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Proof of a finite threshold for the union-find decoder

Satoshi Yoshida, Ethan Lake, Hayata Yamasaki

TL;DR

A rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model is provided and a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size is proved.

Abstract

Fast decoders that achieve strong error suppression are essential for fault-tolerant quantum computation (FTQC) from both practical and theoretical perspectives. The union-find (UF) decoder for the surface code is widely regarded as a promising candidate, offering almost-linear time complexity and favorable empirical error suppression supported by numerical evidence. However, the lack of a rigorous threshold theorem has left open whether the UF decoder can achieve fault tolerance beyond the error models and parameter regimes tested in numerical simulations. Here, we provide a rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model. To this end, we develop a refined error-clustering framework that extends techniques previously used to analyze cellular-automaton and renormalization-group decoders, by showing that error clusters can be separated by substantially larger buffers, thereby enabling analytical control over the behavior of the UF decoder. Using this guarantee, we further prove a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size. We also show that this framework yields a finite threshold for the greedy decoder, a simpler decoder with lower complexity but weaker empirical error suppression. These results provide a solid theoretical foundation for the practical use of UF-based decoders in the development of fault-tolerant quantum computers, while offering a unified framework for studying fault tolerance across these practical decoders.

Proof of a finite threshold for the union-find decoder

TL;DR

A rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model is provided and a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size is proved.

Abstract

Fast decoders that achieve strong error suppression are essential for fault-tolerant quantum computation (FTQC) from both practical and theoretical perspectives. The union-find (UF) decoder for the surface code is widely regarded as a promising candidate, offering almost-linear time complexity and favorable empirical error suppression supported by numerical evidence. However, the lack of a rigorous threshold theorem has left open whether the UF decoder can achieve fault tolerance beyond the error models and parameter regimes tested in numerical simulations. Here, we provide a rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model. To this end, we develop a refined error-clustering framework that extends techniques previously used to analyze cellular-automaton and renormalization-group decoders, by showing that error clusters can be separated by substantially larger buffers, thereby enabling analytical control over the behavior of the UF decoder. Using this guarantee, we further prove a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size. We also show that this framework yields a finite threshold for the greedy decoder, a simpler decoder with lower complexity but weaker empirical error suppression. These results provide a solid theoretical foundation for the practical use of UF-based decoders in the development of fault-tolerant quantum computers, while offering a unified framework for studying fault tolerance across these practical decoders.
Paper Structure (2 sections, 13 theorems, 87 equations, 6 figures)

This paper contains 2 sections, 13 theorems, 87 equations, 6 figures.

Table of Contents

  1. Syndrome extraction circuit for the surface code
  2. Proof of Lemma \ref{['lem:sparsity']}

Key Result

Theorem 3

For the $[[d^2, 1, d]]$ rotated surface code, under the circuit-level local stochastic error model, there exist constants $p_\mathrm{th}>0$ and $\eta>0$ independent of $d$ such that, if the physical error rate $p$ is below the threshold $p<p_\mathrm{th}$, then the logical error rate $p_\mathrm{L}$ o

Figures (6)

  • Figure 1: Illustration of a rotated surface code with distance $d=3$. The $x$- and $y$-axes are used in the definition of the stabilizer extraction circuit in Supplementary Information \ref{['appendix:circuit']}.
  • Figure 2: The logical error can happen only if the extended UF cluster has a diameter larger than or equal to $d$.
  • Figure 3: (a) Illustration of the distance chain in Eq. \ref{['eq:distance_chain']}. (b) The graphical explanation of the upper bound \ref{['eq:fraction_upper_bound']} on the fraction of the edges occupied by level-$k"$ UF clusters.
  • Figure 4: An example of error locations where the greedy decoder causes a logical error for the distance $d=23$.
  • Figure S1: Syndrome extraction circuit for the surface code with distance $d=3$, where time flows from left to right. RX and RZ represent the reset operations to $\ket{+}$ and $\ket{0}$, respectively. MX and MZ represent the measurements in the $X$ and $Z$ bases, respectively.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Definition 1: Syndrome extraction circuit
  • Definition 2: Detector graph
  • Theorem 3: Threshold theorem for the UF decoder
  • Theorem 4: Quasi-polylogarithmic average runtime of parallel UF decoders
  • Theorem 5: Threshold theorem for the greedy decoder
  • Lemma 6: High probability bound on clustering of errors on edges of graphs
  • proof
  • Lemma 7: Threshold theorem on error clustering
  • proof
  • Theorem 8: Spacetime locality of the detector graph
  • ...and 21 more