Research in the field of quantum algorithms and quantum error correction is progressing at an astounding rate. There are many good papers on both subjects, but reading even a few of these may seem a daunting task to the newcomer. The aim of this paper is to give a leisurely introduction to the basic theory of quantum error correcting codes without appealing to even the most basic notions in physics. Thus the article is not a substitute for important papers such as [12] or [7] but rather an advertisement for them. I would be pleased if, in addition, some readers view this as a useful companion article if and when they go on to read more substantial literature on the subject of quantum error correction. I present nothing new here. Rather, I give an elementary account of the important theorems and proofs which appear in these fundamental works using only undergraduate algebra and a bit of classical coding theory. In particular, I give a full proof of the Knill/Laflamme theorem as well as an elementary treatment of stabilizer codes. The goal is to make the literature dealing with this exciting new area more accessible to discrete mathematicians.
Research in the field of quantum algorithms and quantum error correction is progressing at an astounding rate. There are many good papers on both subjects, but reading even a few of these may seem a daunting task to the newcomer. The aim of this paper is to give a leisurely introduction to the basic theory of quantum error correcting codes without appealing to even the most basic notions in physics. Thus the article is not a substitute for important papers such as [12] or [7] but rather an advertisement for them. I would be pleased if, in addition, some readers view this as a useful companion article if and when they go on to read more substantial literature on the subject of quantum error correction. I present nothing new here. Rather, I give an elementary account of the important theorems and proofs which appear in these fundamental works using only undergraduate algebra and a bit of classical coding theory. In particular, I give a full proof of the Knill/Laflamme theorem as well as an elementary treatment of stabilizer codes. The goal is to make the literature dealing with this exciting new area more accessible to discrete mathematicians.
The goal of this paper is to provide a self contained detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes without venturing much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs - quite a few of of them new - that we present here. The connections between (a) quantum stabilizer codes and classical self-orthogonal codes and (b) between error correction and error detection have been established in details. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed here.
The goal of this paper is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves.
This paper serves as an introductory overview of quantum deletion error-correction codes, a burgeoning field within quantum coding theory. Covering foundational concepts, existing research, and open questions, it aims to be the first accessible resource on the subject. This paper contains basic definitions of terms so that readers can read it regardless of their background. This paper invites readers to explore this primer and take their initial steps into the realm of quantum deletion error-correcting codes research.
This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum-mechanical background to be able to understand how quantum error correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.
Error correcting codes is an area of mathematics that combines many theoretical concepts with a wide variety of applications.In this expository thesis, we will begin by introducing many of the fundamental ideas in classical error correcting codes.The goal of this is to introduce the reader to the field of coding theory and discuss a few of the wide range of classical codes that have been studied.The second section will discuss the recent findings in error correcting codes, namely their use in quantum error correcting codes.This section uses much more theoretical areas of math, when compared to the classical section, in the construction of quantum error correcting codes.Most of the research discussed in this section has been completed within the past six years.All of the research discoveries presented in this thesis have been previously known.
Noise is one of the central obstacles to building useful quantum computers, and quantum error correction (QEC) provides the framework for protecting quantum information against it. Unlike classical error correction, QEC must preserve fragile quantum states without copying them, measuring them directly, or destroying the information they encode. Driven by rapid progress in both theory and experiment, this challenge has grown into one of the most active areas of quantum information science. This tutorial gives a guided introduction to modern QEC, developing the core concepts of codes, syndromes, stabilizers, decoding, and fault tolerance before connecting them to major code families and current research directions. We cover both established constructions and newer developments, including topological and subsystem codes, bosonic and qudit codes, dynamical codes, and quantum low-density parity-check (qLDPC) codes. The emphasis is on building operational understanding: explaining not only what the main objects are, but how they are used in code design, error diagnosis, decoding, and fault-tolerant computation. The tutorial is intended for newcomers seeking a first path through QEC, as well as researchers looking for a coherent reference for the concepts, code families, and tools that arise in current work.