and the reference qubit. Thanks to the Jamiołkowski-Choi isomorphism [@J72; @C75], knowledge of this state is equivalent to knowledge of the single-logical-qubit channel $\cE^s_1$ which has been applied to the logical qubit, conditioned on the syndrome $s$ which was observed. The simulation also yields the probability $\Pr(s)$ of the chosen syndrome. This terminates one instance of the simulation.

To simulate concatenation, we repeat the above procedure 7 times, yielding 7 single-qubit logical channels $\cE_1^{s_j}$. We then repeat the above procedure one last time using the noise model $\bigotimes_{j=1}^7 \cE_1^{s_j}$, which describes the noise model seen by the second concatenation layer, conditioned on the syndromes of the first layer. This simulation results in a single-qubit logical channel $\cE^s_2$, where $s$ now denotes the collection of all level 1 syndrome as well as the level 2 syndrome: it comprises $7\times 6 + 6 = 48$ syndrome bits. Thus, we will sometimes refer to $s$ as the syndrome history.

After $\ell$ levels of concatenation, the average channel experienced by a logical qubit is $\overline \cE_{\ell} = \sum_s \Pr(s) \cE^s_{\ell}$. The range of this sum grows exponentially with $\ell$, so even for three levels of concatenation we are forced to sample the distribution $N$ 