Single-Shot Decoding and Fault-tolerant Gates with Trivariate Tricycle Codes

Understand

While quantum low-density parity check (qLDPC) codes are a low-overhead means of quantum information storage, it is valuable for quantum codes to possess fault-tolerant features beyond this resource efficiency.

• In this work, we introduce trivariate tricycle (TT) codes, qLDPC codes that combine several desirable features: high thresholds under a circuit-level noise model, partial single-shot decodability for low-time-overhead decoding, a large set of transversal Clifford gates and automorphisms within and between code blocks, and (for several sub-constructions) constant-depth implementations of a (non-Clifford) $CCZ$ gate.
• TT codes are CSS codes based on a length-3 chain complex, and are defined from three trivariate polynomials, with the 3D toric code (3DTC) belonging to this construction.
• We numerically search for TT codes and find several candidates with improved parameters relative to the 3DTC, using up to 48$\times$ fewer data qubits as equivalent 3DTC encodings.