Quantum LDPC codes with design rate 1/5 and good performance below 1000 physical qubits

This paper presents a new family of quantum low-density parity-check (qLDPC) codes, termed ZSZ-LP codes, designed to bridge the gap between theoretical asymptotic performance and practical, near-term fault-tolerant quantum computing. By utilizing non-abelian metacyclic ZSZ groups (semidirect products of cyclic groups) for the balanced product construction, the author circumvents the low-distance limitations often found in abelian-symmetric codes. These codes achieve a design rate of 1/5 and demonstrate high memory efficiency, with several instances requiring fewer than 1000 physical qubits to reach the gigaquop and teraquop regimes under circuit-level noise.

The construction begins with pairs of classical (3, 6)-regular LDPC codes that share non-abelian ZSZ group symmetries. The author employs a filtered random search to select instances with favorable minimum distances and girth properties. To facilitate practical implementation, the paper develops syndrome extraction circuits tailored for reconfigurable atom arrays using a greedy scheduler. Furthermore, the author constructs equivariant logical Pauli bases—specifically one-generator minimal (OGM) and one-generator symplectic (OGS) bases—which significantly simplify the design of surgery gadgets for logical operations.

Numerical simulations using a GPU-accelerated Relay-BP decoder demonstrate that these codes exhibit a pseudothreshold of approximately 0.5% under idling-free circuit-level noise. The latency of the decoding process is reported at 1-2 ms, which is compatible with the requirements of trapped-ion and neutral-atom hardware. By providing a concrete path to high-rate, finite-length quantum error correction, this work offers a scalable approach for near-term quantum architectures that require both high logical qubit density and efficient error suppression.