Optimal Single-Shot Decoding of Quantum Codes

TL;DR

The paper tackles fault-tolerant quantum decoding with faulty syndrome measurements by recasting single-shot decoding as a joint source-channel coding problem. It introduces a syndrome error-correcting code constructed from low-weight redundant rows added to the CSS parity-check matrix, enabling resilience to measurement errors with a small number of rounds. The authors derive optimal decoding rules, including degenerate MAP and standard MAP, and demonstrate—in two short-code experiments on a $[[16,2]]$ product code and a $[[18,2]]$ toric code—that redundancy-based syndrome correction can outperform repetition while keeping stabilizer weights manageable. The work provides a practical path toward single-shot fault tolerance in CSS codes and highlights that, for the tested setups, degenerate MAP yields little advantage over classical MAP, motivating further exploration with more realistic error models.

Abstract

We discuss single-shot decoding of quantum Calderbank-Shor-Steane codes with faulty syndrome measurements. We state the problem as a joint source-channel coding problem. By adding redundant rows to the code's parity-check matrix we obtain an additional syndrome error correcting code which addresses faulty syndrome measurements. Thereby, the redundant rows are chosen to obtain good syndrome error correcting capabilities while keeping the stabilizer weights low. Optimal joint decoding rules are derived which, though too complex for general codes, can be evaluated for short quantum codes.

Figures (4)

• Figure 1: Classical error model. Pauli $X$ and $Z$ errors are assumed to be independent events, resulting in an associated binary error vector $\bm e_q=[\bm e_Z|\bm e_X]$. It can be seen as the output of a BSC with error probability $\epsilon$, where $\bm e_Z$ and $\bm e_X$ can be inferred independently. Therefore, the figure only shows one component where the subscripts $X$ and $Z$ are omitted. The vector $\bm e$ is never actually observed. Instead, a corrupted version $\tilde{\bm{z}}$ of the encoded syndrome $\bm{z}=\bm{e} \bm{H}^{\intercal} \bm{G}_{s}$ is measured. Measurement errors are modeled by a BSC with error probability $\delta$. The decoder provides an estimate $\hat{\bm{e}}$ of the error vector $\bm e$.
• Figure 2: Decoding failure rate versus error probability $\epsilon$ for the $[[16,2]]$ product code with $q = 0.013$ (red) and $q = 0.021$ (blue).
• Figure 3: Decoding failure rate versus error probability $\epsilon$ for the $[[16,2]]$ product code and $q = 0.013$.
• Figure 4: Decoding failure rate versus error probability $\epsilon$ for the $[[18,2]]$ toric code with $q = 0.013$.