a unitary evolution $U$ involving the qubit in state $\rho$ together with two additional “environmental” qubits initially in state $\ket 0$, i.e., $\cE(\rho) = \Tr_E\{U \rho\otimes \kb 00 U^\dagger\}$.

We are interested in studying a wide range of physical noise models, so we choose to generate random single-qubit CPTP maps $\cE$. Note that there is no natural notion of uniform distribution over the space of CPTP maps[^1]. We can generate random single-qubit noise models $\cE$ using the equivalence to three-qubit unitary matrices $U$ described above. Specifically in this study, we generate a three qubit Hamiltonian $H$ with gaussian distributed unit-variance entries and construct the unitary matrix $U=e^{i\delta H}$ where $\delta$ is a real parameters providing us with some handle on the noise “strength”.

To characterize the response of a fault-tolerant scheme to a given noise model $\cE$, we perform numerical simulations of the concatenated 7-qubit Steane code [@S96]. The Steane code encodes a single logical qubit and has minimal distance $d=3$, i.e., it can correct an arbitrary error on $t=\frac{d-1}2=1$ qubit. In a concatenated code [@F65], we encode each of the 7 qubits making up the code in separate error correcting codes, resulting in a code which encodes a single 