A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound

TL;DR

This paper develops a stabilizer-based framework to analyze quantum error-correcting codes and constructs an infinite family of non-degenerate codes that saturate the quantum Hamming bound for single-qubit errors ($t=1$). By selecting an abelian stabilizer with $a=j+2$ generators for block length $n=2^j$, it achieves $k=n-j-2$ and demonstrates an explicit $n=8$ example encoding 3 qubits, with a method to label all single-qubit Pauli errors via a homomorphism $f:{\cal G}\to({\bf Z}_2)^a$. The results show that the efficiency $k/n$ approaches 1 for large $n$, analogous to classical Hamming codes, and also establish that degenerate codes cannot beat the quantum Hamming bound for $t=1$. The work provides practical, constructive guidance for designing high-rate QECCs and highlights the stabilizer approach as a powerful tool for quantum error correction.

Abstract

I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode $k=n-j-2$ qubits in $n=2^j$ qubits and correct $t=1$ error.