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Efficient Syndrome Decoder for Heavy Hexagonal QECC via Machine Learning

Debasmita Bhoumik, Ritajit Majumdar, Dhiraj Madan, Dhinakaran Vinayagamurthy, Shesha Raghunathan, Susmita Sur-Kolay

TL;DR

The paper tackles decoding for the heavy hexagonal QECC, showing that a gauge-aware FFNN-based syndrome decoder can exceed MWPM by a substantial margin across multiple noise models. By leveraging gauge equivalence, the authors reduce the number of error classes (with linear-search and rank-based variants) and improve both training efficiency and threshold performance, achieving up to about $p_{threshold}\approx 0.0245$ under depolarizing noise. The approach combines a four-layer FFNN trained on large labeled syndrome data with rigorous gauge-based preprocessing, yielding superior logical-error suppression and favorable scalability versus graph-based decoders. These results suggest gauge-aware ML decoders as a practical path toward fault-tolerant quantum computation with the heavy hexagonal code, especially in hardware scenarios favoring subsystem-code structures.

Abstract

Error syndromes for heavy hexagonal code and other topological codes such as surface code have typically been decoded by using Minimum Weight Perfect Matching (MWPM) based methods. Recent advances have shown that topological codes can be efficiently decoded by deploying machine learning (ML) techniques, in particular with neural networks. In this work, we first propose an ML based decoder for heavy hexagonal code and establish its efficiency in terms of the values of threshold and pseudo-threshold, for various noise models. We show that the proposed ML based decoding method achieves $\sim5 \times$ higher values of threshold than that for MWPM. Next, exploiting the property of subsystem codes, we define gauge equivalence for heavy hexagonal code, by which two distinct errors can belong to the same error class. A linear search based method is proposed for determining the equivalent error classes. This provides a quadratic reduction in the number of error classes to be considered for both bit flip and phase flip errors, and thus a further improvement of $\sim 14\%$ in the threshold over the basic ML decoder. Lastly, a novel technique based on rank to determine the equivalent error classes is presented, which is empirically faster than the one based on linear search.

Efficient Syndrome Decoder for Heavy Hexagonal QECC via Machine Learning

TL;DR

The paper tackles decoding for the heavy hexagonal QECC, showing that a gauge-aware FFNN-based syndrome decoder can exceed MWPM by a substantial margin across multiple noise models. By leveraging gauge equivalence, the authors reduce the number of error classes (with linear-search and rank-based variants) and improve both training efficiency and threshold performance, achieving up to about under depolarizing noise. The approach combines a four-layer FFNN trained on large labeled syndrome data with rigorous gauge-based preprocessing, yielding superior logical-error suppression and favorable scalability versus graph-based decoders. These results suggest gauge-aware ML decoders as a practical path toward fault-tolerant quantum computation with the heavy hexagonal code, especially in hardware scenarios favoring subsystem-code structures.

Abstract

Error syndromes for heavy hexagonal code and other topological codes such as surface code have typically been decoded by using Minimum Weight Perfect Matching (MWPM) based methods. Recent advances have shown that topological codes can be efficiently decoded by deploying machine learning (ML) techniques, in particular with neural networks. In this work, we first propose an ML based decoder for heavy hexagonal code and establish its efficiency in terms of the values of threshold and pseudo-threshold, for various noise models. We show that the proposed ML based decoding method achieves higher values of threshold than that for MWPM. Next, exploiting the property of subsystem codes, we define gauge equivalence for heavy hexagonal code, by which two distinct errors can belong to the same error class. A linear search based method is proposed for determining the equivalent error classes. This provides a quadratic reduction in the number of error classes to be considered for both bit flip and phase flip errors, and thus a further improvement of in the threshold over the basic ML decoder. Lastly, a novel technique based on rank to determine the equivalent error classes is presented, which is empirically faster than the one based on linear search.
Paper Structure (29 sections, 14 theorems, 8 equations, 10 figures, 12 tables, 3 algorithms)

This paper contains 29 sections, 14 theorems, 8 equations, 10 figures, 12 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Heavy Hexagonal Code
  4. Gauge generators
  5. Stabilizers
  6. Noise Models
  7. Bit flip error
  8. Phase flip error
  9. Depolarization noise
  10. Measurement error
  11. Stabilizer error
  12. Machine Learning based decoding
  13. Motivation for using Machine Learning based syndrome decoder
  14. Designing ML based decoder for heavy hexagonal code
  15. Methods to reduce error classes for heavy hexagonal code
  16. ...and 14 more sections

Key Result

Lemma 1

Given a codeword $\ket{\psi}$ such that $\ket{\psi} \equiv \Pi_{j} g_j \ket{\psi}$, where $\Pi_{j} g_j$ implies the product of one or more gauge operators $g_j$, any error $e$ acting on the codeword is equivalent to $e (\Pi_{j} g_j)$.

Figures (10)

  • Figure 1: Threshold (black dot) and Pseudo-threshold (red dot) of a MWPM decoder for Heavy Hexagonal QECC, with performance comparison for distance 3 (blue), distance 5 (orange), and distance 7 (green). The cyan straight line $y=x$ is for equal probabilities of physical qubit error and logical error.
  • Figure 2: Distance 3 heavy hexagonal code encoding one logical qubit: (a) the hexagonal structure, (b) the circuit illustration of the heavy hexagonal code with the CNOT gates. Here yellow, white and black circles represents data, flag and ancilla qubits respectively; black ancilla qubits are for measuring the $X$ (red face or plaquette) and $Z$ (blue face or strip) gauge generators. The product of two $Z$ gauge generators at each white plaquette forms a $Z$ stabilizer chamberland2020topological.
  • Figure 3: Circuits for measuring $X$ and $Z$ gauge generators in the heavy hexagonal code where $t_i$ denotes the $i^{th}$ time step. Two flag qubits (white circles) measure a $X$ gauge generator having a weight of 4 and one flag qubit measures a $Z$ gauge generator having a weight of 2 chamberland2020topological.
  • Figure 4: The stabilizers and gauge generators for a distance 5 heavy hexagonal code: the weight-4 $X$ gauge generators are in the red plaquettes, while the weight-2 $X$ gauge generators are on the upper and lower boundaries. Each blue strip denotes weight-2 $Z$ gauge generators. A vertical strip of two adjacent columns with $X$ gauge generators form an $X$ stabilizer. The weight-4 $Z$ gauge generators in the white plaquettes, and weight-2 $Z$ gauge generators on the left and right boundaries are themselves $Z$ stabilizers. chamberland2020topological.
  • Figure 5: (a) $X$ gauge equivalence for bit flip error: simultaneous errors on data qubits 4, 7 and 8 is equivalent to an error on data qubit 5, because by applying $X$ gauge operator $G3$ (consisting of $X$ operators on data qubits 4, 5, 7 and 8) on data qubits 4, 7 and 8, data qubit 5 has an error. (b) $Z$ gauge equivalence for phase flip errors: an error on data qubit 7 is equivalent to an error on data qubit 1 because by applying $Z$ Gauge operator $g4$ (consisting of $Z$ operators on qubits 4 and 7) followed by $Z$ Gauge operator $g1$ (consisting of $Z$ operators on qubits 1 and 4) on data qubit 7, data qubit 1 has an error.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Corollary 1
  • Theorem 2
  • Lemma 6
  • Lemma 7
  • ...and 4 more