01) + |0110110) + |0001111) Quantum error correction Table 24.6 Defining a rule for the codeword bits of C2 (Table 24.5); see text for description. X1 + X3 x1+xx++x [I) = 11111111 + C2) (24.54) + |1000110> |0100101) |0011100) |1110000) + |0101010) + |1001001) + To recall, to construct the Steane code {|0), |I)} we have used a generator matrix G = in systematic form, as shown in Eq. (24.52). As previously mentioned, there exist several other possibilities for defining the parity-check matrix of the (7, 4) Hamming code. It is left as an easy exercise to determine the Steane code {|0), |I)} obtained with a matrix G' = H'T different from that used in this chapter. It does not matter that the resulting codewords |0), |1) differ from those defined in Eqs. (24.53) or (24.54), because in any (n, k) block code the choice of generator and parity-check matrices is only a matter of convention. How can we realize a quantum circuit capable of encoding the above-defined [0), |I) Steane codewords, which represent finely crafted superpositions of eight 7-qubits? Short of a rule to predict the values of the individual qubits within each of these 7-qubit superpositions, the task of designing such a circuit seems rather complex! But here comes a bit of magic in the way we can define a rule for the two codewords. As a starting point, we have seen that each 7-qubit |Y) in the definition |0) refers to the block codeword Y, which is an element of the coset 0 + C2. To find the rule governing the bit values of Y, we first list the codewords in seven columns, as shown in Table 24.6. Because the code is in systematic form, the first four columns 1234 represent parity bits, while the last three columns 567 represent the message bits. If we call X1, x2, X3 these three bits, we observe from the table that the parity bits in columns 1234 are defined by the sums X1 + X2, X1 + X3, x2 + X3, and X1 + X2 + X3, respectively. The Steane codeword |0) is, thus, defined by the three message bits X1, x2, X3, as: (24.55) 24.4 Hadamard-Steane code As shown 