Constructing Quantum Convolutional Codes via Difference Triangle Sets

2026-02-13

AI Summary

This work addresses the construction of quantum convolutional codes with prescribed minimum distance, low memory overhead, and sparse stabilizers. To this end, it introduces difference triangle sets (DTS) into the design of quantum convolutional codes for the first time: a classical self-orthogonal convolutional code is employed to generate the X(D) part, while the Z(D) component is constructed via the reflection indices of a DTS to satisfy the symplectic orthogonality condition. This approach yields an explicit and tunable framework that rigorously guarantees the desired minimum distance. Experimental results demonstrate that the proposed method can flexibly generate quantum convolutional codes of various rates, exhibiting both effectiveness and practicality.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

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.

Problem

• Quantum Convolutional Codes
• Difference Triangle Sets
• Symplectic Orthogonality
• Minimum Distance
• Stabilizer Construction

Innovation

• Quantum Convolutional Codes
• Difference Triangle Sets
• Symplectic Orthogonality
• Self-Orthogonal Convolutional Codes
• Sparse Stabilizers