codeword with a horizontal arrow with $r$ above. The first condition alone allows errors to build up over time as in the top figure, which would quickly lead to a failure. However provided $s<(2t+1)/3$, both conditions together ensure that errors in consecutive error correction rounds do not build up, provided each error correction round introduces no more than $s$ faults, which could remain true for a long time.[]{data-label="fig:ConditionTwoJustification"}](ConditionTwoJustification.png){width="45.00000%"}

Noise model and pseudo-threshold calculations {#subsec:NoiseAndNumerics}
---------------------------------------------

In \[sec:CircuitLevelNoiseFTEC\], we perform a full circuit level noise analysis of various error correction protocols. Unless otherwise stated, we use the following depolarizing noise model:

1.  With probability $p$, each two-qubit gate is followed by a two-qubit Pauli error drawn uniformly and independently from $\{I,X,Y,Z\}^{\otimes 2}\setminus \{I\otimes I\}$.

2.  With probability $\frac{2p}{3}$, the preparation of the ${|0\rangle}$ state is replaced by ${|1\rangle}=X{|0\rangle}$. Similarly, with probability $\frac{2p}{3}$, the preparation of the ${|+\rangle}$ state is replaced by ${|-\rangle}=Z{|+\rangle}$.

3.  With probability $\frac{2p}{3}$, any single qubit measurement has its outcome flipped.

4.  Lastly, with probability $\tilde{p}$, each resting qubit location is followed by a Pauli error drawn uniformly and independently from $\{ X,Y,Z \}$.

Some error correction schemes that we analyze contain a significant number of idle qubit 