We study the notion of $k$-stabilizer universal quantum state, that is, an $n$-qubit quantum state, such that it is possible to induce any stabilizer state on any $k$ qubits, by using only local operations and classical communications. These states generalize the notion of $k$-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and $k$-vertex-minor universal graphs. First, we demonstrate the existence of $k$-stabilizer universal graph states that are optimal in size with $n=\Theta(k^2)$ qubits. We also provide parameters for which a random graph state on $\Theta(k^2)$ qubits is $k$-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of $k$-stabilizer universal graph states on $n = O(k^4)$ qubits. Both rely upon the incidence graph of the projective plane over a finite field $\mathbb{F}_q$. This provides a major improvement over the previously known explicit construction of $k$-pairable graph states with $n = O(2^{3k})$, bringing forth a new and potentially powerful family of multipartite quantum resources.

翻译：我们研究了$k$-稳定子通用量子态的概念，即一种$n$量子比特量子态，能够仅通过局域操作和经典通信在任意$k$个量子比特上诱导出任意稳定子态。这些态推广了Bravyi等人提出的$k$-可配对态概念，并可通过图态和$k$-顶点-小子图通用图从组合学角度进行研究。首先，我们证明了存在最优尺寸为$n=\Theta(k^2)$量子比特的$k$-稳定子通用图态。我们还给出了参数范围，使得$\Theta(k^2)$量子比特上的随机图态以高概率成为$k$-稳定子通用态。我们的第二项贡献是给出了两种显式构造，在$n=O(k^4)$量子比特上构建$k$-稳定子通用图态。这两种构造均依赖于有限域$\mathbb{F}_q$上射影平面的关联图。相较于先前已知的$n=O(2^{3k})$的$k$-可配对图态显式构造，这带来了重大改进，并催生了一类具有潜在强大功能的多方量子资源。