Pauli-Z Gate

The Z gate multiplies the $|1⟩$ amplitude by −1 and leaves the $|0⟩$ amplitude alone. If the state is $\alpha$$|0⟩+\beta$$|1⟩$, it becomes $\alpha$$|0⟩$ − $\beta$$|1⟩$. This sign change is called a phase flip. It does not change measurement probabilities in the computational basis at all, because probabilities depend on the magnitude of the amplitudes, not their signs. The effect only shows up when a later gate (like H) converts the sign difference into a measurable probability change through interference.

Hidden phase revealed by interference

1. 1Start with $|0⟩$. Apply H to get $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$.
2. 2Apply Z. The $|1⟩$ amplitude flips sign. The state becomes ($|0⟩$ − $|1⟩/\sqrt{2}=$ |−⟩.
3. 3Measure now. You still see 50/50 outcomes. The phase change is invisible in this basis.
4. 4Apply H to |−⟩. Hadamard maps |−⟩ to $|1⟩$. Now measurement gives 1 with certainty.
5. 5The hidden phase flip became a visible bit flip through interference. This is the identity HZH = X in action.

Z teaches the central lesson of quantum computing: phase is information. You cannot see it directly in a single measurement, but every quantum algorithm depends on manipulating phase and then using interference to convert phase differences into measurable probability differences.