Constructing Quantum Convolutional Codes via Difference Triangle Sets
Vahid Nourozi, David Mitchell
TL;DR
This work addresses constructing quantum convolutional codes (QCCs) with guaranteed distance under the stringent requirement of symplectic orthogonality between $X(D)$ and $Z(D)$. It introduces a constructive workflow that begins with strong difference triangle sets (DTS) to build a classical self-orthogonal convolutional code (CSOC) described by $X(D)$, then obtains a companion $Z(D)$ via a simple index-reflection map that preserves the DTS structure and ensures commutation by design. The resulting framework yields a search-free method with provable distance properties and low memory, demonstrated through numerical results for rates $R=1/3$, $2/4$, and $3/5$, validating self-orthogonality in the polynomial domain. The approach enables low-latency, streaming-friendly QCCs suitable for quantum communication and fault-tolerant architectures, with scalable encoding/decoding compatible with sliding-window schemes.
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.