All-size onset theorem for constant-rate qLDPC families

Establish an all-size onset theorem for a constant-rate quantum low-density-parity-check code family that characterizes the entropic rigidity depth r in terms of Tanner-graph expansion, bipartite qubit–check incidence-graph expansion, the stabilizer quotient, Pauli-composition spectra, and the growth of low-excess centralizer representatives, thereby determining when configurational entropy first changes the logical decoding decision.

Background

The paper defines the entropic rigidity depth r through the first physical error-weight layer at which maximum-probability decoding and degenerate maximum-likelihood decoding have disjoint logical winner sets. Its exact analyses show that geometry, recursive structure, Pauli purity, and parity-check packing can delay this onset beyond the universal half-distance bound.

For general constant-rate qLDPC families, the paper conjectures that r is jointly controlled by expansion properties, the stabilizer quotient, Pauli-composition spectra, and the proliferation of low-excess centralizer representatives. The paper establishes this principle for particular finite and separable families, but explicitly states that an all-size onset theorem for a constant-rate family is still unresolved.

References

For general constant-rate qLDPC families , we conjecture that $r$ is governed jointly by expansion of the Tanner graph, the bipartite qubit--check incidence graph, the stabilizer quotient, Pauli-composition spectra, and the growth of low-excess centralizer representatives, namely Pauli operators that commute with every stabilizer, considered modulo stabilizers. Expansion and systolic gaps may suppress such factorizations, whereas locally clustered low-weight centralizer classes may proliferate them. An all-size onset theorem for a constant-rate family remains open; balanced factorization provides a concrete, non-sampling diagnostic for testing this design principle.

— Entropic Rigidity in Quantum Memories: How Geometry and Algebra Control the Onset of Degeneracy Corrections  (2608.18420 - Zhao et al., 19 Aug 2026) in Section 5, Discussion and outlook