Deutsch–Jozsa query circuit

The Deutsch–Jozsa problem promises that $f:\{0,1\}^n\to\{0,1\}$ is either constant or balanced (outputs 0 and 1 equally often), and asks which. The circuit prepares the output qubit in $|-\rangle=\frac{1}{\sqrt2}(|0\rangle-|1\rangle)$ and the input register in $H^{\otimes n}|0\rangle^{\otimes n}$, queries the reversible oracle $U_f:|x\rangle|y\rangle\mapsto|x\rangle|y\oplus f(x)\rangle$ once, and applies $H^{\otimes n}$ again to the input register before measuring. Phase kickback turns each query into a sign $(-1)^{f(x)}$ on $|x\rangle$ rather than touching the output qubit, and interference across the $2^n$ branches makes the input register collapse to $|0\rangle^{\otimes n}$ exactly when $f$ is constant, and to a nonzero string when $f$ is balanced — one query suffices, versus up to $2^{n-1}+1$ classical queries in the worst case for a deterministic algorithm. This record reviews the phase-kickback circuit for a two-bit input against that construction; it stores one small educational oracle instance, not a general-$n$ benchmark.