Recursive quantum convolutional encoders are catastrophic: A simple proof

TL;DR

This paper presents a simpler, group-theoretic proof that quantum convolutional encoders cannot be both non-catastrophic and recursive, highlighting a fundamental limitation in quantum coding theory.

Contribution

A simpler, group-theoretic proof demonstrating that recursive quantum convolutional encoders are necessarily catastrophic, clarifying a key theoretical limitation in quantum coding.

Findings

Quantum convolutional encoders cannot be both recursive and non-catastrophic.

The proof uses group theory to relate zero physical-weight cycles to the centralizer of a subgroup.

Entanglement-assisted encoders can bypass this limitation.

Abstract

Poulin, Tillich, and Ollivier discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both non-catastrophic and recursive. Non-catastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly "rather involved," and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic---we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its "finite-memory" subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli-Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is non-recursive if it is non-catastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders---the introduction of shared entanglement as a resource allows the above symmetry to be broken.