Continuum between large-automorphism, low-rate codes and high-rate qLDPC codes

Ascertain whether a continuum of quantum error-correcting codes exists between large-automorphism, low-rate constructions (such as phantom codes) and high-rate qLDPC codes, and determine how resource-optimal operating points are distributed along this continuum in terms of storage, logical gate cost, and overall space–time overhead.

Background

The authors emphasize that optimizing QEC for practical computation requires co-design across two axes: storage efficiency (rate/LDPC structure) and logical gate efficiency. Phantom codes occupy an extreme of vanishing in-block entangling cost yet low rate, whereas many qLDPC codes occupy the opposite extreme.

Understanding whether these extremes are connected by a spectrum of intermediate designs—and where optimal trade-offs lie—would guide future code discovery tailored to hardware constraints and algorithmic workloads.

References

This contrast suggests an existence of a continuum between large-automorphism, low-rate codes and high-rate qLDPC constructions. To what extent this continuum exists, and how resource-optimal operating points are distributed along it, remains an open problem.

— Entangling logical qubits without physical operations  (2601.20927 - Koh et al., 28 Jan 2026) in Section 6 (Discussion and outlook)

Because neither mechanism is optimal alone, the open problem is to determine their best division of labour for a given workload, block size, distance, and hardware platform.

— Achieving the limits of automorphism gates  (2609.19250 - Koh et al., 16 Sep 2026) in Section 10, Discussion and Outlook