Curator's Take
This article shows that planar tile‑code families can be run on the same nearest‑neighbor square lattice used for the surface code, delivering up to four times higher encoding efficiency while still achieving a respectable circuit‑level threshold (≈0.1% with routing). By providing an exhaustive SWAP‑routing method and demonstrating a crossover point near 0.08% physical error where tile codes require fewer qubits per logical qubit than the surface code, it offers a concrete pathway for low‑error hardware to reap the qubit‑saving benefits of qLDPC designs. The work bridges recent theoretical advances in high‑rate LDPC codes with practical hardware constraints, highlighting that the long‑standing surface‑code monopoly may soon be challenged as error rates continue to drop.
— Mark Eatherly
Summary
Tile codes are a family of planar quantum low-density parity-check (qLDPC) codes with weight-6 stabilizers and open boundary conditions, offering an encoding efficiency $kd^2/n$ of up to four times that of the surface code. In this work, we develop an exhaustive search algorithm for finding SWAP-based routing schemes that implement syndrome extraction for four tile-code families using only nearest-neighbor interactions on a two-dimensional square lattice, matching the connectivity of the surface code. Using explicitly constructed routed syndrome-extraction circuits decoded with BP+OSD, we estimate the circuit-level thresholds of these code families. For the SI1000 noise model, the threshold without such a connectivity constraint is obtained in a range 0.23%-0.31%, while it decreases to 0.11%-0.13% with routing, representing a reduction factor of around two to three. Despite this threshold penalty, our resource-footprint analysis shows that routed tile codes require fewer physical qubits per logical qubit than the surface code at sufficiently low physical error rates: Under the SI1000 noise model, we find a crossover near $p^*\approx 0.08\%$, below which routed tile codes become more qubit-efficient, with an advantage that grows monotonically as the physical error rate decreases.