= Cartan decomposition of a two-qubit gate
{c}
{title2=$U=e^{i\chi}(A_1\otimes A_2)e^{-i(c_xX\otimes X+c_yY\otimes Y+c_zZ\otimes Z)}(B_1\otimes B_2)$}
Every <two-qubit gate> has the displayed form with local elements of the <special unitary group> and real interaction parameters. The three Pauli-product generators commute, so the central exponential is the product of three interactions. Local basis changes turn an <Ising coupling> into either of the other Pauli-product couplings. A fixed entangling controlled phase together with arbitrary local rotations can also synthesize each variable interaction, for example through two <CNOT gates> enclosing a variable $z$ rotation.
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