quantum gate directory
XX−YY Interaction
Symbol
$R_{xx-yy}$
Qubits: $2$ Parameters: $1$
Description:
Rotation generated by the $XX - YY$ interaction, acting only on the even-parity subspace.
Alternate notations:
- $R_{xx-yy}(\theta)$
- $\mathrm{XX{-}YY}(\theta)$
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.XXMinusYYGate
ⓘ
|
| PennyLane | — |
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
| Qibo | — |
| pytket | — ⓘ |
| Stim | — |
| OpenQASM | — |
Groups
This gate is contained in the following groups:
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.
$$ R_{xx-yy}(\theta) = \exp\left(-i \frac{\theta}{4} (X \otimes X - Y \otimes Y)\right) = \begin{bmatrix} \cos\frac{\theta}{2} & 0 & 0 & -i\sin\frac{\theta}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -i\sin\frac{\theta}{2} & 0 & 0 & \cos\frac{\theta}{2} \end{bmatrix} $$
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.
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