Strengthened quantum Hamming bound

TL;DR

This paper introduces a new, tighter analytical bound for quantum error-correcting codes by combining and strengthening existing bounds using classical coding theory insights, applicable to both pure and impure codes.

Contribution

It develops a strengthened quantum Hamming bound by integrating classical Lloyd's theorem, providing improved bounds for quantum codes, including impure and stabilizer codes.

Findings

The combined bound is tighter than previous bounds.

The bound applies to impure codes and stabilizer codes.

For certain stabilizer codes, there is a significant improvement in logical qudit count.

Abstract

We report two analytical bounds for quantum error-correcting codes that do not have preexisting classical counterparts. Firstly the quantum Hamming and Singleton bounds are combined into a single tighter bound, and then the combined bound is further strengthened via the well-known Lloyd's theorem in classical coding theory, which claims that perfect codes, codes attaining the Hamming bound, do not exist if the Lloyd's polynomial has some non-integer zeros. Our bound characterizes quantitatively the improvement over the Hamming bound via the non-integerness of the zeros of the Lloyd's polynomial. In the case of 1-error correcting codes our bound holds true for impure codes as well, which we conjecture to be always true, and for stabilizer codes there is a 1-logical-qudit improvement for an infinite family of lengths.