Lifted surgery for non-Abelian two-block group-algebra codes
Abstract
Code surgery measures a logical operator of a quantum LDPC code with O(d) rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to $13$ where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of A4, S4 and SL(2,3) with gain three for a best logical, up to $[[336,26,12]]$. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.
Related
- Lifted surgery: Fast processing with QLDPC codes
- Block Encoding Non-Abelian Lattice Gauge Theory
- Algebra objects in direct limit completions of compact Lie group duals and the classification of c=1 vertex operator algebras
- Improved Quantum Algorithms for Black-Box Abelian Group Decomposition
- Quantum Inversion of Units in Group Rings: Block Dimension, Not Commutativity, Governs Hardness
BibTeX
@misc{pandey2026lifted,
title = {{Lifted surgery for non-Abelian two-block group-algebra codes}},
author = {Tushar Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05307},
eprint = {2610.05307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05307},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lifted surgery for non-Abelian two-block group-algebra codes
AU - Tushar Pandey
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05307
UR - https://arxiv.org/abs/2610.05307
AB - Code surgery measures a logical operator of a quantum LDPC code with O(d) rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to $13$ where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of A_4, S_4 and SL(2,3) with gain three for a best logical, up to $[[336,26,12]]$. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.
ER -