ments as required to reach an unambiguous error diagnostic. The nice feature of the approach is that these projective measurements do not alter the codeword, i.e., they leave it in the same post-measurement state! In the 3-qubit coding example, let us see that it merely suffices to perform two such measurements in a row. Indeed, assume we obtain the two measurements (Z12) and (Z13). If (Z12) (Z13) = 1, there is no error and, therefore, no action is required. If (Z12) = 1 and (Z13) =-1, we have detected an error on the third qubit and, therefore, the corrective action is to flip it. We may reach the same conclusion for the measurement pair (Z12), (Z23) or (Z13), (Z23). Thus, any arbitrary choice of only two successive measurements of this type altogether yields 22 = 4 possible syndrome diagnostics, covering all patterns from zero error to one single error concerning any of the three qubits. The result is the same as using the measurement set Pi=1,2,3,4, except that only two projection operators instead of four have been used. In any case, it is important to recall that the output codeword |q') and the post-measurement states Pilq') or Zij|q'> are identical. It is now only a straightforward technical matter to describe how the Zij projectors defined in Eq. (24.19) can be realized. We recall from Chapter 16 the Z-gate, also referred to as the o3 or Oz Pauli matrix, whose action is to introduce a IT phase flip between the qubit amplitudes (see Table 16.2). It is left as an easy exercise to show that the measurement projector Zij, as defined in Eq. (24.19), in fact summarizes into the simultaneous application of Z onto the qubits i and j in the codeword to be analyzed. Namely, Z12=ZZI2I Z23IZZIZ2 (24.20) Z=ZOIQZ. We may write the above definitions in the compact form: Z12 = ZZI = ZZ, Z23 = IZZ3 = Z2Z3, and Z13 ZIZ3 = ZZ, with the indices in the right-hand side referring to the action of the operator onto the corresponding qubits, and with the identity matrix applying to the remaining qubit being overlooked. Shor code 