Universality versus geometric locality in quantum error-correcting codes

Determine whether universal fault-tolerant quantum computation and geometric locality can be simultaneously achieved in quantum error-correcting codes without incurring the connectivity costs associated with higher-dimensional product-code constructions.

Background

The paper discusses a tension between constructions supporting universal or non-Clifford logical operations and the geometric-locality constraints imposed by planar quantum-computing architectures. Higher-dimensional product codes can support transversal non-Clifford gates, but their implementation on planar hardware requires longer-range couplings. The authors identify resolving this tension between universality and geometric locality as a basic open question.

References

The extra dimension is paid for in connectivity: a 3D product code demands longer-range couplings than a 2D one once laid out on a planar chip, which is the binding constraint for most architectures, and the resulting tension between universality and geometric locality is a basic open question.

— Sipser-Spielman meets Dijkgraaf-Witten: non-Abelian qLDPC codes via twisted sheaf gauge theory and almost-constant-overhead magic state fountain  (2609.31541 - Zhu et al., 25 Sep 2026) in Section 1, Introduction