Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Faster Optimal Decoder for Graph Codes with a Single Logical Qubit

Nirupam Basak, Goutam Paul

TL;DR

This work tackles the challenge of decoding graph codes, a stabilizer-code family built from graph states, where exact maximum-likelihood decoding is generally intractable. By showing that syndrome measurements map arbitrary Pauli noise to Pauli noise, the authors derive a simple Z-based correction that recovers the original state for graph codes with a single logical qubit. They then introduce a hierarchical decoder in which each level can be solved in polynomial time, with the lowest levels achieving optimal decoding. Numerical results on cycle graphs demonstrate substantial speedups over exact MLD (via MIP) while maintaining near-optimal performance at low hierarchy levels, indicating significant practical impact for scalable quantum error correction in networks. The framework sets the stage for extending to multi-qubit logical codes and other stabilizer families, broadening applicability to distributed quantum information processing.

Abstract

In this work, we develop an efficient decoding method for graph codes, a class of stabilizer quantum error-correcting codes constructed from graph states. While optimal decoding is generally NP-hard, we propose a faster decoder exploiting the structural properties of the underlying graph states. Although distinct error patterns may yield the same syndrome, we demonstrate that the post-measurement state follows a well-defined structure determined by the projective syndrome measurement. Building on this idea, we introduce a hierarchical decoder in which each level can be solved in polynomial time. Additionally, this decoder achieves optimal decoding performance at the lower levels of the hierarchy. This strategy avoids the need for full maximum-likelihood decoding of graph codes. Numerical results illustrate the efficiency and effectiveness of the proposed approach.

Faster Optimal Decoder for Graph Codes with a Single Logical Qubit

TL;DR

This work tackles the challenge of decoding graph codes, a stabilizer-code family built from graph states, where exact maximum-likelihood decoding is generally intractable. By showing that syndrome measurements map arbitrary Pauli noise to Pauli noise, the authors derive a simple Z-based correction that recovers the original state for graph codes with a single logical qubit. They then introduce a hierarchical decoder in which each level can be solved in polynomial time, with the lowest levels achieving optimal decoding. Numerical results on cycle graphs demonstrate substantial speedups over exact MLD (via MIP) while maintaining near-optimal performance at low hierarchy levels, indicating significant practical impact for scalable quantum error correction in networks. The framework sets the stage for extending to multi-qubit logical codes and other stabilizer families, broadening applicability to distributed quantum information processing.

Abstract

In this work, we develop an efficient decoding method for graph codes, a class of stabilizer quantum error-correcting codes constructed from graph states. While optimal decoding is generally NP-hard, we propose a faster decoder exploiting the structural properties of the underlying graph states. Although distinct error patterns may yield the same syndrome, we demonstrate that the post-measurement state follows a well-defined structure determined by the projective syndrome measurement. Building on this idea, we introduce a hierarchical decoder in which each level can be solved in polynomial time. Additionally, this decoder achieves optimal decoding performance at the lower levels of the hierarchy. This strategy avoids the need for full maximum-likelihood decoding of graph codes. Numerical results illustrate the efficiency and effectiveness of the proposed approach.
Paper Structure (9 sections, 3 theorems, 33 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 9 sections, 3 theorems, 33 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph States under Single-qubit Noise
  4. Correcting Arbitrary Noise on Graph State by Phase Correction
  5. Decoder for Graph Codes
  6. Numerical Results
  7. Conclusion and Future Work
  8. Conclusion
  9. Future Work

Key Result

Theorem 1

Let $s = (s_1, s_2, \cdots, s_n)$ denote the syndrome obtained from syndrome measurements performed on a graph state affected by arbitrary Pauli errors. The state can be corrected by applying a $Z$ operator to each vertex $v$ for which $s_v = -1$.

Figures (1)

  • Figure 1: Plot of logical error rate $p_L$ for a)$\llbracket5,1,3\rrbracket$, b)$\llbracket9,1,3\rrbracket$, c)$\llbracket11,1,3\rrbracket$ and d)$\llbracket11,1,5\rrbracket$ against physical error rate $p$. Error rates for different levels of the proposed decoder are shown. The logical error rates for the highest level coincide with those of MIP.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof