-1)3|111) (24.25) = |000) + |111). Quantum error correction Bit-flip error The syndrome diagnosis for bit-flip errors is basically the same as described in the earlier section. As in the phase-flip case, the diagnosis may proceed by blocks, considering successively the blocks A, B, C. As we have seen, the procedure consists of comparing qubit pairs i, j and j, k by means of Zij = ZiZ; and Zjk = ZiZk measurements. Considering any block U = A, B, C, assume for instance a bit-flip error happening on the second qubit of U, such that the channel outputs: (24.26) We ought to notice that a single bit-flip error in (|000) + |111)) / 2 corrupts not only one but both qubit terms in the sum, because the state is entangled, as we have mentioned ear- lier. Thus, there is no possibility to have an error pattern of the type (|010) + |111)) / 2 where the second term would remain uncorrupted. This is a nice feature of the Shor code, which is going to be exploited in the syndrome diagnostic. To show this, we first apply Z12 on block U to obtain: Z12/06), = =[[[00) (00] + |11) (11|) x I - + |10) (|00) |101) (24.27) Hence, the measurement (Z12)U (24.28) We notice here that the post-measurement state is the same as the measured state [0'), within an unobservable phase factor ei Hence, the possibility of continuing the 24.2 Shor code Table 24.3 Corrective actions in Shor code from bit- flip syndrome diagnostic based on measurements of (Z12,Z23) in entangled 3-qubit block U, and the prin- ciple of majority logic. (Z12)U (Z23)U Corrective action Flip 3rd qubit Flip 1st qubit Flip 2nd qubit syndrome measurements! Next, we can apply Z23 on block U to obtain, in turn: |101) [101) (24.29) Hence, the measurement Z23)U = (Z12)U = -1. (24.30) The above two measurements, each yielding - 1, are sufficient to diagnose that the second qubit in block U is corrupted. If we are not convinced, we may perform a third measurement with Z13, which, according to expectation, will yield +1 (this being left as an exercise). Such a result indicates that the f