标题： 量子比特稳定子态是复射影3设计 显示英文标题

标题： Qubit stabilizer states are complex projective 3-designs

摘要： 一个复射影$t$-设计是一组向量，它们在某种意义上是在球面上“均匀分布”的，即从它上面均匀采样可以重现到阶数$2t$的哈尔测度的矩。 我们证明了所有$n$-量子比特稳定态的集合在维度$2^n$上构成了一个复射影$3$-设计。此前，稳定态仅被知构成$2$-设计。 主要的技术工具是一个关于稳定态框架势能的一般递归公式。 为了建立这个公式，我们需要计算具有指定内积的稳定态的数量，相对于一个参考态而言。这反过来又归结为一个离散辛向量空间中的计数问题，我们为此找到了一个简单公式。 我们概述了该结果在量子信息和信号分析中的应用。 显示英文摘要

摘要： A complex projective $t$-design is a configuration of vectors which is ``evenly distributed'' on a sphere in the sense that sampling uniformly from it reproduces the moments of Haar measure up to order $2t$. We show that the set of all $n$-qubit stabilizer states forms a complex projective $3$-design in dimension $2^n$. Stabilizer states had previously only been known to constitute $2$-designs. The main technical ingredient is a general recursion formula for the so-called frame potential of stabilizer states. To establish it, we need to compute the number of stabilizer states with pre-described inner product with respect to a reference state. This, in turn, reduces to a counting problem in discrete symplectic vector spaces for which we find a simple formula. We sketch applications in quantum information and signal analysis.