Local Automorphism-Aware Syndrome Compilation for General Quantum LDPC Codes

2026-09-30

AI Summary

This study addresses the high computational complexity of low-depth syndrome extraction for quantum LDPC codes, which requires solving an ordered edge-coloring problem subject to quantum parity constraints. To overcome this challenge, this work proposes the LocalASC algorithm, which replaces full automorphism group computations by constructing translation subgroups and extending symmetry conditions. This approach simplifies Tanner graph constraints into orbit variables that are subsequently lifted, thereby enabling efficient compilation. By integrating local automorphism awareness with the group-algebraic theory of CSS codes, the proposed method derives optimal or lower-bound-saturating syndrome extraction schedules for multiple families of two-block CSS codes and quantum Tanner codes, significantly reducing overall computational complexity.

Abstract

Low-depth syndrome extraction for Calderbank-Shor-Steane (CSS) quantum low-density parity-check codes can be formulated as a proper ordered edge-coloring problem subject to quantum parity constraints. A proper edge-coloring of the CSS Tanner graph ensures that each data or ancilla qubit participates in at most one two-qubit gate per layer, but does not guarantee a valid interleaving of the X- and Z-check measurements as for every overlapping X/Z check pair, the number of shared data qubits on which the X interaction precedes the Z interaction must be even. The minimum number of colors in a proper ordered edge-coloring satisfying the quantum parity constraints equals the minimum two-qubit depth when each stabilizer check is measured with a single ancilla. We introduce local automorphism-aware syndrome compilation (LocalASC), which reduces the constraint system to edge-orbit variables under a subgroup of the Tanner graph automorphisms and lifts each feasible orbit assignment to the full graph. Although the $6$-layer degree lower bound is unattainable for the published weight-$6$ IBM bivariate bicycle codes, we show that this is not universal among two-block CSS codes. Among code instances for which the maximum check weight equals the maximum Tanner graph degree, LocalASC finds depth-optimal syndrome-extraction schedules for several two-block CSS codes with odd component weights, including instances with unequal odd weights. We also obtain lower-bound-saturating syndrome-extraction schedules for several quantum Tanner codes satisfying the same degree condition. To obtain the subgroups used by LocalASC without computing the full automorphism group of the Tanner graph, we construct translation subgroups for two-block group-algebra CSS codes over abelian groups. For quantum Tanner codes, we give conditions under which square-complex symmetries extend to Tanner graph automorphisms.

Problem

• quantum LDPC codes
• syndrome extraction
• edge coloring
• quantum parity constraints
• CSS codes

Innovation

• Local Automorphism-Aware Syndrome Compilation
• Quantum LDPC Codes
• Edge Coloring
• Tanner Graph Automorphisms
• Syndrome Extraction