Timed bit flip during repetition-code syndrome extraction ($s=(e_2\oplus e_3,e_1\oplus e_2\oplus f)$)

Suppose the first ancilla records parity $q_2\oplus q_3$, then a fault of value $f$ flips $q_2$, and the second ancilla records parity $q_1\oplus q_2$. For incoming bit-error pattern $e$, the observed error syndrome is the displayed pair, while the final data error before recovery is $d=e\oplus (0,f,0)$. When $f=1$, the observed syndrome differs from the actual data syndrome by $(1,0)$. Applying the usual minimum-weight recovery leaves a nonzero syndrome, so the state is outside the code space even when the channel itself made no error. This is a timing-dependent failure of this simple extraction circuit, not a general obstruction to fault-tolerant quantum error correction.

For independent channel bit flips of probability $p$ and the specified extraction fault of probability $q$, require recovery of an arbitrary input in the original encoded subspace. Every extraction-fault event leaves nonzero error syndrome; without that fault, channel errors are corrected exactly at weight zero or one. Thus success probability is $(1-q)[(1-p{)}^3+3p(1-p{)}^2]$. Compared with a single unencoded bit-flip channel, strict improvement for $0<p<1/2$ requires $q<p(1-2p)/[(1-p)(1+2p)]$. A different decoder or an added later ideal recovery changes the performance criterion and must be specified separately.