Benchmarking Machine Learning Models for Quantum Error Correction
Abstract: Quantum Error Correction (QEC) is one of the fundamental problems in quantum computer systems, which aims to detect and correct errors in the data qubits within quantum computers. Due to the presence of unreliable data qubits in existing quantum computers, implementing quantum error correction is a critical step when establishing a stable quantum computer system. Recently, machine learning (ML)-based approaches have been proposed to address this challenge. However, they lack a thorough understanding of quantum error correction. To bridge this research gap, we provide a new perspective to understand machine learning-based QEC in this paper. We find that syndromes in the ancilla qubits result from errors on connected data qubits, and distant ancilla qubits can provide auxiliary information to rule out some incorrect predictions for the data qubits. Therefore, to detect errors in data qubits, we must consider the information present in the long-range ancilla qubits. To the best of our knowledge, machine learning is less explored in the dependency relationship of QEC. To fill the blank, we curate a machine learning benchmark to assess the capacity to capture long-range dependencies for quantum error correction. To provide a comprehensive evaluation, we evaluate seven state-of-the-art deep learning algorithms spanning diverse neural network architectures, such as convolutional neural networks, graph neural networks, and graph transformers. Our exhaustive experiments reveal an enlightening trend: By enlarging the receptive field to exploit information from distant ancilla qubits, the accuracy of QEC significantly improves. For instance, U-Net can improve CNN by a margin of about 50%. Finally, we provide a comprehensive analysis that could inspire future research in this field. The code is available in supplementary material.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: **Structure of Introduction**
> We reorganized Section 1 & 2, added Section 2.6 “Related Work” and made the Introduction more self-contained. Thanks!
**Inconsistent mathematical notations**
> We made all the math notations consistent, explain all the notations, and listed them in Table 4 in the Appendix for ease of understanding. Thanks!
**References are too few and outdated**
> We added more recent references. Thanks!
**Missing learning objective**
> We added the learning objective in Section 3.2, which is a cross-entropy loss. Thanks for pointing it out!
**Non-data-driven approaches**
> We report the results of minimum weight perfect matching (MWPM) [1], the state-of-the-art non-data-driven approach. Thanks!
**Explanation of evaluation metric**
> Overall accuracy refers to the accuracy of “all the data qubits” while the error correction rate refers to the accuracy of “all the data qubits with error”. We discussed them in Section 4.2. Thanks!
**Captions are not self-contained**
> We made captions self-contained. For example, in Table 1 and 2, p is the probability of X and Z error in data qubits. Thanks!
**Scalability analysis (distance d grows)**
> We conduct scalability analysis in Section 4.3. We observe that almost all the methods scale very well. For all the approaches, performance (overall accuracy and error correction rate) would not deteriorate as d (distance) grows. The performance would even increase for some GNN methods (e.g., GCN, APPNP, and Multi-GNN). The key reason is it is easier to mine patterns in larger graphs [2]. They also exhibit desirable scalability in terms of inference time. That is, growing $d$ would not increase inference time significantly.
**Availability of code**
> We added code in the supplementary material. Thanks!
**Figure issues**
> We redrawn some figures following your suggestions and fixed them. We also moved Figure 2 to the appropriate place. Thanks!
**Unclear or imprecise statement**
> All were clarified. Thanks!
**Missing analysis on statistical error**
> We conducted hypothesis testing to showcase the statistical significance of various methods over CNN, the best existing method. The results in Table 1,2 show that most methods achieve statistically significant improvement over CNN (pass the t-test, p-value<0.05).
**Broken reference**
> Fixed. Thanks!
**possibility of using parameter sharing to reuse models trained on smaller distances (e.g., smaller surface code settings) for capturing correlations at larger distances**
> We conducted experiment that reused the model trained on smaller distance, but we found the performance degraded significantly. The reason is that graph structures with different sizes exhibit different patterns.
**Required distance (surface code extent) to apply the present algorithms to realistic problems on current hardware**
> We set the distance of surface code from {3, 5, 7} (number of qubits is correspondingly 9, 25, 49) to mimic the realistic setup in the current quantum hardware (# qubits<50). We discuss it in Section 4.2. Thanks!
**Do these systems exhibit any symmetries to enhance the training of better models, possibly on fewer data?**
> Graph neural network (GNN)-based methods (GCN, GCNII, APPNP, MultiGNN) exhibit rotation-invariance while other methods (CNN, UNet) do not. That is, if we rotate the input by 90, 180, or 270 degrees, the GNN’s output would not change. So, the sample efficiency of GNN would be higher, as validated by the empirical results in Table 1&2. Thanks!
**The first work highlighting longer-range correlations between ancilla qubits?**
> Yes. To the best of our knowledge, this is the first work that emphasizes longer-range correlations between ancilla qubits. Thanks!
**Explanation on ‘node representations’**
> The goal of graph neural network is to learn a representation (embedding vector) for each node in the graph. We explained it. Thanks!
# References
1. Ben Criger and Imran Ashraf. Multi-path summation for decoding 2d topological codes. Quantum, 2:102, 2018.
2. Johannes Gasteiger, Stefan Weißenberger, and Stephan Günnemann. Diffusion improves graph learning. In
Advances in Neural Information Processing Systems, 2019.
Assigned Action Editor: ~Cho-Jui_Hsieh1
Submission Number: 1842
Loading