发表机构
University of Waterloo; Kyoto University; IBM Quantum, T.J. Watson Research Center; RIKEN; Perimeter Institute for Theoretical Physics（滑铁卢大学; 京都大学; IBM量子，T.J.沃森研究中心; 理化学研究所; 理论物理珀塞尔研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明环面码中电磁对偶性无法在微观上实现为二阶克利福德操作，并推广至$\mathbb{Z}_N$情形，揭示群法则与微观实现之间的不匹配。
AI 中文摘要
环面码中的电磁对偶性在任意子类型层面具有二阶性，并对编码的量子信息实施类似阿达玛型的逻辑变换。在此，我们探讨该对偶性是否也能在微观层面实现为二阶克利福德操作。对于$\mathbb{Z}_2$环面码，我们证明任何保持局域性的电磁交换克利福德实现都不可能具有二阶性。我们的证明完全通用，既不要求平移对称性及受限的系统尺寸族，也不要求顶点与面稳定子之间的固定配对。在几何上，该证明局部模拟了电弦与磁弦之间不可避免的交叉，类似于不可定向流形中的交叉帽。将问题扩展到$\mathbb{Z}_N$环面码，我们发现对于奇数$N$存在二阶克利福德实现，而对于偶数$N$则不可能。我们的结果表明，涌现电磁对偶性的群法则未必能忠实地提升到其微观克利福德实现。
英文摘要
Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the $\mathbb{Z}_2$ toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the $\mathbb{Z}_N$ toric code, we find that an order-two Clifford realization exists for odd $N$, whereas it is impossible for even $N$. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.
Comments36 pages, many figures
