# S Gate (Phase Gate)

**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 $|1\rangle$).

## Properties

- **$S^2 = Z$**: 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)

## Relation to Z

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

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

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

## Uses

- **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

## Implementation

- **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