# S† gate (inverse S gate)

**S† gate** (also written S-dagger or S inverse) applies a $-90°$ phase to the $|1\rangle$ state. It is the inverse of the [[quantum-gate-s|S gate]], with $S^\dagger S = I$ and $S^\dagger = S^3$ (since $S^4 = I$).

Matrix:

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

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

## Properties

- **Inverse of S**: $S^\dagger S = S S^\dagger = I$
- **Self-adjoint modulo phase**: $(S^\dagger)^\dagger = S$
- **$(S^\dagger)^2 = Z$**: applying S† twice gives the Z gate
- **(S†)^4 = I**: applying S† four times returns to identity
- **Part of Clifford group**: S† is a Clifford gate (can be simulated classically with stabilizer formalism)

## Relation to S and Z gates

$$S^\dagger = S^{-1} = S^3 = R_Z(-\pi/2) = e^{i\pi Z/4}$$

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

$$S S^\dagger = I$$

## Uses

- **Phase correction**: undo S gate phase; adjust phases in quantum circuits
- **Clifford group**: with Hadamard and CNOT, forms Clifford group (efficient classical simulation)
- **Circuit synthesis**: reverse phase operations added by S gates
- **Algorithm building**: phase gates appear in quantum algorithms and ansatze

## Implementation

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

## Relations

- [[quantum-gate-phase|Phase gates]]: category containing S† and other phase gates
- [[quantum-gate-s|S gate]]: inverse relationship; $S^\dagger = S^{-1}$
- [[quantum-gate-z|Z gate]]: $(S^\dagger)^2 = Z$; related via phase accumulation
- [[quantum-gate-rz|RZ gate]]: $S^\dagger = R_Z(-\pi/2)$
- [[quantum-gate-clifford|Clifford gates]]: S† is a Clifford gate
- [[quantum-gate-single-qubit|Single-qubit gates]]: fundamental building block