Quantum information processing is now a well-evolved field of study with roots to quantum physics that has significantly evolved from pioneering works over almost more than a century. Today, we are at a stage where elementary forms of quantum computers and communication systems are being built and deployed. In this paper, we begin with a historical background into quantum information theory and coding theory for both entanglement-unassisted and assisted quantum communication systems, motivating the need for quantum error correction in such systems. We then begin with the necessary mathematical preliminaries towards understanding the theory behind quantum error correction, central to the discussions within this article, starting from the binary case towards the non-binary generalization, using the rich framework of finite fields. We will introduce the stabilizer framework, build upon the Calderbank-Shor-Steane framework for binary quantum codes and generalize this to the non-binary case, yielding generalized CSS codes that are linear and additive. We will survey important families of quantum codes derived from well-known classical counterparts. Next, we provide an overview of the theory behind entanglement-assisted quantum ECCs along with encoding and syndrome computing architectures. We present a case study on how to construct efficient quantum Reed-Solomon codes that saturate the Singleton bound for the non-degenerate case. We will also show how positive coding rates can be realized using tensor product codes from two zero-rate entanglement-assisted CSS codes, an effect termed as the coding analog of superadditivity, useful for entanglement-assisted quantum communications. We discuss how quantum coded networks can be realized using cluster states and modified graph state codes. Last, we will motivate fault-tolerant quantum computation from the perspective of coding theory. We end the article with our perspectives on interesting open directions in this exciting field.

量子信息处理现在是一个发展良好的研究领域，其根源在于量子物理学，在近一个多世纪的开创性工作中取得了显着的发展。今天，我们正处于构建和部署基本形式的量子计算机和通信系统的阶段。在本文中，我们从无纠缠和辅助量子通信系统的量子信息理论和编码理论的历史背景开始，激发了此类系统中量子纠错的需求。然后，我们从必要的数学基础开始，以理解量子纠错背后的理论，这是本文讨论的核心，从二元情况开始到非二元泛化，使用丰富的有限域框架。我们将介绍稳定器框架，以二进制量子代码的 Calderbank-Shor-Steane 框架为基础，并将其推广到非二进制情况，从而产生线性和可加性的广义 CSS 代码。我们将调查源自著名经典对应物的重要量子代码系列。接下来，我们概述了纠缠辅助量子 ECC 背后的理论以及编码和校正子计算架构。我们提出了一个关于如何构造有效的量子里德-所罗门码的案例研究，该码使非简并情况下的单例边界饱和。我们还将展示如何使用来自两个零速率纠缠辅助CSS代码的张量积代码来实现正编码率，这种效应被称为超可加性的编码模拟，对于纠缠辅助量子通信很有用。我们讨论如何使用集群状态和修改的图状态代码来实现量子编码网络。最后，我们将从编码理论的角度推动容错量子计算。我们以对这个令人兴奋的领域中有趣的开放方向的看法来结束本文。