The mathematics of error correcting quantum codes

Kalyanapuram Rangachari Parthasarathy

Markus Grassl

Abstract We show how quantum algorithms for encoding and decoding of quantum error-correcting codes (QECC) can be derived from the underlying mathematical structure. For this, we first characterize general quantum error-correcting codes, followed by the discussion of CSS codes and additive quantum error-correcting codes. For the last two families, examples are given that illustrate the constructions.

Markus Grassl

Yuichiro Fujiwara

Suppressing the effect of decoherence plays a vital role in the theory of quantum information processing.Despite many skepticisms about its feasibility, the existence of error correction schemes in the quantum do- main was proved in the last century, giving the birth to the field of quantum error correction.Since then, the new field has seen remarkably rapid progress in various directions including experimental realizations of quantum error-correcting codes.While quantum information science draws on many branches of physics, computer science, and mathematics, combinatorics is rapidly becoming an indispensable mathematical tool to the study of quantum error correction.In this note, we review some of the latest developments in quantum error correction where combinatorial design theory and coding theory play the central role.

Peter M Kosek

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.

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