Balanced Hadamard interaction-free measurement ($P_{\mathrm{success}}=1/4,P_{\mathrm{absorb}}=1/2,P_{\mathrm{inconclusive}}=1/4$)

With two Hadamard gates as mixers, an unobstructed zero-phase arm gives only port 0 for upper-port incidence. An ideal absorber in arm 1 acts on the intermediate superposition $(|0⟩+|1⟩)/\sqrt{2}$. It absorbs with probability $1/2$; its no-absorption branch is the unnormalized state $|0⟩/\sqrt{2}$. Recombination gives amplitude $1/2$ at each surviving output, so the dark-port success probability is $1/4$. The result assumes a calibrated transparent alternative and an ideal absorber; a dark-port event alone does not establish those apparatus assumptions.