quantum gate directory
Phase
Symbol
$S$
Qubits: $1$
Description:
$\pi/2$ phase gate, equal to $\sqrt{Z}$.
Alternate notations:
- $S$
- $\sqrt{Z}$
- $Z^{1/2}$
- $P(\pi/2)$
- $K$
Controlled variant: Controlled-S
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.SGate
|
| PennyLane |
pennylane.S
|
| Cirq |
cirq.S
|
| Q# |
Std.Intrinsic.S
|
| PyQuil |
pyquil.gates.S
|
| Braket |
braket.circuits.gates.S
|
| BQSKit |
bqskit.ir.gates.SGate
|
| Qibo |
qibo.gates.S
|
| pytket |
pytket.circuit.OpType.S
|
| Stim |
S
ⓘ
|
| OpenQASM |
stdgates.inc: s
|
Groups
This gate is contained in the following groups:
The $S$ gate applies a relative phase of $i$ to the $|1\rangle$ component.
$$S = \begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix} = \sqrt{Z}$$
Properties
- Clifford and diagonal.
- $S^2 = Z$ and $S^4 = I$.
- Conjugates $X$ into $Y$: $S X S^\dagger = Y$.
Usage
- Phase corrections and basis changes in Clifford circuits.
- Building block for $T$ and controlled-phase operations.
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