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A High-Performance List Decoding Algorithm for Surface Codes with Erroneous Syndrome

Jifan Liang, Qianfan Wang, Lvzhou Li, Xiao Ma

TL;DR

Numerical results demonstrate that the proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.

Abstract

Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.

A High-Performance List Decoding Algorithm for Surface Codes with Erroneous Syndrome

TL;DR

Numerical results demonstrate that the proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.

Abstract

Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.
Paper Structure (20 sections, 1 theorem, 16 equations, 10 figures, 3 tables)

This paper contains 20 sections, 1 theorem, 16 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Surface codes and their decoding algorithms
  3. Motivation
  4. Contributions
  5. Organization
  6. Preliminaries
  7. Stabilizer codes and surface codes
  8. MWPM decoder
  9. The BP-OSD algorithm
  10. System model
  11. Proposed method
  12. Enhancing BP with a priori syndrome information
  13. OSD with refined LLRs
  14. Extracting quantum/syndrome errors
  15. Parameters and details
  16. ...and 5 more sections

Key Result

Proposition 1

Assume that the quantum channel error pattern $\bm{e}$ and the syndrome error pattern $\bm{s}$ are extracted from a valid virtual codeword $\bm{c}$. Then, the syndrome of the Pauli error pattern $\bm{e}$ matches the estimated syndrome $\bm{s}$, i.e., where $\mathbf{H}$ is the parity-check matrix of the quantum code, and $\odot$ denotes the symplectic inner product.

Figures (10)

  • Figure 1: The $[[n = 41, k = 1, d = 5]]$ surface code. In the lattice representation, $X$-type stabilizers are shown on the faces, while $Z$-type stabilizers are shown on the vertices.
  • Figure 2: System model. The quantum information is sent through a depolarizing channel, and the ideal measured syndrome is corrupted by a bit-flip channel (the single/double lines denote the quantum/classical information).
  • Figure 3: The error model of the quantum channel with depolarizing rate $p$ and syndrome channel with bit-flip rate $q$.
  • Figure 4: The procedure of quantum error correction with erroneous syndrome (the single/double lines denote the quantum/classical information).
  • Figure 5: The framework of the proposed BP-OSD algorithm.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Example 1
  • Example 2
  • Example 3
  • Proposition 1
  • proof