{ h ′ $_{1}$, · · · , h ′ r } in order to preserve the commutation relations (3)-(6). We call V$_{1}$$_{,}$$_{2}$ a unitary row operator . Later we will need unitary row operators that perform the multiplication of g ′ i to h ′ $_{j}$, the multiplication of ¯ Z ′ i to h ′ $_{j}$, and the multiplication of ¯ X ′ i to h ′ $_{j}$, respectively. These four types of unitary row operators are summarized in Table 1. When a different set of generators of the stabilizer group is chosen instead of { g ′ $_{1}$, · · · , g ′ r } , this is equivalent to performing a unitary row operator V , which comprises a series of unitary row operators of type 1 on { g ′ $_{1}$, · · · , g ′ r } . The operation of V on the simplified check matrix H ′ corresponding to { g ′ $_{1}$, · · · , g ′ $_{r}$, h ′ $_{1}$, · · · , h ′ r } is to multiply H ′ by a (2 n − 2 k ) × (2 n − 2 k ) matrix of the form If where R ′ $_{i}$s are elementary row operations, then It can be checked that MH ′ satisfies (12). If a set T = { t$_{1}$, · · · , t$_{c}$ } of c < r ancilla qubits are replaced by ebits, it is possible that after the operation of V , the group S ′ I = 〈 g$_{j}$ : j / ∈ T 〉 22 M =    M$_{Z}$ O O M$_{X}$    . M$_{X}$ = R$_{m}$R$_{m}$ − 1 · · · R$_{1}$, M$_{Z}$ = R T $_{m}$R T $^{m}$− 1 · · · R T $_{1}$. Type 1 . V h ′ $_{j}$V † = { h ′ $_{l}$h ′ $_{m}$, if j = l ; h ′ $_{j}$, if j = l . V g ′ $_{j}$V † = { g ′ $_{m}$g ′ $_{l}$, if j = m ; g ′ $_{j}$, if j = m . Type 2 . V h ′ $_{j}$V † = h ′ $_{l}$g ′ $_{m}$, if j = l ; h ′ $_{j}$, if j = l . V h ′ $_{j}$V † = h ′ $_{m}$g ′ $_{l}$, if j = m ; h ′ $_{j}$, if j = m . Type 3 . V h ′ $_{j}$V † = { h ′ l ¯ Z ′ $_{m}$, if j = l ; h ′ $_{j}$, if j = l . V ¯ X ′ $_{j}$V † = { g ′ l ¯ X ′ $_{m}$, if j = m ; X ′ $_{j}$, if j = m . Type 4 . V h ′ $_{j}$V † = h ′ l ¯ X ′ $_{m}$, if j = l ; h ′ $_{j}$, if j = l . V ¯ Z ′ $_{j}$V † = g ′ l ¯ Z ′ $_{m}$, if j = m ; Z ′ $_{j}$, if j = m . Table 1: Four types of unitary row operators