in Eq. (24.54), the other Steane codeword |I) is generated by the bitwise addition of Z = 1111111 to each of the elements in C2/0 + C2, which yields the alternate definition: (24.56) + 1) x1 + + + 1)). Letting the bit variable a = 0, 1 we finally obtain a common definition for the Steane codewords: x3+a)x1+a)lx2+a)x3+a)). (24.57) We may simplify the above definition even further by setting the bit variables y1 = y2 X2 X3 which gives (24.58) (where the modulo-2 addition property U + = 0 is applied for any bit U = Xi, y, a = 0, 1). As we shall see next, the general definition of |à) in Eq. (24.57) represents the "magic formula" needed to build a complete Steane encoder circuit! Indeed, consider the first qubit in là), namely y1 + y2 + a). The most elementary subcircuit that can be implemented to generate ly1 + y2 + a) with y1, y2 = 0, 1 is shown in Figure 24.7. It is seen from the figure that the two Hadamard gates produce two |yi)=|+> states that can act as control qubits on the target qubit |a) through CNOT gates (recalling that ly) CNOT|x) = ly+x) = |x+y)). We observe that |a), which represents the originator's message qubit to be encoded, is not limited to the basis states |0) or |1); most generally, it can be of the form a) = a|0) + B|1), where, as usual, a, B are complex amplitudes satisfying |a|2 + 1B12 = 1. Based on the above, it is not at all difficult to conceive the full Steane-code encoding circuit shown in Fig. 24.8, which represents only one possible implementation out of many other variants, let alone the flexibility associated with the many other choices for generator matrix G in the (7,4) code. Note that the same circuit, as traversed from right to left, can be used for decoding, i.e., retrieving the message qubit |a) = a|0) + B|1) from the 7-qubit |â), as obtained from the recipient after implementing error correction. This concludes this chapter on quantum error correction. An eerie feeling may rise from realising that the above-described CSS codes only represent another conceptual subspace withi