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 $A_4$, $S_4$ 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.