All the stabilizer codes of distance 3

TL;DR

This paper characterizes all stabilizer codes of distance 3 for qubits, providing necessary and sufficient conditions for their existence, and offers explicit constructions for optimal codes at any length.

Contribution

It establishes exact existence criteria for stabilizer codes of distance 3 and provides explicit constructions for optimal codes at arbitrary lengths.

Findings

Necessary and sufficient conditions for stabilizer codes of distance 3.

Explicit construction methods for optimal codes.

Complete classification of such codes based on length.

Abstract

We give necessary and sufficient conditions for the existence of stabilizer codes $[[n,k,3]]$ of distance 3 for qubits: $n-k\ge \lceil\log_2(3n+1)\rceil+\epsilon_n$ where $\epsilon_n=1$ if $n=8\frac{4^m-1}3+\{\pm1,2\}$ or $n=\frac{4^{m+2}-1}3-\{1,2,3\}$ for some integer $m\ge1$ and $\epsilon_n=0$ otherwise. Or equivalently, a code $[[n,n-r,3]]$ exists if and only if $n\leq (4^r-1)/3, (4^r-1)/3-n\notin\lbrace 1,2,3\rbrace$ for even $r$ and $n\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n\not=1$ for odd $r$. Given an arbitrary length $n$ we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.