1,2,3,4 defined in Eq. (24.4), but this time with an ambiguity in favor of a single-error event. This shows that the quantum repetition code is inadequate for higher-order error detection and correction. How about implementing the quantum repetition code in the phase-flip channel? In Chapter 11, we saw that the phase-flip channel transforms a qubit lq) = a|1) + B|0) into lq - or introduces a dephasing factor eir between the two complex amplitudes a, B. Clearly, the effect of this channel is to flip the states ++ and |-> into each other. Therefore, we can apply the same repetition-code principle as described previously, using, this time (and for instance), the 4-qubit encoding scheme: +)=1++++) (24.17) where it), 1-> have the meaning of the logical 1+), |-) qubits. In the basis representation {l+), |-)}, there is strictly no difference between this quantum repetition code and that applying to the bit-flip channel in the basis representation {|0), |1)}. But we may use the code in Eq. (24.17) in the latter basis by means of Hadamard gates. As we have seen in Chapter 15, the Hadamard gate H performs the following transformations: H|0) |+) H|1) =|-> (24.18) H|+)==00 H|-> = |1). The encoding circuit for this repetition code is similar to the one concerning the bit-flip channel (Fig. 24.1) with the inclusion of three Hadamard gates to effect this conversion, as shown in Fig. 24.3. The originator qubit 1q) = a|1) + B|0) is, thus, transformed into Quantum error correction [q)=a(0)+B1) (q)=a/t)+B-> Figure 24.3 Quantum circuit for encoding a qubit lq) into a 4-qubit repetition codeword 19) before transmission to a phase-flip channel. The error syndrome measurement can be performed with the same projector set Pi=1,2,3,4 as defined in Eq. (24.4), but with the 1, 0 symbols inside the bra-kets <-1/1.> being switched into +, -, namely, P and SO on. Another possibility is to pass each of the received codeword qubits through a Hadamard gate, corresponding to the operation which allows one to perform the error syndrome measurement 