标题： 半普遍性框架：3-qudit SU(d)不变门的半普遍性 显示英文标题

标题： A framework for semi-universality: Semi-universality of 3-qudit SU(d)-invariant gates

摘要： 具有对称性尊重门的量子电路在量子信息科学中引起了广泛关注。 尽管近期的研究已经发展出针对具有阿贝尔对称性的电路的理论，揭示了阿贝尔和非阿贝尔情况之间的关键区别，但缺乏一个全面的非阿贝尔对称性的框架。 在本工作中，我们开发了新颖的技术和一个强大的框架，该框架特别有助于理解具有非阿贝尔对称性的电路。 使用这个框架，我们解决了关于具有SU(d)对称性的量子电路的一个开放问题。 我们证明，3-qudit SU(d)不变门是半通用的，即生成所有SU(d)不变的幺正变换，除了在不同表示的对称性子空间之间的相对相位上有一些约束。 此外，我们证明当补充3个辅助qudits时，这些门可以实现完全通用性。 有趣的是，我们发现研究具有3-qudit门的电路也有助于更好地理解具有2-qudit门的电路。 特别是，我们证明尽管2-qudit SU(d)不变门本身不是半通用的，但最多需要11个辅助qudits就可以变得通用。 此外，我们还研究了由随机SU(d)不变门组成的电路的统计特性。 我们的发现表明，虽然具有2-qudit门的电路不形成SU(d)不变幺正变换上的哈尓测度的2设计，但具有3-qudit门的电路可以生成t设计，其中t与qudits数量的平方成比例。 显示英文摘要

摘要： Quantum circuits with symmetry-respecting gates have attracted broad interest in quantum information science. While recent work has developed a theory for circuits with Abelian symmetries, revealing important distinctions between Abelian and non-Abelian cases, a comprehensive framework for non-Abelian symmetries has been lacking. In this work, we develop novel techniques and a powerful framework that is particularly useful for understanding circuits with non-Abelian symmetries. Using this framework we settle an open question on quantum circuits with SU(d) symmetry. We show that 3-qudit SU(d)-invariant gates are semi-universal, i.e., generate all SU(d)-invariant unitaries, up to certain constraints on the relative phases between sectors with inequivalent representation of symmetry. Furthermore, we prove that these gates achieve full universality when supplemented with 3 ancilla qudits. Interestingly, we find that studying circuits with 3-qudit gates is also useful for a better understanding of circuits with 2-qudit gates. In particular, we establish that even though 2-qudit SU(d)-invariant gates are not themselves semi-universal, they become universal with at most 11 ancilla qudits. Additionally, we investigate the statistical properties of circuits composed of random SU(d)-invariant gates. Our findings reveal that while circuits with 2-qudit gates do not form a 2-design for the Haar measure over SU(d)-invariant unitaries, circuits with 3-qudit gates generate a t-design, with t that is quadratic in the number of qudits.