Quantum convolutional codes: fundamentals

TL;DR

This paper introduces the theory of quantum convolutional codes, providing formalism, encoding circuits, error analysis, and decoding algorithms to enhance long-distance quantum communication.

Contribution

It presents a comprehensive framework for quantum convolutional codes, including polynomial formalism, efficient encoding, error propagation analysis, and maximum likelihood decoding.

Findings

Polynomial formalism for stabilizer groups

Linear gate complexity encoding circuit

Linear classical complexity error estimation algorithm

Abstract

We describe the theory of quantum convolutional error correcting codes. These codes are aimed at protecting a flow of quantum information over long distance communication. They are largely inspired by their classical analogs which are used in similar circumstances in classical communication. In this article, we provide an efficient polynomial formalism for describing their stabilizer group, derive an on-line encoding circuit with linear gate complexity and study error propagation together with the existence of on-line decoding. Finally, we provide a maximum likelihood error estimation algorithm with linear classical complexity for any memoryless channel.