Note also that a two-qubit Pauli error $P_{1}\otimes P_{2}$ arising at a two-qubit gate location counts as a single fault.

It is well known that with only a single measurement ancilla, a single fault in a blue CNOT of the stabilizer measurement circuit shown in \[fig:StabNonFT\] can result in a multi-weight error on the data block. This could cause a distance-$3$ code to fail, or more generally could cause a distance-$d$ code to fail due to fewer than $(d-1)/2$ total faults. We therefore say the blue CNOTs are *bad* according to the following definition:

A circuit location in which a single fault can result in a Pauli error $E$ on the data block with $\mathrm{wt}(E) \ge 2$ will be referred to as a bad location. \[Def:BadErrorDef\]

[0.25]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of 