controlled 2 qubits
Controlled sqrt(X)
Applies $\sqrt{X}$ to the target when the control qubit is $|1\rangle$.
The controlled-$\sqrt{X}$ gate applies $\sqrt{X}$ to the target qubit conditioned on the control being $|1\rangle$.
$$ \mathrm{C}\sqrt{X} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1 + i}{2} & \frac{1 - i}{2} \\ 0 & 0 & \frac{1 - i}{2} & \frac{1 + i}{2} \end{bmatrix} $$
Properties
- $(\mathrm{C}\sqrt{X})^2 = \mathrm{C}X$, but there are other gates $A$ that satisfy $A^2 = \mathrm{C}X$ such as $(\mathrm{C}\sqrt{X})^\dag$.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis.
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
Weyl chamber
Where this gate sits among all two-qubit gates, up to single-qubit rotations before and after.
Coordinates:
not entangling entangling perfect entangler
$(a, b, c)$ means the gate equals $e^{i(a\,XX + b\,YY + c\,ZZ)}$ up to single-qubit gates, reduced to the chamber $\pi/4 \ge a \ge b \ge |c|$ (the convention of the canonical gate, Qiskit and Cirq). The $a = \pi/4$ face is a mirror: $(\pi/4, b, c)$ and $(\pi/4, b, -c)$ are the same gate. Drag the figure to rotate it, double-click to reset.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| SDK | Name |
|---|---|
| Qiskit | qiskit.circuit.library.CSXGate |
| PennyLane | — ⓘ |
| Cirq | cirq.CNotPowGate ⓘ |
| Q# | — ⓘ |
| PyQuil | — |
| Braket | braket.circuits.gates.CV ⓘ |
| BQSKit | bqskit.ir.gates.SqrtCNOTGate ⓘ |
| Qibo | qibo.gates.CSX |
| pytket | pytket.circuit.OpType.CSX |
| Stim | — |
| OpenQASM | — ⓘ |