Threshold theorem for quantum computation

Threshold theorem for quantum computation. There are a threshold $p_{\mathrm{th}}>0$ and an exponent $c\in\mathbb N$ such that for every physical fault rate $0\le p<p_{\mathrm{th}}$ there is a constant $A$ with the following property.

For every number of logical qubits $n$, every depth $D$ and every target accuracy $0<\varepsilon<1$, there are a block size $b$ and a block decoder $\mathrm{dec}:\{0,1\}^b\to\{0,1\}$ such that every ideal circuit $C$ on $n$ qubits of depth $D$ can be simulated by a physical circuit $C'$ on $n$ blocks of $b$ qubits satisfying

and, for every independent stochastic noise model with fault rate at most $p$,

Here $|C|$ is the number of locations of $C$ (gates, preparations and idle steps), $P_{\mathrm{ideal}}(C)$ is the outcome distribution of measuring all qubits of the noiseless run of $C$, and $\mathrm{dec}_*P_{\mathrm{noisy}}(C')$ is the blockwise-decoded outcome distribution of the noisy run of $C'$.

This is the boxed statement of the source: a circuit with $p(n)$ gates can be simulated with error probability at most $\varepsilon$ using $O(\mathrm{poly}(\log(p(n)/\varepsilon))\,p(n))$ gates on hardware whose components fail with probability at most $p$, provided $p$ is below a constant threshold.

Formalization Note "Gates" of the source are counted as locations, idle steps included, as in the standard analyses where waiting qubits are also noisy. The source's $O(\log^c(p(n)/\varepsilon)\,p(n))$ is written non-asymptotically as $A(\log(|C|/\varepsilon)+1)^c|C|$. The block size and the decoder may depend on $n$, $D$, $\varepsilon$ and $p$ but not on the circuit $C$, which rules out decoders that encode the answer. The "reasonable assumptions about the noise" of the source are fixed as independent faults with arbitrary local fault channels.