What is the matrix of the iSwap gate?

Question

Mostly I'm confused over whether the common convention is to use +$i$ or -$i$ along the anti-diagonal of the middle $2\times 2$ block.

Answer 1

Mostly I'm confused over whether the common convention is to use +i or -i along the anti-diagonal of the middle 2x2 block.

The former. There are two $+i$'s along the anti-diagonal of the middle $2\times 2$ block of the iSWAP gate. See page 95 here^[$\dagger$].

[$\dagger$]: Explorations in Computer Science (Quantum Gates) - Colin P. Williams

Answer 2

Whether you use $+i$ or $-i$ is entirely up to you. After all, your definition of $\pm i$ is merely a convention. On the other hand, I think I've only ever seen it with $+i$.

On a more general footing, you can consider that iSWAP is the gate obtained by time-evolving with an XX interaction ($H=-\sigma_x\otimes\sigma_x - \sigma_y\otimes\sigma_y$), in which case it depends on which sign of $i$ in the Schrödinger equation and for the Hamiltonian you prefer. (You get $+i$ if you evolve with $\exp[-iHt]$, $t=\pi/4$, and chose the minus sign in the Hamiltonian as above, i.e. a ferromagnet).