Theory Behind Quantum Error Correcting Codes: An Overview

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.