if for sufficiently weak noise, the effective noise on the logical qubits is even weaker. More precisely, we say that an error correction protocol is a $t$-FTEC if the following is satisfied:

For $t = \lfloor (d-1)/2\rfloor$, an error correction protocol using a distance-$d$ stabilizer code $C$ is $t$-fault-tolerant if the following two conditions are satisfied:

1.  For an input codeword with error of weight $s_{1}$, if $s_{2}$ faults occur during the protocol with $s_{1} + s_{2} \le t$, ideally decoding the output state gives the same codeword as ideally decoding the input state.

2.  For $s$ faults during the protocol with $s \le t$, no matter how many errors are present in the input state, the output state differs from a codeword by an error of at most weight $s$.

\[Def:FaultTolerantDef\]

Here ideally decoding is equivalent to performing fault-free error correction. By codeword, we mean any state ${|\overline{\psi}\rangle} \in C$ such that $g{|\overline{\psi}\rangle} = {|\overline{\psi}\rangle} \thinspace  \forall \thinspace g \in \mathcal{S}$ where $\mathcal{S}$ is the stabilizer group for the code $C$. Note that for the second criteria in \[Def:FaultTolerantDef\], the output and input codeword can differ by a logical operator.

The first criteria in \[Def:FaultTolerantDef\] ensures that correctable errors 