Measurement based quantum computing is preformed by adding non-Clifford measurements to a prepared stabilizer states. Entangling gates like CZ are likely to have lower fidelities due to the nature of interacting qubits, so when preparing a stabilizer state, we wish to minimize the number of required entangling states. This naturally introduces the notion of CZ-distance. Every stabilizer state is local-Clifford equivalent to a graph state, so we may focus on graph states $\left\vert G \right\rangle$. As a lower bound for general graphs, there exist $n$-vertex graphs $G$ such that the CZ-distance of $\left\vert G \right\rangle$ is $\Omega(n^2 / \log n)$. We obtain significantly improved bounds when $G$ is contained within certain proper classes of graphs. For instance, we prove that if $G$ is a $n$-vertex circle graph with clique number $\omega$, then $\left\vert G \right\rangle$ has CZ-distance at most $4n \log \omega + 7n$. We prove that if $G$ is an $n$-vertex graph of rank-width at most $k$, then $\left\vert G \right\rangle$ has CZ-distance at most $(2^{2^{k+1}} + 1) n$. More generally, this is obtained via a bound of $(k+2)n$ that we prove for graphs of twin-width at most $k$. We also study how bounded-rank perturbations and low-rank cuts affect the CZ-distance. As a consequence, we prove that Geelen's Weak Structural Conjecture for vertex-minors implies that if $G$ is an $n$-vertex graph contained in some fixed proper vertex-minor-closed class of graphs, then $\left\vert G \right\rangle$ has CZ-distance at most $O(n\log n)$. Since graph states of locally equivalent graphs are local Clifford equivalent, proper vertex-minor-closed classes of graphs are natural and very general in this setting.

翻译：基于测量的量子计算是通过对已制备的稳定子态施加非克利福德测量来完成的。由于相互作用量子比特的特性，像CZ这样的纠缠门很可能具有较低的保真度，因此在制备稳定子态时，我们希望最小化所需纠缠门的数量。这自然引入了CZ距离的概念。每个稳定子态都局部克利福德等价于一个图态，因此我们可以专注于图态 $\left\vert G \right\rangle$。作为一般图的下界，存在 $n$ 顶点图 $G$，使得 $\left\vert G \right\rangle$ 的 CZ 距离为 $\Omega(n^2 / \log n)$。当 $G$ 包含在某些特定的真图类中时，我们获得了显著改进的界。例如，我们证明如果 $G$ 是一个团数为 $\omega$ 的 $n$ 顶点圆图，那么 $\left\vert G \right\rangle$ 的 CZ 距离至多为 $4n \log \omega + 7n$。我们证明如果 $G$ 是一个秩宽至多为 $k$ 的 $n$ 顶点图，那么 $\left\vert G \right\rangle$ 的 CZ 距离至多为 $(2^{2^{k+1}} + 1) n$。更一般地，这是通过我们为孪生宽至多为 $k$ 的图所证明的界 $(k+2)n$ 得到的。我们还研究了有界秩扰动和低秩割如何影响 CZ 距离。作为推论，我们证明了关于顶点子式的Geelen弱结构猜想意味着：如果 $G$ 是一个包含在某个固定的真顶点子式封闭图类中的 $n$ 顶点图，那么 $\left\vert G \right\rangle$ 的 CZ 距离至多为 $O(n\log n)$。由于局部等价图的图态是局部克利福德等价的，真顶点子式封闭图类在此背景下是自然且非常一般的。