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Spanning Tree Matching Decoder for Quantum Surface Codes

Diego Forlivesi, Lorenzo Valentini, Marco Chiani

TL;DR

The spanning tree matching (STM) decoder for surface codes is introduced, which guarantees the error correction capability up to the code’s designed distance by first employing an instance of the minimum spanning tree on a subset of ancilla qubits within the lattice.

Abstract

We introduce the spanning tree matching (STM) decoder for surface codes, which guarantees the error correction capability up to the code's designed distance by first employing an instance of the minimum spanning tree on a subset of ancilla qubits within the lattice. Then, a perfect matching graph is simply obtained, by selecting the edges more likely to be faulty. A comparative analysis reveals that the STM decoder, at the cost of a slight performance degradation, provides a substantial advantage in decoding time compared to the minimum weight perfect matching (MWPM) decoder. Finally, we propose an even more simplified and faster algorithm, the Rapid-Fire (RFire) decoder, designed for scenarios where decoding speed is a critical requirement.

Spanning Tree Matching Decoder for Quantum Surface Codes

TL;DR

The spanning tree matching (STM) decoder for surface codes is introduced, which guarantees the error correction capability up to the code’s designed distance by first employing an instance of the minimum spanning tree on a subset of ancilla qubits within the lattice.

Abstract

We introduce the spanning tree matching (STM) decoder for surface codes, which guarantees the error correction capability up to the code's designed distance by first employing an instance of the minimum spanning tree on a subset of ancilla qubits within the lattice. Then, a perfect matching graph is simply obtained, by selecting the edges more likely to be faulty. A comparative analysis reveals that the STM decoder, at the cost of a slight performance degradation, provides a substantial advantage in decoding time compared to the minimum weight perfect matching (MWPM) decoder. Finally, we propose an even more simplified and faster algorithm, the Rapid-Fire (RFire) decoder, designed for scenarios where decoding speed is a critical requirement.
Paper Structure (12 sections, 4 theorems, 2 equations, 3 figures, 1 table)

This paper contains 12 sections, 4 theorems, 2 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Quantum Stabilizer Error-Correcting Codes
  4. Minimum Weight Perfect Matching
  5. Spanning Tree Matching Decoder
  6. Minimum Spanning Tree Phase
  7. Tree Matching Phase
  8. Error Correction
  9. Distance-preserving decoding
  10. Rapid-fire decoder
  11. Numerical Results
  12. Conclusions

Key Result

Lemma 1

Given a surface code, let us call $\mathcal{C}$ an arbitrary Pauli $\bm{Z}$ error chain connecting two sites defects $v_1$ and $v_2$. Then, the intersection of $\mathcal{C}$ with any column between $v_1$ and $v_2$ has cardinality $1 \pmod 2$. The intersection with any of the other columns has cardin

Figures (3)

  • Figure 1: Spanning tree matching decoder with a $[[85,1,7]]$ surface code. a) Three $\bm{Z}$ channel errors occur on the lattice. Exited ancillas are depicted in red. b) Two alternative MST obtained with the nearest ghost ancilla to the left (above) and to the right (below) boundary, respectively. c) Resulting $\mathcal{E}$ from the tree matching procedure.
  • Figure 2: Spanning tree matching decoder with a $[[85,1,7]]$ surface code. Both matched spanning trees have $w > t + 1$. Hence, the error correction is performed according to \ref{['eq:metricCol']}.
  • Figure 3: Logical error probability vs. physical error probability of the channel. Rotated $[[9,1,3]]$, and standard $[[13,1,3]]$, $[[41,1,5]]$, and $[[85,1,7]]$ surface codes over depolarizing channel.

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Theorem 1
  • proof