Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 58 / 3 / c

Use the probability of failing to restore an arbitrary unknown input in the original encoded subspace as the error criterion. With no extraction fault, the bit-flip repetition code succeeds exactly on channel-error patterns of weight zero or one. Independent physical flips giveThe probability $B$ is the logical bit-flip probability for the ideal recovery. The preceding calculation shows that every occurrence of the specified internal fault leaves the code space. Independence of that fault therefore gives exact encoded-state failure with a faulty parity check:A single unencoded qubit has bit-flip error probability $p$. Thus the encoded scheme improves this error criterion precisely whenFor $0<p<1/2$, $p-B=p(1-p)(1-2p)>0$ and $A=(1-p{)}^2(1+2p)>0$. Rearranging givesEquality gives no improvement. At $p=0$ the unencoded channel is already perfect; at $p=1/2$ ideal recovery only ties it. For $1/2<p\le 1$, the stipulated majority correction is no better even without the internal fault, and this fault cannot improve it. Hence there is no nonnegative $q$ giving a strict improvement in those ranges.

The criterion concerns restoration of every possible logical state: a logical bit flip can accidentally preserve a particular logical $X$ eigenstate, but that is not protection of an unknown input. The threshold assumes the displayed single fault location and ideal encoding, measurement and conditional correction. Adding another recovery circuit or comparing a different average-fidelity metric would define a different noise model or performance criterion.