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 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.
Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.
Quantum error correction (QEC) and fault-tolerant quantum computation represent one of the most vital theoretical aspects of quantum information processing. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic obstacle to the development of large-scale quantum computers. The introduction of quantum error correction in 1995 showed that active techniques could be employed to mitigate this fatal problem. However, quantum error correction and fault-tolerant computation is now a much larger field and many new codes, techniques, and methodologies have been developed to implement error correction for large-scale quantum algorithms. In response, we have attempted to summarize the basic aspects of quantum error correction and fault-tolerance, not as a detailed guide, but rather as a basic introduction. The development in this area has been so pronounced that many in the field of quantum information, specifically researchers who are new to quantum information or people focused on the many other important issues in quantum computation, have found it difficult to keep up with the general formalisms and methodologies employed in this area. Rather than introducing these concepts from a rigorous mathematical and computer science framework, we instead examine error correction and fault-tolerance largely through detailed examples, which are more relevant to experimentalists today and in the near future.
The presentation file consists of two parts. In Part 1: Overlooked incompatibility between finite fields and quantum error correction with its remedies, we give a simple counterexample to the popular definition of quantum stabilizer codes invoving finite fields, and review two mathematically correct ways to define them. In Part 2: Introduction to entanglement-assisted quantum error correction from pre-quantum mathematical literature, based on Part 1 we introduce entanglement-assisted quantum error correction by using as much finite fields and as little complex vectors/matrices as possible, which is hopefully more friendly to classical coding theorests. We also clarify the connection to pre-quantum mathematical literature. These slides will be used in a workshop held on November 2026 at Castelló University Institute of Mathematics and Applications (IMAC), Universitat Jaume I, Spain. Version 1 has redundant assumptions in the Witt extension theorem, which are removed in version 2. The other file consists of mathematical proofs omitted from the presentation. It is generated by AI.
The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.
Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles \nto surmount before it can become a reality. Among these hurdles is that of protection of \ninformation from random errors. A potential solution for this challenge is stabilizer codes, \nwhich are the analog of classical linear error-correcting codes, however, with an additional \naxis of error possibility. By and large, quantum computing is discussed in the standard \nqubit, or two-level, language, however, it is worth considering the case of qudits, or more \nthan two-level systems. Often times results follow simply from some form of algebraic \nextension: typically group theory or linear algebra. In this work we consider some features \nthat are not immediately apparent from that approach, more often appealing to physical \nintuition to guide our mathematical ideas, then proving these ideas using the language of \nqudit operators. \n \nHere we consider two particular previously unexplored ideas. The first idea is that of \nembedding and inscribing of codes into spaces of different sizes than the stabilizer code \nwas originally designed for. Here, we show that all codes can be embedded, and that for \ninfinitely many primes we can in fact guarantee that the distance is at least preserved–a \nsomewhat surprising result. The second idea is, in a way an application of those presented \nin the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage \nof relabelling of syndromes and the rapid increase in syndrome space upon using codes in \nlarge spaces. Both of these are somewhat useful in their current forms, however, with some \nadditional mathematical work could be turned into potentially very powerful tools for the \nprotection of quantum information. We finish off by, and along the way, mentioning various \nfuture directions to carry work on this.