Asymptotics of the quantum Hamming bound for subsystem codes

TL;DR

This paper demonstrates that impure subsystem codes, such as Bacon-Shor codes, can violate the quantum Hamming bound, unlike pure stabilizer codes, revealing new limits on quantum error correction performance.

Contribution

It establishes that impure subsystem codes do not obey the quantum Hamming bound asymptotically, contrasting with pure codes and expanding understanding of quantum code limits.

Findings

Impure subsystem codes can violate the quantum Hamming bound.

Existence of arbitrarily long Bacon-Shor codes that violate the bound.

Pure codes of large length obey the quantum Hamming bound.

Abstract

Ashikhmin and Litsyn showed that all binary stabilizer codes - pure or impure - of sufficiently large length obey the quantum Hamming bound, ruling out the possibility that impure codes of large length can outperform pure codes with respect to sphere packing. In contrast we show that impure subsystem codes do not obey the quantum Hamming bound for pure subsystem codes, not even asymptotically. We show that there exist arbitrarily long Bacon-Shor codes that violate the quantum Hamming bound.