Benchmarking Machine Learning Models for Quantum Error Correction

TL;DR

A new perspective to understand machine learning-based quantum error correction is provided, and a machine learning benchmark is curate to assess the capacity to capture long-range dependencies 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.

Figures (8)

• Figure 1: Illustration of syndrome extraction inspired by roffe2019quantum. We first add a redundancy qubit $|0\rangle_2$ on $|\psi\rangle_1$ to create a logical state $|\psi\rangle_L$. Then some errors (denoted $E$) are applied on ${}{|}\psi\rangle_L$ to get $E|\psi\rangle_L$. After that, we use stabilizer $Z_1Z_2$ controlled by ancilla qubit $A$ to extract the syndrome.
• Figure 2: Illustration of surface code (Section \ref{['sec:surface_code']}). The circles and squares represent the data and ancilla qubits, respectively. The red solid edges and blue dashed edges represent the controlled-X and controlled-Z operations controlled by ancilla qubits and acting on data qubits, respectively. As shown on the right, only $A_6$ and $A_7$ exhibit syndrome. For instance, $A_1$ does not have the syndrome, meaning the data qubits connected to $A_1$ are error-free. Based on the process of elimination, we can ascertain that only $D_7$ has an error.
• Figure 3: Formulation of Quantum error correction. (a). Real data contains two kinds of qubits: data qubits (circle) and ancilla qubits (square). (b). Only the ancilla qubits are observed. We use the grid structure and the values of ancilla qubits as the input feature; (c). We are interested in inferring the error types of data qubits (no error, X error, Z error or X&Z error).
• Figure 4: Illustration of U-Net ronneberger2015u. By utilizing convolution and downsampling in U-Net, we effectively capture information from distant ancilla qubits for QEC. Notably, the input and output dimensions remain consistent due to upsampling in U-Net.
• Figure 5: Illustration of GCN kipf2016semi and GCNII chen2020simple (purple dashed line). GCN stacks graph convolution layers and enlarges the receptive field to capture information from distant ancilla qubits for QEC. In each layer of GCN, each node aggregates messages from its neighbors. Based on GCN, GCNII incorporates the raw feature (purple dashed line).