Phase Kickback

Every controlled gate has a classical story: "apply the operation to the target if and only if the control is $|1\rangle$." Phase kickback is the quantum twist: when the target is an eigenstate of the gate, the associated eigenvalue does not stay on the target — it travels back and multiplies the control.

Eigenstates and eigenvalues

A unitary gate $U$ has eigenstates $|\lambda\rangle$ satisfying

The gate leaves the state's shape unchanged and only multiplies it by a complex number of magnitude

1. For the $Z$ gate (and therefore also for $CZ$), the eigenstates are $|0\rangle$ and $|1\rangle$ with eigenvalues $+1$ and $-1$ respectively:

The kickback mechanism

Consider a two-qubit system where the target qubit is prepared in an eigenstate $|\lambda\rangle$ of $U$ with eigenvalue $e^{i\phi}$. Let the control start in an arbitrary superposition $\alpha|0\rangle + \beta|1\rangle$. The joint state before the controlled-$U$ gate is

Applying the controlled-$U$ gate acts $U$ on the target only in the $|1\rangle$ branch of the control:

Factor out $|\lambda\rangle$ because the target state is the same in both branches:

The target is unchanged. The phase $e^{i\phi}$ has moved onto the control's $|1\rangle$ amplitude — it has "kicked back." The control qubit now carries information about the gate's eigenvalue even though it was the target that the gate acted on.

A concrete example: CZ with $|+\rangle$ control and $|1\rangle$ target

The $CZ$ gate multiplies the $|11\rangle$ component by $-1$ and leaves $|00\rangle$, $|01\rangle$, $|10\rangle$ unchanged. Prepare the control in $|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and the target in $|1\rangle$:

After $CZ$, the $|11\rangle$ component picks up a factor of $-1$:

Factor the target qubit out:

The target is still $|1\rangle$. The control, however, has flipped from $|+\rangle$ to $|{-}\rangle$ — a phase difference of $\pi$ on the Bloch sphere's equator. The $-1$ eigenvalue of $Z$ acting on $|1\rangle$ has been transferred to the control qubit.

Try it

Put qubit 0 into $|+\rangle$ with a Hadamard, put qubit 1 into $|1\rangle$ with an X gate, then apply $CZ$. The grader checks the full state vector, which should be $\tfrac{1}{\sqrt{2}}(|01\rangle - |11\rangle)$ — equivalently $|{-}\rangle|1\rangle$.

After running, inspect the Amplitudes tab: the $|01\rangle$ amplitude should be $\approx +0.707$ and the $|11\rangle$ amplitude $\approx -0.707$, while $|00\rangle$ and $|10\rangle$ remain zero. That sign change on the control is phase kickback in action.

Sign in on the full site to ask questions and join the discussion.