Phase Kickback
Phase kickback transfers a controlled operation's eigenphase onto the control qubit, a mechanism behind many quantum algorithms.
The trick
Phase kickback is a recurring pattern in which the action of a controlled gate shows up as a relative phase on the control qubit rather than on the target. It is the mechanism at the heart of phase estimation, Deutsch-Jozsa, and Shor's algorithm.
How it works
Suppose the target is in an eigenstate |u> of a unitary U, so U|u> = e^{i phi}|u>. Apply a controlled-U with the control in superposition (|0>+|1>)/sqrt(2). When the control is |1>, U acts and multiplies by e^{i phi}; when it is |0>, nothing happens. The target |u> is unchanged, but the control becomes (|0> + e^{i phi}|1>)/sqrt(2). The phase has been kicked back onto the control.
The CNOT example
Take CNOT with the target prepared in |-> = (|0>-|1>)/sqrt(2), an eigenstate of X with eigenvalue -1. If the control is |1>, the target picks up a factor of -1 that appears as a phase on the control. So CNOT on a |-> target flips the control's phase — the simplest instance of kickback, and the reason X-basis targets are used to inject phases.
Why it matters
Kickback converts information stored in a target operator's eigenphase into control-qubit phases that can then be processed by interference and a Fourier transform. This is exactly how quantum phase estimation reads out eigenvalues, and phase estimation in turn powers Shor's factoring and quantum simulation.
- Control in superposition, target in an eigenstate of U
- Eigenphase e^{i phi} appears on the control, not the target
- Feeds phase-estimation and the Fourier transform