Constructing Quantum Convolutional Codes via Difference Triangle Sets

TL;DR

This paper presents a new method for constructing quantum convolutional codes using difference triangle sets, ensuring sparse stabilizers, small memory, and guaranteed minimum distance, with demonstrated numerical results.

Contribution

The paper introduces a novel construction of quantum convolutional codes based on difference triangle sets, linking classical CSOC design to quantum code properties.

Findings

Constructed QCCs with guaranteed minimum distance.

Demonstrated the construction for various code rates.

Provided numerical results validating the approach.

Abstract

In this paper, we introduce a construction of quantum convolutional codes (QCCs) based on difference triangle sets (DTSs). To construct QCCs, one must determine polynomial stabilizers $X(D)$ and $Z(D)$ that commute (symplectic orthogonality), while keeping the stabilizers sparse and encoding memory small. To construct Z(D), we show that one can use a reflection of the DTS indices of X(D), where X(D) corresponds to a classical convolutional self-orthogonal code (CSOC) constructed from strong DTS supports. The motivation of this approach is to provide a constructive design that guarantees a prescribed minimum distance. We provide numerical results demonstrating the construction for a variety of code rates.