Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 / 2 / d / Solution

Under independent physical phase flips, an inner block suffers an effective phase flip of its encoded qubit exactly when it contains an odd number of errors. The binomial parity identity givesand hence the effective outer error probability isFor every $p<1/2$ and finite $m$, one has $q_m<1/2$, so outer majority-vote decoding succeeds with probability tending to one as $n\to \infty$. At $p=1/2$, one has $q_m=1/2$. The physical threshold is therefore still

For fixed $p\in (0,1/2)$, however, $q_m$ increases monotonically to $1/2$ as $m$ grows. Its distance from threshold isConsequently the majority-vote standardized separation is only of order $(1-2p{)}^m\sqrt{n}$, and the large-deviation exponent is of order $n(1-2p{)}^{2m}$. Thus larger $m$ leaves the threshold unchanged but makes the logical phase-error probability decay more slowly with $n$.