\rangle$ close to 1 (see inset of Fig. \[Zav\_90thresh\_int\]).

Our analytical results describe the numerical data remarkably well. By fitting $f_c$ in Eq. to the simulation data, excellent agreement is found for an Ohmic bath (top panel of Fig. \[Zav\_90thresh\_int\]), while for a super-Ohmic bath (lower panel), analytics and numerics agree well for $L \gtrsim 64$. Furthermore, the fit yields values for $f_c$ of about $0.01 - 0.02$, which is reasonable in comparison to the upper bound $f_c = 0.11$ found for a model of uncorrelated errors (dashed-dotted lines in Fig. \[Zav\_90thresh\_int\]) [@Dennis2002]. See also the discussion in Sec. \[sec:lifetime\_formula\].

The lifetime $\tau$ can be compared to the physical time scales of single spin flips, $1/\gamma(0)$ and $1/\gamma(-2J)$. For instance, for the $L=256$ super-Ohmic case in Fig. 3 we obtain $\tau \gamma(-2J)\simeq 5\times 10^5 $, i.e., already for a moderate system size the lifetime $\tau$ of the memory is about a $10^6$ times longer than the single-spin lifetime. For quantum dots, the latter is typically in the range of milliseconds to seconds at about $100\,{\rm mK}$ [@Golovach2004; @Amasha2008].

![\[Zav\_90thresh\_int\] (Color online.) Thermal stability of the interacting memory. The data in the top (bottom) panel were obtained for an Ohmic (super-Ohmic, 