= XY exchange interaction
{c}
{title2=$E=(X\otimes X+Y\otimes Y)/2$}
The isotropic XY exchange interaction between two <qubits> is $E=(X\otimes X+Y\otimes Y)/2$. Its action is
$$
E|00\rangle=E|11\rangle=0,\qquad E|01\rangle=|10\rangle,\qquad E|10\rangle=|01\rangle.
$$
Indeed, $X|0\rangle=|1\rangle$, $X|1\rangle=|0\rangle$, $Y|0\rangle=i|1\rangle$, and $Y|1\rangle=-i|0\rangle$. The two terms cancel on equal bits and add on unequal bits. Thus a network <Hamiltonian operator> $H=\sum_{a,b}K_{ab}E_{ab}$ preserves excitation number; in the <single-excitation subspace>, it is the weighted adjacency <matrix>, moving an excitation from $a$ to $b$ with hopping coefficient $K_{ab}$.
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