idle qubits fail with probability $\tilde{p}$ and all other circuit operations (gates, preparations and measurements) fail with probability $p$, which recovers standard circuit noise when $\tilde{p}=p$. A detailed description is given in \[subsec:NoiseAndNumerics\].

The weight of a Pauli operator $E$ ($\text{wt}(E)$) is the number of qubits on which it has non-trivial support. We first make some definitions,

$$\begin{aligned}
    \mathcal{E}_{t} = \{ E \in \mathcal{P}_{n} | \text{wt}(E) \le t \},
    \end{aligned}$$ where $\mathcal{P}_{n}$ is the $n$-qubit Pauli group. \[Def:EpsilontSet\]

Given a stabilizer group $\mathcal{S} = \langle g_{1}, \cdots, g_{m} \rangle$, we define the syndrome $s(E)$ to be a bit string, with i’th bit equal to zero if $g_i$ and $E$ commute, and one otherwise. Let $E_{\text{min}}(s)$ be a minimal weight correction $E$ where $s(E)=s$. We say operators $E$ and $E'$ are logically equivalent, written as $E \sim E'$, iff $E' \propto g E$ for $g \in \mathcal{S}$. \[Def:LogEquivDef\]

An error correction protocol typically consists of a sequence of basic operations to infer syndrome measurements of a stabilizer code $C$, followed by the application of a Pauli operator (either directly or through Pauli frame tracking [@DA07; @Barbara15; @CIP17]) intended to correct errors in the system. Roughly speaking, a given protocol is fault-tolerant 