Phase Kickback (Explained)

April 6, 2025

Phase kickback is one of the most useful “little tricks” in quantum computing. It explains how a controlled operation—apparently acting on the target—can end up putting a phase on the control.

This idea shows up everywhere: oracle phase marking in Grover, and especially in quantum phase estimation (which relies on the QFT).

The key setup

You need three ingredients:

A control qubit in superposition: $\alpha\lvert 0\rangle + \beta\lvert 1\rangle$
A target register in a special state $\lvert \psi\rangle$
A unitary $U$ such that $\lvert \psi\rangle$ is an eigenstate of $U$ :

U\lvert\psi\rangle = e^{i\theta}\lvert\psi\rangle.

Now apply the controlled- $U$ gate (control on the first qubit, target is $\lvert\psi\rangle$ ).

The “kickback” algebra (why it works)

Start with the joint state:

(\alpha\lvert 0\rangle + \beta\lvert 1\rangle)\otimes\lvert\psi\rangle.

Controlled- $U$ acts as:

if control is $\lvert 0\rangle$ : do nothing
if control is $\lvert 1\rangle$ : apply $U$ to the target

So the output is:

$$ \alpha\lvert 0\rangle\lvert\psi\rangle + \beta\lvert 1\rangle,U\lvert\psi\rangle

\alpha\lvert 0\rangle\lvert\psi\rangle + \beta\lvert 1\rangle,e^{i\theta}\lvert\psi\rangle. $$

And here’s the punchline: because $\lvert\psi\rangle$ is unchanged (only picks up a phase), the state factors:

\lvert\psi\rangle\left(\alpha\lvert 0\rangle + \beta e^{i\theta}\lvert 1\rangle\right).

The target returns to the same physical state, while the control has acquired a relative phase $e^{i\theta}$ on its $\lvert 1\rangle$ component. That phase is now “stored” on the control qubit, where it can interfere with other amplitudes.

If the target is not an eigenstate of

U

, then

U\lvert\psi\rangle

changes the target in a state-dependent way, and control+target typically become entangled. Then you don’t get a clean, single phase on the control.

From bit-oracle to phase-oracle (the practical version)

Many algorithms start from a reversible “bit oracle” for a Boolean function $f$ :

O_f:\lvert x\rangle\lvert y\rangle \mapsto \lvert x\rangle\lvert y\oplus f(x)\rangle.

If you prepare the ancilla as $\lvert -\rangle=\frac{1}{\sqrt{2}}(\lvert 0\rangle-\lvert 1\rangle)$ , then flipping it applies a phase:

O_f\lvert x\rangle\lvert -\rangle = (-1)^{f(x)}\lvert x\rangle\lvert -\rangle.

So without measuring anything, the function value $f(x)$ ends up encoded as a phase on $\lvert x\rangle$ .

This is exactly the “marking” behavior used in Grover-style phase oracles.

Tiny example in Qiskit (controlled-Rz)

The $R_z(\phi)$ rotation has eigenstates $\lvert 0\rangle$ and $\lvert 1\rangle$ :

$R_z(\phi)\lvert 0\rangle=\lvert 0\rangle$
$R_z(\phi)\lvert 1\rangle=e^{i\phi}\lvert 1\rangle$

If you use $\lvert 1\rangle$ as the target eigenstate, then a controlled- $R_z(\phi)$ kicks back a phase onto the control:

import numpy as np
from qiskit import QuantumCircuit

phi = np.pi / 3
qc = QuantumCircuit(2)

# control in |+>
qc.h(0)

# target eigenstate |1>
qc.x(1)

# controlled unitary
qc.crz(phi, 0, 1)

qc.draw("text")

See how phase marking appears in Grover: What is an oracle?
See where phase information gets read out: Quantum Fourier Transform (QFT)

What Is a Hadamard Gate?Eigenvectors and Eigenvalues (Linear Algebra for Quantum Computing)