# Is a generalized 2D cluster state still a universal resource?

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 $$\theta >0$$?

• There is quite some work on "computational phases of matter" (by Raussendorf, Tzu-Chieh Wei, David Stephen, and others). Among others, this asks how much you can deform the cluster state to still have a resource for MBQC. One key insight is that this is related to certain symmetries (subsystem symmetries). I don't think your deformation respects them, so I suspect the answer is no, but I didn't check carefully. -- But that literature might be a good place to check. Sep 4, 2022 at 11:32

I think that a generalized 2D weighted graph state is a resource with $$\textbf{specific}$$ values in angles $$\theta$$ of the entangling gates. Not any pattern of weighted graph state is a universal resource. This paper may give insights to your question. https://arxiv.org/abs/1704.06504