Minimum-Weight Parity Factor Decoder for Quantum Error Correction

TL;DR

HyperBlossom presents a certifying framework for MLE decoding of quantum LDPC codes by formulating decoding as a Minimum-Weight Parity Factor problem on decoding hypergraphs and extending the blossom algorithm to hypergraphs. It couples a primal MWPF solver with a dual LP solver, augmented by clustering and relaxing to achieve near-linear average-time decoding, while providing proximity bounds on optimality. The software Hyperion demonstrates substantial improvements in logical error rate over MWPM and BPOSD across several codes and noise models, along with scalable runtime up to large code distances. This work unifies existing graph-based decoders (UF, MWPM, HUF) under a single mathematical framework, enabling certifiable, high-accuracy decoding with tunable speed-accuracy trade-offs suitable for diverse QEC architectures.

Abstract

Fast and accurate quantum error correction (QEC) decoding is crucial for scalable fault-tolerant quantum computation. Most-Likely-Error (MLE) decoding, while being near-optimal, is intractable on general quantum Low-Density Parity-Check (qLDPC) codes and typically relies on approximation and heuristics. We propose HyperBlossom, a unified framework that formulates MLE decoding as a Minimum-Weight Parity Factor (MWPF) problem and generalizes the blossom algorithm to hypergraphs via a similar primal-dual linear programming model with certifiable proximity bounds. HyperBlossom unifies all the existing graph-based decoders like (Hypergraph) Union-Find decoders and Minimum-Weight Perfect Matching (MWPM) decoder, thus bridging the gap between heuristic and certifying decoders. We implement HyperBlossom in software, namely Hyperion. Hyperion achieves a 4.8x lower logical error rate compared to the MWPM decoder on the distance-11 surface code and 1.6x lower logical error rate compared to a fine-tuned BPOSD decoder on the $[[90, 8, 10]]$ bivariate bicycle code under code-capacity noise. It also achieves an almost-linear average runtime scaling on both the surface code and the color code, with numerical results up to sufficiently large code distances of 99 and 31 for code-capacity noise and circuit-level noise, respectively.

Figures (30)

• Figure 1: Example decoding hypergraph of rotated surface code with depolarizing noise. Black circles are data qubits. Red and white circles are ancilla qubits.
• Figure 2: The HyperBlossom framework is based on relaxing the \ref{['eqs:ILP']} formulation (Left) of \ref{['eqs:MWPF']} (§\ref{['ssec:mwpf']}) into an \ref{['eqs:LP']} problem (Middle) and solving the latter's dual formulation (\ref{['eqs:DLP']}) (Right) along with the \ref{['eqs:MWPF']} problem. This approach is inspired by the blossom algorithm but formulates the problem on the decoding hypergraph, instead of the syndrome graph.
• Figure 3: Visualization of \ref{['eqs:DLP']} solution $\vec{y}$ on the decoding hypergraph $G = (V, E)$ with a uniform edge weight of $w_e$. For each \ref{['def:invalid']} subgraph $S \in \mathcal{O}$, we visualize its dual variable $y_S$ as colored segments occupying a $y_S/w_e$ portion of each edge $e \in \delta(S)$. By definition, \ref{['def:tight-edges']} are those fully occupied, e.g., $T = \{e_0, e_2, e_4\}$ highlighted in solid color instead of the greyed colors.
• Figure 4: Overview of the HyperBlossom algorithm. The Primal phase solves the \ref{['eqs:MWPF']} problem, while the Dual phase solves the \ref{['eqs:DLP']} problem. They exchange information through a narrow interface. This interaction is inspired by the blossom algorithm.
• Figure 5: An example of batch relaxing. The radius of the circle centered at a vertex $v_i$ represents the corresponding dual variable $y_S, S = (\{v\}, \varnothing)$. (a) The initial \ref{['eqs:DLP']} solution has \ref{['def:tight-edges']}$T_1 = T = \{ e_0, e_1, e_2, e_3 \}$. We find a \ref{['def:relaxer']}$R_1[T_1]$ that increases $y_{S_0}$, decreases $y_{S_1}$, and leaves others unchanged. We have $\mathcal{R}_1 = \{ e_1 \}$. (b) Given the remaining \ref{['def:tight-edges']}$T_2 = T_1 \setminus \mathcal{R}_1 =\{ e_0, e_2, e_3 \}$, we find a \ref{['def:relaxer']}$R_2[T_2]$ with $\mathcal{R}_2= \{ e_3 \}$. (c) For the remaining \ref{['def:tight-edges']}$T_3 = T_2 \setminus \mathcal{R}_2 = \{ e_0, e_2 \}$, we find a \ref{['def:trivial-direction']}$\Delta\vec{y}[T_3]$ to increase the dual sum: growing $y_{S_4}$. (d) We compose these \ref{['def:directions']}$R_1[T_1]$, $R_2[T_2]$ and $\Delta\vec{y}[T_3]$ into a single \ref{['def:useful-direction']}$\Delta'\vec{y}[T]$ using \ref{['theorem:cascaded-relaxing']} (\ref{['algo:compose']}).

Theorems & Definitions (38)

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