Havlicek et al. propose a feature map for embedding $n$-dimensional classical data on $n$ qubits: $U_{\phi(x)}H^{\otimes n}$, where $$ U_{\phi(x)} = \exp (i \sum_{S \subseteq [n]} \phi_S(x) \prod_{i \in S} Z_i) \\ \phi_i(x) = x_i, \; \phi_{\{i, j\}}(x) = (\pi - x_0)(\pi - x_1) $$ and $Z_i$ is a $Z$-gate on the $i$-th qubit.
I'm currently considering the 2-dimensional, 2-qubit case, $$ U_{\phi(x)} = \exp(i x_0 Z_0 + i x_1 Z_1 + i (\pi - x_0) (\pi - x_1) Z_0 Z_1), $$ and trying to find out, how do I get from the above definition to the circuit, as it is implemented in Qiskit:
┌───┐┌──────────────┐
q_0: ┤ H ├┤ U1(2.0*x[0]) ├──■─────────────────────────────────────■──
├───┤├──────────────┤┌─┴─┐┌───────────────────────────────┐┌─┴─┐
q_1: ┤ H ├┤ U1(2.0*x[1]) ├┤ X ├┤ U1(2.0*(π - x[0])*(π - x[1])) ├┤ X ├
└───┘└──────────────┘└───┘└───────────────────────────────┘└───┘
My reasoning for the first 2 $U1$ rotations is that since $\exp(i \theta Z) = RZ(2 \theta)$ up to global phase and $RZ$ is equivalent to $U1$ up to global phase, we might as well use $U1(2 \phi_i(x))$. Is that correct?
And if so, how do we then get $CX \; U1(2 \phi_{\{0, 1\}}(x)) \; CX$ from $\exp(i \phi_{\{0, 1\}}(x) Z_0 Z_1)$?