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

Abstract: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. We construct syndrome-extraction circuits for these codes and numerically demonstrate single-shot decoding in $Z$ memory, in both phenomenological and circuit-level noise models. We also demonstrate that the codes have good performance under single-shot decoding even in $X$ memory. Under circuit-level noise, TT codes have a crossing point of $0.3\%$ in $X$ memory and $1\%$ in $Z$ memory (with single-shot decoding). All TT codes possess several transversal $CZ$ gates that can partially address logical qubits between two code blocks. The codes possess a large set of automorphisms that can perform Clifford gates within a code block. Finally, we establish several TT code polynomial constructions that allow for a constant-depth implementation of logical $CCZ$ gates. We find codes with $CCZ$ gates that use up to $5.7\times$ fewer data qubits as equivalent-distance 3DTC encodings, and others with encoding rates as high as $1/4$.

33 pages, 18 figures. Responding to reviewer feedback

arXiv:2508.08191 [quant-ph]

(or arXiv:2508.08191v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2508.08191 Focus to learn more arXiv-issued DOI via DataCite

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