Unencoded circuits: output error at most $p\,|C|$

Errors accumulate at most linearly (unencoded circuits). Let $C$ be any circuit on a finite register and let $0\le p\le 1$. For every independent stochastic noise model with fault rate at most $p$,

where $P_{\mathrm{noisy}}(C)$ and $P_{\mathrm{ideal}}(C)$ are the outcome distributions of measuring all qubits after the noisy and the noiseless runs of $C$, and $|C|$ is the number of locations of $C$.

Without error correction, a circuit with $p(n)$ components is only reliable when $p\cdot p(n)$ is small; this is the baseline that fault tolerance improves on, and the same bound is the level-$0$ case of the concatenated construction.