Citation: Schumacher, Benjamin, and Michael D. Westmoreland. "Approximate quantum error correction." Quantum Information Processing 1 (2002): 5-12.
Web: https://arxiv.org/abs/quant-ph/0112106
Tags: Computer-scientific, Error-correcting-codes, Approximate-structure
In this article, the authors make the first systematic study of approximate quantum error correction. By approximate quantum error correction, we mean a code for which there is a recovery map which approximately composes with the encoding and noise channels to give the identity. One of the main contributions is to argue that in natural setting, such approximate quantum error correction is always possible. The way they do this is by considering the "coherent information" passes through the noise channel. It was proved earlier by Schumacher and Nielsen that a recovery channel exists if and only if this coherent information is maximized. The authors show that if it is approximately maximized then there exists an approximate recovery scheme.
This paper is quite early, and as such the authors still need to spend some time laying the foundations. For instance, they need to lower bound the relative entropy in terms of the trace distance. This is now a standard result, but it was introduced in this paper.
Prior to this work, there had been one article broadly related to approximate quantum error correction, where a specific code was introduced:
> Barnum, Howard, and Emanuel Knill. "Reversing quantum dynamics with near-optimal quantum and classical fidelity." Journal of Mathematical Physics 43.5 (2002): 2097-2106.
There has since been a good amount of work related to approximate quantum error correction:
> Faist, Philippe, et al. "Continuous symmetries and approximate quantum error correction." Physical Review X 10.4 (2020): 041018.
> Sang, Shengqi, Timothy H. Hsieh, and Yijian Zou. "Approximate quantum error correcting codes from conformal field theory." Physical Review Letters 133.21 (2024): 210601.
> Kim, Isaac H., and Michael J. Kastoryano. "Entanglement renormalization, quantum error correction, and bulk causality." Journal of High Energy Physics 2017.4 (2017): 1-19.
> Liu, Shuwei, et al. "Noise-strength-adapted approximate quantum codes inspired by machine learning." arXiv preprint arXiv:2503.11783 (2025).