Introduction to Quantum Error Correction with Stabilizer Codes

TL;DR

This paper introduces quantum error correction using stabilizer codes, covering basic concepts, examples, mathematical formalism, topological codes, and recent decoding approaches with neural networks, aimed at computer scientists and mathematicians.

Contribution

It provides an accessible introduction to stabilizer codes, including practical implementations and recent decoding techniques, bridging theory and application for newcomers.

Findings

Examples of simple error correcting codes without formalism

Mathematical details of stabilizer codes explained for non-mathematicians

Implementation of codes using OpenQASM and neural network decoding methods

Abstract

We give an introduction to the theory of quantum error correction using stabilizer codes that is geared towards the working computer scientists and mathematicians with an interest in exploring this area. To this end, we begin with an introduction to basic quantum computation for the uninitiated. We then construct several examples of simple error correcting codes without reference to the underlying mathematical formalism in order to develop the readers intuition for the structure of a generic code. With this in hand, we then discuss the more general theory of stabilizer codes and provide the necessary level of mathematical detail for the non-mathematician. Finally, we give a brief look at the elegant homological algebra formulation for topological codes. As a bonus, we give implementations of the codes we mention using OpenQASM, and we address the more recent approaches to decoding using neural networks. We do not attempt to give a complete overview of the entire field, but provide the reader with the level of detail needed to continue in this direction.