We give necessary and sufficient conditions for the existence of stabilizer codes $[[n,k,3]]$ of distance 3 for qubits: $n-kge lceillog_2(3n+1)rceil+epsilon_n$ where $epsilon_n=1$ if $n=8frac{4^m-1}3+{pm1,2}$ or $n=frac{4^{m+2}-1}3-{1,2,3}$ for some
integer $mge1$ and $epsilon_n=0$ otherwise. Or equivalently, a code $[[n,n-r,3]]$ exists if and only if $nleq (4^r-1)/3, (4^r-1)/3-n otinlbrace 1,2,3rbrace$ for even $r$ and $nleq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n ot=1$ 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.
We provide a systematic way of constructing entanglement-assisted quantum error-correcting codes via graph states in the scenario of preexisting perfectly protected qubits. It turns out that the preexisting entanglement can help beat the quantum Hamm
ing bound and can enhance (not only behave as an assistance) the performance of the quantum error correction. Furthermore we generalize the error models to the case of not-so-perfectly-protected qubits and introduce the quantity infidelity as a figure of merit and show that our code outperforms also the ordinary quantum error-correcting codes.
