Adds a quarter-turn of phase to the |1⟩ branch.
The Phase gate (also called the S gate) is Z’s gentler cousin. Where Z adds a full 180° phase flip, Phase adds only a 90° turn — it multiplies |1⟩ by i instead of −1. This is a great gate for seeing that quantum amplitudes are truly complex numbers, not just real-valued weights. The factor of i is invisible in computational-basis measurements, but it changes how the state interferes with other states later.
The Phase gate is diagonal with entries (1, i). It is a special case of the RZ rotation: S = RZ(π/2) up to global phase, and S² = Z. The eigenvalues are 1 and i, with eigenstates |0⟩ and |1⟩. On the Bloch sphere, it is a 90° rotation around the Z axis. Together with H, the S gate generates the Clifford group for a single qubit.
Measurement odds stay the same in the computational basis.
Quarter-turn phase accumulates — visible only after a mixing gate like H.
Rotates 90° around the Z axis. Quarter-turn of equatorial phase. The poles stay fixed while equatorial states rotate by 90°.
Because the Phase gate does not change probabilities, it can look useless. But quantum algorithms use accumulated phase to create interference patterns that bias the final measurement.
The Phase gate demonstrates that quantum information is richer than probabilities. The factor of i is as physical as any bit flip — it just lives in a dimension (phase) that you need interference to access.
How many S gates does it take to equal one Z gate?