quantum gate directory
Controlled-S
Symbol
$\mathrm{C}S$
Qubits: $2$
Description:
Applies a phase of $i$ to the $|11\rangle$ state, the square root of controlled-$Z$.
Alternate notations:
- $\Lambda(S)$
- $\sqrt{\mathrm{C}Z}$
- $\mathrm{CP}(\pi/2)$
Controlled version of: Phase
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.CSGate
|
| PennyLane | — ⓘ |
| Cirq |
cirq.CZPowGate
ⓘ
|
| Q# | — ⓘ |
| PyQuil | — ⓘ |
| Braket | — ⓘ |
| BQSKit |
bqskit.ir.gates.CSGate
|
| Qibo | — ⓘ |
| pytket |
pytket.circuit.OpType.CS
|
| Stim | — |
| OpenQASM | — ⓘ |
Groups
This gate is contained in the following groups:
The controlled-$S$ gate is a two-qubit diagonal gate, and the special case $\phi = \pi/2$ of the controlled phase gate.
$$ \mathrm{C}S = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & i \end{bmatrix} $$
Properties
- Diagonal and symmetric: control and target are interchangeable.
- Squares to controlled-$Z$: $\mathrm{C}S^2 = \mathrm{C}Z$, hence the notation $\sqrt{\mathrm{C}Z}$.
- Not Clifford. Like the $T$ gate, it sits in the third level of the Clifford hierarchy, so it is a genuine resource beyond stabilizer circuits.
Decompositions
- In terms of $T$ gates and CNOTs: $$\mathrm{C}S = (T \otimes T) \, \mathrm{CNOT} \, (I \otimes T^\dagger) \, \mathrm{CNOT}$$ Three $T$ gates is the minimum for an exact, ancilla-free circuit.
Usage
- Appears throughout the QFT, which is built from controlled phase gates of decreasing angle: $\mathrm{C}Z$, $\mathrm{C}S$, $\mathrm{C}T$, and so on.
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