Quantum oracles and phase kickback

Make a function reversible

A Boolean function $f(x)$ may discard information. Its reversible oracle keeps the input and XORs the output into a target:

$$U_f\lvert x,y\rangle=\lvert x,y\oplus f(x)\rangle.$$

This map is its own inverse. The oracle is a specified operation, not a magical free answer source; its implementation can dominate an algorithm’s cost.

Use an eigenstate of X

The state $\lvert-\rangle=(\lvert0\rangle-\lvert1\rangle)/\sqrt2$ satisfies $X\lvert-\rangle=-\lvert-\rangle$.

So a function-controlled X on this target gives

$$U_f\bigl(\lvert x\rangle\lvert-\rangle\bigr)=(-1)^{f(x)}\lvert x\rangle\lvert-\rangle.$$

The target factors out; the input receives a function-dependent phase. This is phase kickback.

A two-input example

Let $f(x_0x_1)=x_1$. The values on 00, 01, 10 and 11 are 0, 1, 0 and 1. A uniform input register receives signs $+,-,+,-$. Applying H to both input qubits converts that phase pattern into a nonzero bitstring.

The Deutsch–Jozsa lab calculates this phase-kickback input register directly. It explicitly factors out the minus-state ancilla; it does not pretend to show an extra simulated wire.

Check yourself: if $f$ is constant one, every component acquires the same minus sign. Why is this global phase unobservable, while a balanced pattern is not?

Reference: IBM Quantum — Fundamentals of quantum algorithms.