Level-$k$ concatenated simulation: error $(cp)^{2^k}/c$ per location, size $O(d^k)$

Concatenated simulation at every level. There are constants $c>0$, $A$ and $d\in\mathbb N$ such that for every number of levels $k$ there are a block size $b$ and a decoder $\mathrm{dec}:\{0,1\}^b\to\{0,1\}$ with the following property: every ideal circuit $C$ on $n$ qubits has a physical simulation $C'$ on $n$ blocks of $b$ qubits with

and for every $0\le p$ with $cp<1$ and every independent stochastic noise model with fault rate at most $p$,

This is the statement that better gates can be used recursively to build even better gates: after $k$ levels each location fails with probability at most $(cp)^{2^k}/c$, at a cost of a factor $d^k$ in size. For $k=0$ it is the unencoded bound, and $k=1$ is the one-level statement.