Quantum Convolutional Codes

Conversations and writings about quantum error correction (QEC) usually apply to quantum computation, specifically the correction of the bit flip and phase flip errors that arise on the current generation of quantum processors. For an overview of that application, our “Introduction to Quantum Error Correction” page summarizes a The Quantum Insider (TQI) article which addresses the need for QEC, the types of errors that arise, the difference between error mitigation and error correction, the concept of logical qubits, prototypical QEC codes, and more.

“Quantum Error Correction (QEC) refers to a set of  techniques used to protect quantum information from errors due to decoherence and other quantum noise.”
- from our “Quantum error correction” page

Despite the overwhelming popularity of that particular application, there are actually three high-level classifications of quantum code, which is short for quantum error correction code (QECC). Each is derived from quantum code theory, and these are:

• Stabilizer codes, which are generalized Shor codes such as the Steane code
• Topological codes, which use two-dimensional lattices of qubits  
• Convolutional codes, which have a memory structure and are used in communication

A quantum convolutional code, also known as a convolutional encoder linear code, is a type of quantum error correction code that is applied to quantum communication. Like the other types of codes, these codes protect quantum information from errors resulting from noise and decoherence. The fundamental difference between convolutional codes and other codes is that convolutional codes have a memory structure that allows the reuse of qubits during the encoding process. This is valuable when transmitting large numbers of qubits, because correcting the same number of errors with fewer qubits is more efficient. Furthermore, the encoding process is known to be less complicated, and using the code does not require any special network infrastructure.

What are Quantum Convolutional Codes?

The error rates of quantum computers are well known. But as the IEEE paper “Description of Quantum Convolutional Codes” notes, quantum networks are plagued by environmental noise throughout the entire process of transmitting qubits. As a result, quantum communication requires error correction, as well. Quantum convolutional codes are the quantum error correction codes for quantum communication.

The difference between convolutional codes and the other codes is the memory structure. Instead of a large block code, there is a simple repetitive pattern. This repeated encoding unitary gives rise to the memory structure, while reducing encoding and decoding complexity and enhancing error correction efficiency.

For further reading, a HAL open science paper titled “Quantum convolutional codes: fundamentals” delves into the theory behind these codes. The paper is also freely downloadable on arXiv.

Origins of Quantum Convolutional Codes

The theory behind these codes goes back more than two decades. They were first introduced by H. F. Chau in 1998. Within five years, Ollivier and Tillich added the stabilizer framework, encoding methods, and decoding methods for these codes.

These codes have since continued to evolve. Forney later introduced a “tail-biting” variant with greater efficiency and less complexity. At least two more variants, based on direct limit constructions, have been derived from Reed-Muller and generalized Reed-Solomon codes.  

Applications of Quantum Convolutional Codes

Convolutional codes might have applicability in quantum computing, after all they are a method for encoding quantum information. However, they are generally referenced with quantum communication and the transmission of quantum information. Their relative simplicity, though, might still make them practical for computing applications.

For more specific applicability, classical convolutional codes might serve as the best guide. Classical variants assure reliable data transfer across a wide variety of transmissions, including video, radio, mobile, and satellite. It is reasonable to assume that quantum convolutional codes might be applicable to the quantum analogues of those.