My question is a simple one: Can 2 XY gates commute ?
XY gate is a 2-qubit XX+YY interaction also sometimes referred to as iSwap gate.
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If I denote $U_{ij}(t)=e^{-it(X_iX_j+Y_iY_j)}$, then we have $$ [U_{ij},U_{kl}]=0 $$ if $(i,j)=(k,l)$, i.e. the two gets act on the same pair of qubits, or if $(i,j)\neq(k,l)$, meaning they act on completely distinct pairs.
Generically, if there's partial intersection between $(i,j)$ and $(k,l)$, the two won't commute. Of course, there are some exceptions - trivial cases (e.g. if one of the times is an integer multiple of $\pi$, $U$ is just the identity).
