Examples of minimal-memory, non-catastrophic quantum convolutional encoders

TL;DR

This paper introduces a technique to find minimal-memory, non-catastrophic quantum convolutional encoders, demonstrating significant memory reduction and enabling practical simulations of quantum turbo codes.

Contribution

We develop a method to construct minimal-memory, non-catastrophic quantum convolutional encoders, improving efficiency and enabling performance simulations of quantum turbo codes.

Findings

Successfully encoded the FGG code with just one memory qubit

Developed an online decoder for the minimal-memory encoder

Enabled simulation of quantum turbo code performance

Abstract

One of the most important open questions in the theory of quantum convolutional coding is to determine a minimal-memory, non-catastrophic, polynomial-depth convolutional encoder for an arbitrary quantum convolutional code. Here, we present a technique that finds quantum convolutional encoders with such desirable properties for several example quantum convolutional codes (an exposition of our technique in full generality will appear elsewhere). We first show how to encode the well-studied Forney-Grassl-Guha (FGG) code with an encoder that exploits just one memory qubit (the former Grassl-Roetteler encoder requires 15 memory qubits). We then show how our technique can find an online decoder corresponding to this encoder, and we also detail the operation of our technique on a different example of a quantum convolutional code. Finally, the reduction in memory for the FGG encoder makes it feasible to simulate the performance of a quantum turbo code employing it, and we present the results of such simulations.