# Matrix definition of the Ising XX gate

On the English Wikipedia, $$XX$$ Ising gates are defined in matrix form as :

$$XX(\phi) = \begin{bmatrix}\cos(\phi)&0&0& -i \sin(\phi)\\ 0&\cos(\phi)&-i \sin(\phi) & 0 \\ 0 & -i \sin(\phi) & \cos(\phi) & 0 \\ -i \sin(\phi) & 0 & 0 & \cos(\phi) \\ \end{bmatrix}$$

However, on the French Wikipedia, the $$XX$$ Ising gates are defined as :

$$XX(\phi) = \frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0& -i e^{i \phi}\\ 0&1&-i & 0 \\ 0 & -i & 1 & 0 \\ -i e^{i \phi} & 0 & 0 & 1 \\ \end{bmatrix}$$

From some sources, I believe the right one is the first one. I would like some confirmations about the second definition being an error, and I would be glad to take any explanation on it. In particular, does this matrix in the second definition corresponds to the definition of another gate?

Your first option is the correct one, being related to $$e^{-i\phi X\otimes X}$$, which is $$XX(\phi) = \begin{bmatrix}\cos(\phi)&0&0& -i \sin(\phi)\\ 0&\cos(\phi)&-i \sin(\phi) & 0 \\ 0 & -i \sin(\phi) & \cos(\phi) & 0 \\ -i \sin(\phi) & 0 & 0 & \cos(\phi) \\ \end{bmatrix}.$$

The second option doesn't make a whole lot of sense; it's not even unitary, so it's really not right. It cannot be the definition of a gate.

• thank you, I forgot the minus sign in the first definition. Should I edit it and let you edit your answer accordingly ? Otherwise it might be confusing for someone reading this if they assume I did no mistake in copying the Wikipedia definition – nathan raynal Apr 16 at 8:28
• @nathanraynal Sure. I'll edit mine now... – DaftWullie Apr 16 at 8:47