Let $G = (V, E)$ be a $\sqrt{n} \times \sqrt{n}$ grid graph. Consider preparing a 2D cluster state as follows.
$$|\psi_{\mathsf{C}}\rangle = \underset{(i, j) \in E}{\Pi} (\mathsf{CZ})_{i, j} | + \rangle^{\otimes n},$$
where $\mathsf{CZ}$ is the controlled-$\mathsf{Z}$ operator acting on qubits $i$ and $j$. We know that this state is a universal resource for measurement based quantum computing (using postselection, we can implement any $\mathsf{BQP}$ circuit with this state.)
Let's say we generalize this state.
Let's say for a particular angle $\theta$, I replace $\mathsf{CZ}$ with the following:
$$|\psi_{\mathsf{C, \theta}}\rangle = \underset{(i, j) \in E}{\Pi} e^{i \theta~ \mathsf{Z}_i \mathsf{Z}_j} | + \rangle^{\otimes n},$$
where $\mathsf{Z}_i$ and $\mathsf{Z}_j$ are $n$ qubit Pauli-$\mathsf{Z}$ operators each, acting non-trivially on qubits $i$ and $j$ respectively.
Does $|\psi_{\mathsf{C, \theta}}\rangle$ also remain a universal resource for measurement based quantum computing, for any value of $\theta >0$?
