S gate applies a 90° phase to the $|1\rangle$ state, also called the phase gate or $\sqrt{Z}$ (since $S^2 = Z$).

Matrix:

$$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$$

Action: $S|0\rangle = |0\rangle$, $S|1\rangle = i|1\rangle$ (adds $\pi/2$ phase to $|1\rangle$).

• $S^2 = Z$: applying S twice gives the Z gate
• $S^4 = I$: applying S four times returns to identity
• Inverse: $S^\dagger = -iZ S = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}$
• Part of Clifford group: S is a Clifford gate (can be simulated classically with stabilizer formalism)

$$S = R_Z(\pi/2) = e^{-i\pi Z/4}$$

$$Z = S^2 = R_Z(\pi)$$

$$T = S/\sqrt{-i} \approx S^{1/2}$$

• Phase correction: adjust phase in quantum circuits
• Clifford group: with Hadamard and CNOT, forms Clifford group (efficient classical simulation)
• T gate building: $T^2 = S$; intermediate between identity and T
• Algorithm building: phase gates appear in variational algorithms

• Superconducting qubits: RZ($\pi/2$) gate (virtual on most systems, no pulse needed)
• Trapped ions: controlled phase via detuned laser
• Photonic: optical phase shifter set to $\pi/2$
• Cost: free if RZ is virtual; otherwise ~20 ns like single-qubit rotations