Description of Quantum Convolutional Codes

Quantum communication system is influenced by environment noise in the whole process. Quantum error-correcting codes can solve this problem efficiently. The polynomial representation of a quantum state is defined. Based on the structure of the classical convolutional codes, the basis state of the quantum convolutional codes is transformed into the multiplication of an information polynomial by the generator polynomial. A new method is proposed to encode and decode these quantum codes and the networks can be realized with the polynomial multiplication circuits. Finally, we provide a maximum likelihood error estimation algorithm with complexity growing linearly.

I. Introduction

Quantum error-correction is a basic technique for transmitting quantum information reliably over a noisy quantum channel. Many explicit constructions of quantum error-correcting codes[1], [2], [3], [4] have been proposed so far. In classical coding theory, practical systems have mostly used convolutional codes[5] rather than block codes, because convolutional codes are usually superior in terms of their performance-complexity tradeoff So, people also pay attention to construct quantum convolutional codes (QCCs)[6]. These codes are aimed at protecting a flow of quantum information over long distance communication. Grassl and Beth gave a general method to construct quantum shift registers in 2000[7]. In this paper, we present a technique for encoding and decoding tailored to quantum convolutional codes. The resulting quantum circuits are based on the quantum version of polynomial multiplication. Finally, a simple error estimation algorithm inspired by their classical analogue is provided.