How does the kickback phase work... And what breaks it

The Hidden Life of Pauli Eigenstates

In quantum computing, information is often hidden in a phase. But what happens if you want to extract this phase without collapsing or disturbing the target qubit?

Simple.

You use phase kickback.

Phase kickback is one of those phenomena that feels like magic when it works. You prepare a control qubit in the ∣+⟩ state, apply a controlled-unitary gate, and voilà: the control qubit picks up a phase. And even though the unitary has acted on the target qubit, that remains untouched.

Sounds like a dream.

But here's the secret that no one will tell you right away.

Phase kickback only works if the target qubit is in an eigenstate of the controlled unitary.