A Complete and Natural Rule Set for Multi-Qudit Clifford Circuits in All Odd Prime Dimensions
Abstract
We present a complete set of $16$ rewrite rules for multi-qudit Clifford circuits in all odd prime dimensions. Completeness means that any two Clifford circuits representing the same linear map can be rewritten into each other using these rules. Each rule involves at most three qudits and admits an intuitive interpretation. To establish completeness, we first work at the symplectic level, using the isomorphism between the symplectic group Sp(2n,ℤp) and the quotient of the Clifford group by the Pauli group. We construct a circuit normal form that captures the stabiliser tableau of a Clifford operator and is unique up to Pauli correction. Using this normal form, we derive a complete set of symplectic relations, which we then lift to Clifford relations by incorporating Pauli corrections. We also formally verify completeness in the Agda proof assistant.
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BibTeX
@misc{bian2026complete,
title = {{A Complete and Natural Rule Set for Multi-Qudit Clifford Circuits in All Odd Prime Dimensions}},
author = {Xiaoning Bian and Sarah Meng Li and Neil J. Ross and John van de Wetering and Yuming Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.40106},
eprint = {2609.40106},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40106},
}
RIS
TY - UNPB
TI - A Complete and Natural Rule Set for Multi-Qudit Clifford Circuits in All Odd Prime Dimensions
AU - Xiaoning Bian
AU - Sarah Meng Li
AU - Neil J. Ross
AU - John van de Wetering
AU - Yuming Zhao
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.40106
UR - https://arxiv.org/abs/2609.40106
AB - We present a complete set of $16$ rewrite rules for multi-qudit Clifford circuits in all odd prime dimensions. Completeness means that any two Clifford circuits representing the same linear map can be rewritten into each other using these rules. Each rule involves at most three qudits and admits an intuitive interpretation. To establish completeness, we first work at the symplectic level, using the isomorphism between the symplectic group Sp(2n,ℤ_p) and the quotient of the Clifford group by the Pauli group. We construct a circuit normal form that captures the stabiliser tableau of a Clifford operator and is unique up to Pauli correction. Using this normal form, we derive a complete set of symplectic relations, which we then lift to Clifford relations by incorporating Pauli corrections. We also formally verify completeness in the Agda proof assistant.
ER -