interactions 2 qubits 1 parameter
XX−YY Interaction
Rotation generated by the $XX - YY$ interaction, acting only on the even-parity subspace.
The $XX - YY$ gate is the counterpart of the XY gate: where XY rotates the single-excitation subspace $\{|01\rangle, |10\rangle\}$, this gate rotates the even-parity subspace $\{|00\rangle, |11\rangle\}$ and leaves the rest alone.
$$ R_{xx-yy}(\theta) = \exp\left(-i \frac{\theta}{4} (X \otimes X - Y \otimes Y)\right) $$
In the computational basis:
$$ \begin{bmatrix} \cos\tfrac{\theta}{2} & 0 & 0 & -i\sin\tfrac{\theta}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -i\sin\tfrac{\theta}{2} & 0 & 0 & \cos\tfrac{\theta}{2} \end{bmatrix} $$
Properties
- Coherently converts $|00\rangle \leftrightarrow |11\rangle$: a two-photon (pair-creation) process, in contrast to the excitation-conserving hopping of XY.
- Commutes with the XY gate, since the two act on disjoint subspaces.
- Together the pair decomposes the standard two-qubit rotations, for example $$R_{xx}(\theta) = \mathrm{XY}(-\theta) \, R_{xx-yy}(\theta)$$ because $X \otimes X = \tfrac{1}{2}(XX + YY) + \tfrac{1}{2}(XX - YY)$.
Usage
- Appears in bosonic and superconducting-circuit contexts as the two-mode-squeezing analog, and in quantum chemistry circuits that need controlled pair excitations.
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. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\theta = 4\pi t$).
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.XXMinusYYGate ⓘ |
| PennyLane | — |
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
| Qibo | — |
| pytket | — ⓘ |
| Stim | — |
| OpenQASM | — |