Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers \(U^{\otimes t}\) of all unitaries \(U\in U(d)\) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications:

• We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group.

• We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well).

• We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem.

• We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Schur-Weyl 对偶是量子信息中无处不在的工具。它的核心是这样的陈述，即与所有幺正\(U\in U(d)\)的t倍张量幂\(U^{\otimes t}\)交换 的算子空间由置换跨越的吨张量的因素。在这项工作中，我们描述了Clifford 幺正的张量幂的类似对偶理论。Clifford 群是量子信息许多子领域的中心对象，在容错理论中最为突出。二元论在有限几何方面有一个简单而清晰的描述。我们在几个应用中证明了它的有效性：

• 我们解决了量子特性测试中的一个悬而未决的问题，表明“稳定器”是可有效测试的：有一个协议，可以访问未知状态的六个副本，可以确定它是否是稳定器状态，或者它是否远离来自稳定器状态的集合。我们对 Clifford 小组进行了相关的成员资格测试。

• 我们发现稳定器状态的张量幂具有增加的对称群。相反，我们提供了相应的de Finetti 定理，表明具有这种对称性的任意状态的减少可以通过稳定器张量幂的混合很好地近似（在某些情况下，指数很好）。

• 我们表明，就其 Wigner 函数的和负性而言，纯态到稳定器集的距离可以是下界的。这为负和（以及相关的法力）赋予了新的定量意义——一种与容错量子计算相关的度量。结果构成了离散哈德逊定理的稳健推广。

• 我们表明，可以从有限数量（独立于量子点的数量）的 Clifford 轨道中获得任意阶的复杂投影设计。为了证明这个结果，我们给出了随机稳定器状态的任意矩的明确公式。