC005 B1: Action in the Fourier Basis

Next: Reversible Logic and Oracles