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where f ( l ) is a noise parameter of a long wavelength. Although this noise is inevitable since we need the su- perposition of macroscopically distinct logical bases, we can still show that | E$_{0}$ ( λ < 1) 〉 has no other instability than this inevitable one. In Fig. 3, we plot the second largest eigenvalue e$_{2}$ of the VCM versus N for | E$_{0}$ (0 . 5) 〉 . From this figure we obtain 

which means that | E$_{0}$ (0 . 5) 〉 is stable against all noises of the type 

except for the case of Eq. (4). 

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In short, we have seen that the Z$_{2}$ quantum memory can have superposition of macroscopically distinct states and it is stable against any local noise except for the inevitable one. 

As is seen in Fig. 4, the energy gap Δ E ≡ E$_{1}$ − E$_{0}$ be- tween the exact ground state and the first excited state for λ = 0 . 5 decays exponentially fast as N → ∞ . There- fore, it is physically allowed to take a linear combination of the exact ground state and the first excited state for sufficiently large N . 

From Eq. (3) and the analogy with the physics of the single two-level atom system, the rough picture of the first excited state | E$_{1}$ (0 . 5) 〉 is expected to be 

and therefore the superposition 

of the exact ground state and the first excited state is expected to have no superposition of macroscopically dis- tinct states. 

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Indeed, we plot e$_{1}$ versus N for | E ′ 0 〉 in Fig. 5. This figure shows that e$_{1}$ = O ( N $^{0}$), which means that the superposition of macroscopically distinct states in quan- tum memory disappears for sufficiently large system size. Therefore, the Z$_{2}$ quantum memory is of limited use as a small quantum memory. 

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It is worth mentioning that such a fast decay of the energy gap is a natural consequence of the locality of the Hamiltonian. According to Eqs. (3) and (5), the differ- ence between the exact ground state and the first excited state is the relative phase of | φ$_{±}$ 〉 . If the energy gap does not go to 0 for large N , we can distinguish the exact ground state and the first excited state by measuring the energy. If the Hamiltonian is a local one, i.e., it does not contain many-point correlation, this means that we can know the relative phase by measuring only few-point correlations, which obviously contradicts to the common sense of decoherence [19]. 

