Rotates the phase of the amplitude by 90°, multiplying it by i.
The Phase gate (also called the S gate) multiplies the amplitude by i (the imaginary unit), while leaving unchanged. This is a 90° phase rotation, compared to the Z gate’s full 180° phase flip. Like Z, it does not change measurement probabilities in the computational basis. The factor of i is invisible if you measure right away, but it shifts how the state interferes with other states later. The Phase gate demonstrates concretely that quantum amplitudes are complex numbers, not just real-valued weights.
The Phase gate is a diagonal matrix with entries (1, i). A diagonal matrix acts on each basis state independently without mixing them. Phase is a special case of the RZ rotation: S = RZ up to a global phase factor. Applying it twice gives Z, because i² = −1, so S² = Z. Its eigenvalues (the factors each eigenstate gets multiplied by) are 1 and i, with eigenstates and . 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 (the set of gates that map Pauli operators to Pauli operators under conjugation).
No change to measurement probabilities in the computational basis.
A 90° phase shift accumulates on the amplitude. This becomes visible only after a mixing gate like H converts phase into probability.
Rotates the Bloch vector 90° around the Z axis. The poles stay fixed. Equatorial states rotate by a quarter-turn.
It is tempting to think the Phase gate does nothing useful because it does not change measurement probabilities. That sounds plausible because the histogram looks identical before and after. What actually happens is that the phase shift changes how the state interferes with other states, and quantum algorithms rely on these accumulated phase shifts to bias the final measurement toward the correct answer.
The Phase gate shows that quantum information is richer than probabilities alone. The factor of i is physically real. It lives in the phase dimension, and you need interference (via a gate like H) to access it. Understanding this is essential for grasping how quantum algorithms extract answers.
How many S gates does it take to equal one Z gate?