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{ 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. 

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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 

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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 〉 

