Efficient Approximate Degenerate Ordered Statistics Decoding for Quantum Codes via Reliable Subset Reduction

TL;DR

The study introduces Reliable Subset Reduction (RSR), a BP-driven preprocessing step that fixes highly reliable qubits to drastically shrink the decoding problem for quantum stabilizer codes. It further integrates an approximate degenerate OSD (ADOSD) that leverages code degeneracy to prune the candidate set, and extends the pipeline to circuit-level noise via detector error models, yielding a scalable BP+RSR+ADOSD framework. The approach achieves strong performance across a range of codes (CSS and non-CSS, topological, and lifted constructions), often outperforming MWPM and LSD, with substantial reductions in problem size (often below 5% of the original) that enable higher-order OSD even for problems with >$10^4$ error variables. Of particular note are the circuit-level demonstrations where MBP$_4$+ADOSD$_4$ attains competitive thresholds (e.g., around 0.76%) and improves decoding efficiency, highlighting its practical relevance for robust quantum error correction in realistic settings.

Abstract

Efficient and scalable decoding of quantum codes is essential for high-performance quantum error correction. In this work, we introduce Reliable Subset Reduction (RSR), a reliability-driven preprocessing framework that leverages belief propagation (BP) statistics to identify and remove highly reliable qubits, substantially reducing the effective problem size. Additionally, we identify a degeneracy condition that allows high-order OSD to be simplified to order-0 OSD. By integrating these techniques, we present an ADOSD algorithm that significantly improves OSD efficiency. Our BP+RSR+ADOSD framework extends naturally to circuit-level noise and can handle large-scale codes with more than $10^4$ error variables. Through extensive simulations, we demonstrate improved performance over MWPM and Localized Statistics Decoding for a variety of CSS and non-CSS codes under the code-capacity noise model, and for rotated surface codes under realistic circuit-level noise. At low physical error rates, RSR reduces the effective problem size to less than 5\%, enabling higher-order OSD with accelerated runtime. These results highlight the practical efficiency and broad applicability of the BP+ADOSD framework for both theoretical and realistic quantum error correction scenarios.

Figures (8)

• Figure 1: A tree with nodes consisting of all bit strings of length 4 and with at most weight 2. The root node is 0000. A child node is obtained by flipping one of the zero bits from its parent node that come after the last bit that is one.
• Figure 2: MBP$_4$+ADOSD$_4$ decoding with various values of $\alpha$ for the $[[175,7,7]]$ LCS code. The curve MLE is taken and rescaled from ORM24. The notation $\alpha\in(1.6,-0.01,0.5)$ means that $\alpha$ is tested in the sequence $1.60,1.59,1.58,\dots,0.51,0.50$ with ${\mathrm{T}}=100$ and $\theta=0.999995$.
• Figure 3: MBP$_4$+ADOSD$_4$ decoding with various maximum number of iterations ${\mathrm{T}}$ for the $[[175,7,7]]$ LCS code with $\theta=0.999995$.
• Figure 4: AMBP$_4$ and MBP$_4$+ADOSD$_4$ decoding performance for various BB codes with ${\mathrm{T}}=100$ and $\theta=0.999995$.
• Figure 5: MBP$_4$+OSD$_4$-2, MBP$_4$+ADOSD$_4$, and AMBP$_4$ decoding performance for various $[[d^2,1,d]]$ rotated surface codes with ${\mathrm{T}}=100$ and $\theta=0.999995$.

Theorems & Definitions (11)

• Definition 1
• Proposition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 7
• proof
• Theorem 8
• proof