A No-Go Theorem for Order-Two Clifford Electric-Magnetic Duality
Abstract
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 ℤ2 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 ℤN 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.
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BibTeX
@misc{takahashi2026nogo,
title = {{A No-Go Theorem for Order-Two Clifford Electric-Magnetic Duality}},
author = {Shunta Takahashi and Zhi Li and Beni Yoshida},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02097},
eprint = {2610.02097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02097},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A No-Go Theorem for Order-Two Clifford Electric-Magnetic Duality
AU - Shunta Takahashi
AU - Zhi Li
AU - Beni Yoshida
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02097
UR - https://arxiv.org/abs/2610.02097
AB - 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 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 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.
ER -