Self-attention U-Net decoder for toric codes

TL;DR

The paper tackles decoding toric codes in the NISQ era by introducing SU-NetQD, a multilevel self-attention U-Net decoder that combines a low-level data-qubit error corrector with a high-level logical-error detector. It represents syndromes and recovery operations as 3D tensors on the torus, uses circular padding and data augmentation to respect toroidal topology, and applies transfer learning to scale across code distances. Empirically, SU-NetQD outperforms the MWPM decoder across depolarizing, biased depolarizing, and circuit-level noise, achieving higher thresholds (up to $p_c\approx0.231$ under extreme bias) and lower logical error rates, with the high-level decoder enhancing MWPM performance. These results indicate a practical, scalable approach to quantum error correction that could improve the reliability of near-term quantum devices.

Abstract

In the NISQ era, one of the most important bottlenecks for the realization of universal quantum computation is error correction. Stabiliser code is the most recognizable type of quantum error correction code. A scalable efficient decoder is most desired for the application of the quantum error correction codes. In this work, we propose a self-attention U-Net quantum decoder (SU-NetQD) for toric code, which outperforms the minimum weight perfect matching decoder, especially in the circuit level noise environments. Specifically, with our SU-NetQD, we achieve lower logical error rates compared with MWPM and discover an increased trend of code threshold as the increase of noise bias. We obtain a high threshold of 0.231 for the extremely biased noise environment. The combination of low-level decoder and high-level decoder is the key innovation for the high accuracy of our decoder. With transfer learning mechanics, our decoder is scalable for cases with different code distances. Our decoder provides a practical tool for quantum noise analysis and promotes the practicality of quantum error correction codes and quantum computing.

Figures (12)

• Figure 1: (a) The structure of toric code and its stabilizer operators. The toric code is a square lattice with periodic boundary conditions, where each edge holds a qubit (represented by a circle). Solid lines represent the original lattice, and dashed lines represent the dual lattice. $A_v$ and $B_p$ are the toric code generators composed of tensor products of the Pauli operators $X$ (red) or $Z$ (blue), centered on the vertices and plaquettes of the original lattice, respectively. (b) Syndrome Measurement Circuit for toric code. Top: circuit for Z-type measurement. Bottom: circuit for X-type measurement.
• Figure 2: The illustration of toric code. The logical operators are global loops and cannot be shrunk to a point through continuous deformations. The local loops are trivial and can be shrunk to a single vertex or removed entirely by local operations.
• Figure 3: An example of (a) QEC errors, (b) the correct recovery chain, (c) the incorrect recovery chain Errors (left). A non-closed error chain causes the eigenvalues of the stabilizers at its ends to become -1, which results in the measurement outcomes on the corresponding measurement qubits becoming $|1\rangle$ as shown in (a). The quantum decoder needs to infer the recovery chain operations from the syndrome to eliminate it, but incorrect inference can lead to logical errors as shown in (b) and (c).
• Figure 4: The scheme of our low-level decoder's two-round iterative decoding process. The input is the original syndrome. The 1st round iteration provides a suggested recovery chain R1 and a syndrome S1 after the 1st round of error corrections. The 2nd round iteration provides a suggested recover chain R2 and the syndrome without errors S2. The output of the final suggested recovery chain is the xor of R1 and R2.
• Figure 5: The syndromes and recoveries map of toric code under bit flip noise. The measurement results on the measurement qubits form the training dataset for the low-level decoder (top right), with the blue color noting the Z measurements and the pink color noting the X measurements, which are measured in sequential time steps; the recovery operations on the data qubits form the dataset labels for the low-level decoder (bottom right).