S† gate (also written S-dagger or S inverse) applies a $-90°$ phase to the $|1\rangle$ state. It is the inverse of the 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 $|1\rangle$).

• 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)

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

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

$$S S^\dagger = I$$

• 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

• 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

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