Single-Shot and Few-Shot Decoding via Stabilizer Redundancy in Bivariate Bicycle Codes

TL;DR

This work addresses single-shot decoding for coprime bivariate bicycle (BB) quantum LDPC codes by uncovering an exact algebraic link between the gcd polynomial $g(z)$ and stabilizer redundancy. It shows that $r_X=r_Z=\deg g(z)=k/2$ and that the X/Z syndrome codes are cyclic codes generated by $g(z)$, with stabilizer-relations governed by $h(z)=(z^N-1)/g(z)$, yielding BCH-type bounds on $d_S$ and a $t_S\le\lfloor k/4\rfloor$ limit that couples rate to measurement robustness. The paper proves a one-round decoding correctness condition and compares single-shot performance to repeated-round baselines, demonstrating that enhancing the BCH-design distance of $g(z)$ can achieve single-shot fidelity comparable to multi-round schemes at moderate blocklengths, while highlighting a structural bottleneck: high quantum rate reduces stabilizer redundancy. Numerical results on small coprime BB codes validate the algebraic bounds and show practical single-shot advantages under BP+OSD decoding, informing algebraic design rules for future 2BGA codes in measurement-limited architectures.

Abstract

Bivariate bicycle (BB) codes are a prominent class of quantum LDPC codes constructed from group algebras. While the logical dimension and quantum distance of \emph{coprime} BB codes are known to be determined by a greatest common divisor polynomial $g(z)$, the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial $g(z)$ dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance ($d_S$) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.

Figures (5)

• Figure 1: Schematic view of a CSS code and its X-syndrome code. The rows of $H_X$ define X-type stabilizers $S_i^X$ acting on data qubits. Noiseless syndrome patterns $s$ lie in the code $C_{\mathrm{S}}^X=\mathrm{im}(H_X)$ and satisfy additional relations $R_X s=0$ coming from stabilizer redundancy.
• Figure 2: Comparison of decoding schedules. Left: Standard repetition ($R=3$) relies on temporal majority voting. Right: Single-shot ($R=1$) uses spatial redundancy to clean errors immediately, eliminating wait times.
• Figure 3: Performance of the classical syndrome codes $C_{\mathrm{S}}^X$.
• Figure 4: Logical error rate $P_L$ vs physical error rate $p$ (with $q=p$).
• Figure 5: Logical error rate as a function of measurement rounds $R$.

Theorems & Definitions (14)

• Definition 1: Syndrome codes
• Lemma 1: Basic properties
• proof
• Remark 1: Noise model vs circuit-level noise
• Theorem 1: Redundancy and syndrome codes
• proof
• Theorem 2: BCH bound for $C_{\mathrm{S}}^X$
• proof
• Theorem 3: Singleton bound for $d_{\mathrm{S}}$
• proof