Phase: the part you can't measure directly
An amplitude is not a chance. It carries a sign (more generally, a phase) that does not change what you see when you measure that state. The phase only shows up when amplitudes combine and interfere. H then Z then H reads 1 on every shot, while H then H reads 0 on every shot, and the only difference is one sign.
Look at two states: (|0⟩ + |1⟩)/√2 and (|0⟩ − |1⟩)/√2. Measure either one and you get the same odds, 50/50. The only difference is the minus sign. That sign is called a relative phase. It is real information. A machine can set it and change it. But no direct measurement of 0 versus 1 can see it.
Here is an everyday picture. Two swings can move at the same height and speed. One can be going forward while the other goes back. A photo that only shows height can't tell them apart. But push them into each other and the difference matters a lot.
In general, amplitudes are complex numbers and the relative phase can be any angle. A sign flip is the special case of a half turn. This page sticks to the sign, because it is enough to show how the whole thing works.
Can you see the sign directly?
How do you make phase visible?
You make the amplitudes combine again. A second H turns (|0⟩ + |1⟩)/√2 into |0⟩. It turns (|0⟩ − |1⟩)/√2 into |1⟩. In other words, it changes a phase difference into a difference in odds.
Here is the arithmetic for the minus version. H sends |0⟩ to (|0⟩ + |1⟩)/√2 and |1⟩ to (|0⟩ − |1⟩)/√2. For the answer 0 you get ½ − ½ = 0. For the answer 1 you get ½ + ½ = 1. So it reads 1 every time.
This is interference (amplitudes adding up or canceling) used as a measuring tool. It is the only way any quantum computer ever reads a phase. It sets things up so the phase changes the odds of what you measure.
What does H, Z, H give?
And without the Z?
Why does this matter for algorithms?
Keep two ideas apart. A global phase means multiplying the whole state by one shared number. It never changes anything you can see. A relative phase is a difference between amplitudes, and it is data. This experiment stored one bit in a sign and read it back with certainty.
Most quantum algorithms follow this pattern. Deutsch–Jozsa writes facts about a function into phases. Then one layer of H gates makes them interfere into a sure answer. Grover's search flips the phase of the item you want, over and over. Each time, interference piles more chance onto that item.
No phase control means no quantum advantage. That is why hardware engineers worry most about phase getting scrambled.
What does this look like on real hardware?
Phase is where hardware is both strongest and weakest.
Strongest: on superconducting chips, turns around the Z axis are done as virtual gates. The control software just updates its reference for later pulses. It takes no time and adds almost no error. So the Z in this lesson is free on such machines, while every H costs a real microwave pulse.
Weakest: phase is exactly what the surroundings mess up. This is called dephasing, and the T2 time measures how long a qubit resists it. Dephasing scrambles relative phases. It ruins interference long before the qubit loses its energy. Think of a choir slowly drifting out of time with each other, even while every singer keeps singing.
Compare T2 across platforms on the QPU pages and the comparison view. Holding on to phase, even more than holding a bit value, is the main engineering challenge of quantum hardware.