If there exists a standard [[ n + c, k, d ]] quantum code, then we may not be achieving the maximum boost to our error correcting power from the c ebits of shared entanglement. We expect added entanglement in general to increase the error-correcting ability of a quantum error-correcting code such that the EAQEC code is not equivalent to any standard quantum stabilizer code, and indeed this turns out to be possible by our encoding optimization pro- cedure. (Note that this is not always possible—the smallest examples of the [[3 , 1 , 3; 2]] and [[4 , 1 , 3; 1]] codes are both equivalent to the standard [[5 , 1 , 3]] QECC, and this is the best that can be done.) 

We now consider how added entanglement affects standard quantum stabilizer codes. Suppose { g ′ $_{1}$, g ′ $_{2}$, · · · , g ′ r } forms a set of independent generators of a stabilizer group S for an [[ n, k, d ]] quantum stabilizer code C ( S ), and U$_{E}$ is the encoding unitary operator obtained by Wilde’s encoding circuit algorithm [12], such that U$_{E}$Z$_{1}$U † E = g ′ $_{1}$, · · · , U$_{E}$Z$_{r}$U † E = g ′ $_{r}$. The logical operators are ¯ Z ′ 1 = U$_{E}$Z$_{r}$$_{+1}$U † $_{E}$, · · · , ¯ Z ′ k = U$_{E}$Z$_{n}$U † $_{E}$, ¯ X ′ 1 = U$_{E}$X$_{r}$$_{+1}$U † $_{E}$, · · · , ¯ X ′ k = U$_{E}$X$_{n}$U † $_{E}$. Suppose entanglement between the encoder and decoder is available, and a set T of c ancilla qubits are replaced by ebits. An EAQEC code is obtained. This introduces c simplified generators h ′$_{j}$ = U$_{E}$X$_{t}$$_{j}$U † $_{E}$’s, t$_{j}$ ∈ T , j = 1 , · · · , c , to the generating set of the stabilizer group S . As we will examine in detail below, the encoding unitary operator is not uniquely defined. The EAQEC code defined by S ′ = 〈 g ′ $_{1}$, · · · , , g ′ $_{r}$, h ′ $_{1}$, · · · , , h $^{′$_{c}$}$〉 may gain higher error-correcting ability by modifying the encoding operator. 

We first discuss the case c = r , where the generators h ′$_{i}$ ’s are symplectic partners of g ′ $_{i}$’s, respectively. We will treat the case c < r later, by optimizing the choice of c linearly independent generators from the group 〈 h ′ $_{1}$, · · · , , h $^{′$_{r}$}$〉 . 

## 3.1 Selecting Symplectic Partners and Logical Operators 

Since the symplectic partners of g ′ $_{1}$, · · · , g ′ r are not unique, we now explain how to select these partners such that the minimum distance of the EAQEC codes is higher than the 

