and They work as the logical bit flip and logical phase, re- spectively. The code space is also specified as the lowest- energy eigenspace of the two-body Hamiltonian It is obvious that two degenerate separable ground states of this Hamiltonian are macroscopically distinct with each other, and therefore the indistinguishability of logi- cal bases, | ˜ 0 〉 and | ˜ 1 〉 , is not destroyed by local errors. Ideally, the logical Hadamard operation which is the essential ingredient of various quantum in- formation processings, is realized by using the logical ˆ X operation, Eq. (1). However, such non-local operation is not easy to experimentally implement. Therefore, it is reasonable to try to manipulate the quantum memory in the local way: where λ is an external control parameter. From the Perron-Frobenius theorem [17], the ground state of this Hamiltonian is non-degenerate if λ = 0. It is also known that this Hamiltonian exhibits the quan- tum phase transition at λ = 1 [18]. Let us denote the exact ground state of ˆ H corresponding to the external parameter λ by | E$_{0}$ ( λ ) 〉 , and evaluate index p of | E$_{0}$ ( λ ) 〉 for various λ . In Fig. 1, the largest eigenvalue e$_{1}$ of the VCM versus N is plotted by changing the value of λ . This figure shows that which means that the Z$_{2}$ quantum memory has superpo- sition of macroscopically distinct states for values λ < 1 of the control parameter λ . In order to see the structure of the superposition, the probability distribution P ( M$_{z}$ ) of ˆ M$_{z}$ in | E$_{0}$ (0 . 5) 〉 is plot- ted in Fig. 2 for N = 13. This figure suggests that the ground state is, in a rough picture, a superposition of two macroscopically distinct states: where | φ$_{±}$ 〉 are some states satisfying 〈 φ$_{±}$ | ˆ M$_{z}$ | φ$_{±}$ 〉 ≃ ± N , respectively. According to Ref. [5], this means that | E$_{0}$ ( λ < 1) 〉 is un- stable against the local noise described by the interaction Hamiltonian By seeing the eigenvector of the VCM corresponding to the largest eigenvalue e$_{1}$ , we can also know that the additive operator which gives the maximum fluctuation in | E$_{0}$ ( λ < 1) 〉 is 3 ˆ Z ≡ ˆ σ$_{z}$ (1) . ˆ H$_{0}$ = − N ∑ l =1 ˆ B$_{l}$. | ˜ 0 〉 → 1 √$_{2}$ ( | ˜ 0 〉 + | ˜ 1 〉 ) | ˜ 1 〉 → 1 √$_{2}$ ( | ˜ 0 〉 − | ˜ 1 〉 ) , ˆ H = ˆ H$_{0}$ + λ N ∑ l =1 ˆ σ$_{x}$ ( l ) , (2) | E$_{0}$ (0 . 5) 〉 ≃ | φ$_{+}$ 〉 + | φ$_{−}$ 〉 , (3) • | E$_{0}$ ( λ ≥ 1) 〉 has p < 2 • | E$_{0}$ ( λ < 1) 〉 has p = 2, ˆ H$_{int}$ ≡ N ∑ l =1 f ( l )ˆ σ$_{z}$ ( l ) , (4) ˆ M$_{z}$ ≡ N ∑ l =1 ˆ σ$_{z}$ ( l ) . FIG. 2. The probability distribution P ( M z ) of ˆ M z in | E 0 (0 . 5) 〉 with N = 13. Similar structures are obtained for other values of N . FIG. 1. (Color online) e 1 versus N for | E 0 ( λ ) 〉 with various λ . From the bottom, λ =1.5, 1.4, 1.3, 1.2, 1.1, 1.0, 0.9, 0.8, 0.7, 0.6, and 0.5, respectively. Lines are guides to the eye.  2  3  4  5  6  7  8  9  10  11  3  4  5  6  7  8  9  10  11 e 1 N  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45 -15 -10 -5  0  5  10  15 P(M z ) M$_{z}$