Phase Kickback

Phase kickback is the trick where a controlled operation pushes an eigenphase back onto the control qubit — the mechanism behind most quantum speedups.

Where

Deutsch–Jozsa, BV, Grover oracle, QPE

Requires

target in an eigenstate of the operation

What it does

If the target qubit is an eigenstate of the applied gate, a 'controlled' version doesn't change the target — instead the eigenphase appears as a relative phase on the control. Putting the target in |−⟩ turns an oracle's bit-answer into a phase the algorithm can interfere on.

Where it's used

The conceptual key to oracle-based algorithms and phase estimation.

In code (Qiskit)

# ancilla in |->; controlled-U with U|->=-|-> flips control phase