A phase gate can make a later basis measurement come out differently. 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 creates a y-basis state
After the phase gate, the state has equal direct probabilities but a different phase relation.
S∣+x⟩=∣+y⟩
The gate keeps direct balance and changes the y readout
The scan separates direct balance from the later y-basis result.
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The y-basis reveals that phase relation
Measured in the y basis, the probabilities are 1 and 0.