quantum gate directory
Canonical entangling
Symbol
Description:
Parameterizes every two-qubit interaction up to local unitaries.
SDK Support
| SDK | Name |
|---|---|
| Qiskit | — |
| PennyLane | — |
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
The canonical gate generates every two-qubit interaction (up to single-qubit rotations) through three real parameters $(a, b, c)$:
$$ \mathrm{CAN}(a, b, c) = \exp(i(a X{\otimes}X + b Y{\otimes}Y + c Z{\otimes}Z)) $$
In the computational basis:
$$ \begin{bmatrix} \mathrm{e}^{ic}\cos(a-b) & 0 & 0 & i\mathrm{e}^{ic}\sin(a-b) \\ 0 & \mathrm{e}^{-ic}\cos(a+b) & i\mathrm{e}^{-ic}\sin(a+b) & 0 \\ 0 & i\mathrm{e}^{-ic}\sin(a+b) & \mathrm{e}^{-ic}\cos(a+b) & 0 \\ i\mathrm{e}^{ic}\sin(a-b) & 0 & 0 & \mathrm{e}^{ic}\cos(a-b) \end{bmatrix} $$
Properties
- Every $U \in SU(4)$ equals $\mathrm{CAN}(a,b,c)$ sandwiched by local single-qubit unitaries for some $a, b, c \in \mathbb{R}$ satisfying $\pi/4 \geq a \geq b \geq |c|$. This region defines the Weyl chamber.
- Two gates are locally equivalent if and only if they share the same $(a, b, c)$ coordinates.
- The Hamiltonian $aXX + bYY + cZZ$ block-diagonalizes in the Bell basis.
Special Cases
| $(a, b, c)$ | Locally equivalent |
|---|---|
| $(0, 0, 0)$ | Identity |
| $(\pi/4, 0, 0)$ | CNOT / CZ |
| $(\pi/8, \pi/8, 0)$ | $\sqrt{i\mathrm{SWAP}}$ |
| $(\pi/4, \pi/4, 0)$ | iSWAP |
| $(\pi/8, \pi/8, \pi/8)$ | $\sqrt{\mathrm{SWAP}}$ |
| $(\pi/4, \pi/4, \pi/4)$ | SWAP (global phase) |
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