Phase: the part you can't measure directly
Amplitudes are not probabilities: they carry a sign — more generally a phase — that leaves the measurement statistics of that state completely unchanged. Phase becomes visible only when amplitudes recombine 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 a sign.
Consider two states: (|0⟩ + |1⟩)/√2 and (|0⟩ − |1⟩)/√2. Measure either directly and the statistics are identical — 50/50. The minus sign is a relative phase: real, physical, machine-manipulable information that no direct measurement in the computational basis can see.
In general amplitudes are complex numbers and the relative phase can be any angle; the sign flip is the special case of a half-turn. This page sticks to the sign, because it is enough to show the whole mechanism.
Can you see the sign directly?
How do you make phase visible?
Recombine the amplitudes. A second H maps (|0⟩ + |1⟩)/√2 to |0⟩ and (|0⟩ − |1⟩)/√2 to |1⟩ — it converts a phase difference into a probability difference. This is interference used as an instrument: the only way any quantum computer ever reads a phase is by arranging for that phase to change measurement probabilities.
What does H, Z, H give?
And without the Z?
Why does this matter for algorithms?
Keep two distinctions crisp. A global phase — multiplying the entire state by one common factor — changes nothing observable, ever. A relative phase between amplitudes is data: this experiment stored one bit in a sign and read it back with certainty.
That is the template most quantum algorithms follow. Deutsch–Jozsa writes a function's structure into phases and then interferes them into a deterministic answer with one layer of H gates; Grover's search repeatedly flips the phase of the marked item and interferes to concentrate probability on it. No phase manipulation, no quantum advantage — which is why the corruption of phase is the failure mode hardware engineers care about most.
What does this look like on real hardware?
Phase is where hardware is simultaneously strongest and weakest. Strongest: on superconducting processors, Z-axis rotations are implemented as virtual gates — a software update to the pulse frame that takes zero time and introduces essentially zero error. The Z in this lesson's circuit is free on such machines, while every H costs a real microwave pulse.
Weakest: uncontrolled phase is exactly what the environment corrupts. Dephasing — quantified by the T2 time — randomises relative phases and destroys interference long before the qubit's energy decays. Compare T2 across platforms on the QPU pages and the comparison view: holding phase, more than holding a bit value, is the defining engineering challenge of quantum hardware.