Degenerate Quantum Codes for Pauli Channels

TLDR

Degenerate quantum error‑correcting codes can correct more errors than they uniquely identify, yet their properties remain poorly understood. The authors propose a heuristic to design degenerate quantum codes for high‑noise Pauli channels, enabling communication at rates unattainable by nondegenerate codes. The heuristic constructs codes by selecting stabilizer generators that exploit degeneracy to correct many errors at high noise rates. The study demonstrates a substantial performance gap between degenerate and nondegenerate codes, and identifies a channel where degenerate codes do not outperform the best nondegenerate code yet differs markedly from channels where nondegenerate codes are optimal.

Abstract

A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal.