Some Progress on Quantum Error Correction for Discrete and Continuous Error Models

Quantum computing has increasingly gained attention for decades since it can surpass classical computing in various aspects. A crucial issue of quantum computing is how to protect information from noise interference such that the error rate during quantum information processing can be limited within an acceptable bound. The technique to address this issue is quantum error correction (QEC). Developing QEC technologies needs to define noise as formal error models and design QEC approaches for these error models. Discrete error models and continuous error models are two kinds of definitions of noise. The former assumes noise occurs independently and can be discretized into a set of basic errors. The former also has a tool, named quantum operations, to describe a specific error model. The latter describes that noise is continuous in time by using differential equations. In this paper, we categorize QEC approaches into three types according to the different error models: discrete error models, specific error models, and continuous error models. We also analyze the state-of-the-art QEC approaches and discuss some future directions. Furthermore, we propose the perturbed error models and their possible definitions, aiming to find the effect of the perturbation during quantum information processing.