论文标题
Clifford Tensor Powers的二元性理论
Duality theory for Clifford tensor powers
论文作者
论文摘要
克利福德小组的表示理论在量子信息理论中起着越来越重要的作用，包括在诸如构建量子系统认证协议，量子模拟和量子加密等不同用例中。在这些应用中，定义表示形式的张量似乎尤为重要。这些张量力的代表理论在两个制度中被理解。 1。对于奇数，如果功率t不大于系统n的数量：在这里，可以使用Clifford组与某些离散正交组之间的双重性理论来对发生的IRREPS发表相当明确的陈述（该理论与Howe Duality duality and duality and duality and the eta-corressive有关）。 2。对于量子位：张量功率为t = 4的功率已通过具体情况进行分析。在本文中，我们为二元方法提供了一个统一的框架，该框架也涵盖了量子系统。为此，我们将符号表示形式的等级概念转化为Qubit Clifford组的表示，并将符号和正交组之间的ETA对应关系概括为Clifford与某些正交组织组之间的对应关系。作为示例应用程序，我们提供了一个协议，以有效地实现黑盒Clifford统一进化的复杂共轭。
The representation theory of the Clifford group is playing an increasingly prominent role in quantum information theory, including in such diverse use cases as the construction of protocols for quantum system certification, quantum simulation, and quantum cryptography. In these applications, the tensor powers of the defining representation seem particularly important. The representation theory of these tensor powers is understood in two regimes. 1. For odd qudits in the case where the power t is not larger than the number of systems n: Here, a duality theory between the Clifford group and certain discrete orthogonal groups can be used to make fairly explicit statements about the occurring irreps (this theory is related to Howe duality and the eta-correspondence). 2. For qubits: Tensor powers up to t=4 have been analyzed on a case-by-case basis. In this paper, we provide a unified framework for the duality approach that also covers qubit systems. To this end, we translate the notion of rank of symplectic representations to representations of the qubit Clifford group, and generalize the eta correspondence between symplectic and orthogonal groups to a correspondence between the Clifford and certain orthogonal-stochastic groups. As a sample application, we provide a protocol to efficiently implement the complex conjugate of a black-box Clifford unitary evolution.
