quantum gate directory

The XY gate is the one-parameter family generated by the exchange (hopping) interaction $XX + YY$. It is a partial iSWAP: it rotates within the $\{|01\rangle, |10\rangle\}$ subspace and leaves $|00\rangle$ and $|11\rangle$ untouched.

$$ \mathrm{XY}(\theta) = \exp\left(i \frac{\theta}{4} (X \otimes X + Y \otimes Y)\right) $$

In the computational basis:

$$ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\tfrac{\theta}{2} & i\sin\tfrac{\theta}{2} & 0 \\ 0 & i\sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

Special values

Properties

• Number-preserving: it commutes with the total excitation number, like Givens rotations and FSim. In fact $\mathrm{FSim}(\theta, 0) = \mathrm{XY}(-2\theta)$.
• Forms a one-parameter group: $\mathrm{XY}(\alpha) \, \mathrm{XY}(\beta) = \mathrm{XY}(\alpha + \beta)$.
• The complementary rotation on the even-parity subspace is the $XX - YY$ gate; the two commute, and together they compose $R_{xx}$ and $R_{yy}$.
• Beware conventions: Rigetti, Braket, and PennyLane use the matrix above, while Qiskit's XXPlusYYGate rotates with the opposite sign and adds a phase parameter $\beta$.

Usage

• Native entangling gate on Rigetti hardware (as a calibrated pulse family), and a natural primitive for fermionic and hard-core boson simulation, where hopping terms are exactly of this form.