标题： 多量子比特 Clifford 群是三设计 显示英文标题

标题： Multiqubit Clifford groups are unitary 3-designs

摘要： 酉算符 $t$-设计是许多研究领域中的通用工具，包括随机基准测试、量子过程层析成像和 scrambing。 尽管许多研究人员付出了巨大努力，但关于酉算符 $t$-设计的文献中知之甚少，其中 $t\geq3$。 我们证明了任意偶数幂维度上的多量子比特 Clifford 群不仅是一个酉算符 2-设计，还是一个 3-设计。 此外，除了维度为 4 的情况外，它是最小的 3-设计。 作为直接结果，多量子比特 Clifford 群的任何纯态轨道都构成一个复射影 3-设计；特别是，稳定子态集合构成一个 3-设计。 此外，我们的研究有助于研究 Clifford 群的更高矩，这些矩在从量子信息科学到信号处理的许多研究领域中非常有用。 此外，我们揭示了酉算符 3-设计与离散相空间物理之间的令人惊讶的联系，并因此简单地解释了为什么没有离散 Wigner 函数相对于多量子比特 Clifford 群具有协变性，这对研究量子计算具有内在兴趣。 显示英文摘要

摘要： Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.