A phase gate three-row scan applies S, Z, and H to the same balanced plus-x state and compares the z-basis populations afterward. The S and Z rows both leave the populations at one half and one half, changing only the phase between branches; the H row genuinely redistributes population to probability one and zero exactly for the stated toy gate rules; real gate operations have finite fidelity, leakage to other states, and calibration drift.
Three gates act on the same balanced input; only one of them genuinely redistributes the z-basis populations. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A phase gate can leave the direct populations fixed
The S gate applied to the plus-x state keeps the z-basis populations balanced at one half each, exactly as before the gate. Only the phase between the branches changed.
P↑=21,P↓=21
Three gates on the same input separate phase from population
S and Z only ever touch phase in this model: both leave the z-basis populations balanced. H is not a phase gate here; it genuinely moves population, collapsing the balanced input onto one definite branch.
gateSZHP↑21211P↓21210
H genuinely redistributes population, not just phase
The H gate output is the definite up branch: probability one up, probability zero down. This is the one row in the scan where the population itself changed, not only the phase between branches.