Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 51 / 4 / d / Solution

Independent applications of the phase-flip channel are a classical mixture of the eight physical Pauli Z gate error patterns. A specified weight-$k$ pattern has probability ${ϵ}^k(1-ϵ{)}^{3-k}$. The recovery from the preceding part succeeds on weight zero and one, and yields the logical Pauli X gate on weight two and three. There are three weight-two patterns and one weight-three pattern, so the logical failure probability isAveraging the corrected branches gives the complete decoded quantum channelIn particular, the leading logical error is $3{ϵ}^2$, rather than an error of first order in $ϵ$. For the stated range,Thus the failure probability is smaller than the physical error probability. The decoded channel is a bit-flip channel; it is not $D_{p_L}$, whose error operator would be $Z$. This distinction follows from the logical codeword exchange in the phase-flip repetition code.