Concatenated Threshold Under Local-Stochastic Correlated Noise

Apply [F3] to each level-$r$ rectangle. Its proof counts fault-witness sets and uses only the local-stochastic subset bound, so it requires no independence between child fault indicators. Thus each top-level circuit rectangle is bad with probability at most $p_r=A^{-1\mathrm{/}q}(A^{1\mathrm{/}q}\eta {)}^{(q+1{)}^r}$.

If $\eta =0$, each single-location fault event has probability zero by the local-stochastic subset bound; the finite union bound gives no faults almost surely, so level zero suffices. Otherwise put $\theta =A^{1\mathrm{/}q}\eta \in (0,1)$ and $c=-\log \theta >0$. Any $r$ with $(q+1{)}^r\ge c^{-1}\log (N{A}^{-1\mathrm{/}q}\mathrm{/}\delta )$ is sufficient; if the logarithm is nonpositive or its positive value is at most $c$, $r=0$ suffices. For fixed parameters, the least qualifying $r$ is $O(\log \log (N\mathrm{/}\delta ))$ as $N\mathrm{/}\delta \to \mathrm{\infty }$.

By [F11], the probability $\alpha$ that at least one of the $N$ top-level circuit rectangles is bad satisfies $\alpha \le N{p}_r$. For every fault pattern, [F9] gives a conditional channel, so its output on the fixed joint data input and external reference is a density operator. On the complementary event, recursive goodness gives at most $q$ bad children in every stage. The one-level contract [F8] and finite hypothesis [A1] permit the simultaneous recursive correction guarantee [F4] for the per-location channels allowed by [F1]. Its recursive final decoder takes every such hierarchical residual instrument to the ideal logical instrument. Thus every pattern with no bad top-level rectangle has decoded output ${\rho }_{\mathrm{i}\mathrm{d}}$. If $\alpha >0$, the normalized finite mixture of bad-pattern outputs is a density operator $\tau$; if $\alpha =0$, choose any density operator for $\tau$. Then ${\rho }_{\mathrm{o}\mathrm{u}\mathrm{t}}=(1-\alpha ){\rho }_{\mathrm{i}\mathrm{d}}+\alpha \tau$, and [F10] confirms that the weighted schedule output is a channel.

By homogeneity of trace distance and [F6], $D({\rho }_{\mathrm{o}\mathrm{u}\mathrm{t}},{\rho }_{\mathrm{i}\mathrm{d}})=\alpha D(\tau ,{\rho }_{\mathrm{i}\mathrm{d}})\le \alpha \le N{p}_r$. Applying [F7] to the final POVM bounds its output total-variation distance by $N{p}_r$, and every success event changes by at most that total-variation distance.

Because the gadget family is fixed and finite, choose a constant $\kappa \ge 1$ bounding both the number of elementary locations and the scheduled depth of every replacement gadget, including preparation, control, reset, and idle locations. Replacing each location of a schedule of size $S$ and depth $D$ gives size at most $S\kappa$ and depth at most $D\kappa$: replace disjoint locations of each original layer in parallel and pad their gadgets to the common depth bound. Induction gives size at most $S{\kappa }^r$ and depth at most $D{\kappa }^r$ at level $r$. Thus the multiplicative size and depth overheads are at most ${\kappa }^r$, which is a fixed power of $\log (N\mathrm{/}\delta )$ by step 1.2.