Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 5 / b / ii

A two-query network begins in $|0{⟩}_x|0{⟩}_y$, calls $U_f$, flips the input with $X_x$, calls $U_f$ again, and measures the answer wire in Z. The successive states areThe measured answer is zero for constant functions and one for balanced functions. The circuit is drawn in i.

Explicit oracle circuits follow from the form $f(x)=a\oplus bx$:Thus the four circuits, respectively for the truth tables in i, are the identity on both wires; a NOT on the answer wire; a CNOT from input to answer; and a CNOT together with an answer NOT. The latter commute, so their order is immaterial. These circuits leave the input wire unchanged and XOR precisely $f(x)$ into the answer.