quantum gate directory
B Gate
Symbol
$B$
Description:
Two-qubit entangling gate that can synthesize arbitrary elements of $U(4)$ with only two applications.
SDK Support
| SDK | Name |
|---|---|
| Qiskit | — |
| PennyLane | — |
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
The B gate introduced in Zhang, Vala, Sastry, and Whaley is
$$ \begin{align*} B & = \exp\left[-i\left(\frac{\pi}{4}X\otimes X + \frac{\pi}{8}Y\otimes Y\right)\right] \\ & = \begin{bmatrix} \alpha^{+} & 0 & 0 & i\alpha^{-} \\ 0 & \alpha^{-} & i\alpha^{+} & 0 \\ 0 & i\alpha^{+} & \alpha^{-} & 0 \\ i\alpha^{-} & 0 & 0 & \alpha^{+} \end{bmatrix} \end{align*} $$
Where $$ \begin{align*} \alpha^{\pm} & = \tfrac{\sqrt{2}}{2}\left(\cos \tfrac{\pi}{8} \pm \sin \tfrac{\pi}{8}\right) \\ & = \tfrac{\sqrt{2}}{4}\left(\sqrt{2 + \sqrt{2}} \pm \sqrt{2 - \sqrt{2}}\right) \end{align*} $$
Properties
- Any $U \in U(4)$ can be realized with only two B gates plus local single-qubit gates.
- This improves on common alternatives (for example CNOT and DCNOT), which need three applications, and many other two-qubit gates that need three or more to synthesize arbitrary $U(4)$ operations.
References
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