n a grander space of the so-called stabilizer codes. Such codes, which are, by and large, not limited to error correction but rather are at the root of advanced Quantum error correction y, a a 0)+11 2 = y) Figure 24.7 Elementary circuit to generate the quantum state +y2++ a) given bit values y2 = 0, 1, as based on H (Hadamard) and CNOT gates. The states obtained at intermediate stages are shown and pointed to by dotted arrows. y1+y2+a) y1+y+a) y1+y2+y3 0) - H 0) - H 0) - H Figure 24.8 Possible circuit implementation for encoding the Steane code example, as defined in Eq. (24.58). quantum computing, represent yet another field by itself. Within the framework of these chapters, hopefully the reader has grasped the essentials to exercise his or her curiosity towards this direction. If the reader has gone this far in these chapters, there are no more conceptual difficulties involved, except for the never-ending work of specialization in the field. The last chapter, concerning quantum cryptography should complete our basic training towards the impossible or ill-defined grander picture, whether or not we elect to become a specialist! 24.5 Exercises Exercises 24.1 (B): Establish the following three operator definitions 12=ZZI=(00)(001+11)(11))0I- = - (|01) (01 + |10) (10|) I, - x <01 + |10) (10|), Z13 Z=0)(00I00)(0+1)(10I01)(1 = where Z is the Pauli matrix, z=(6-1) 24.2 (M): Given the 9-qubit quantum error-correction (Shor) code |000) show that a single qubit phase-flip error in any of the three blocks A, B, C can be detected through the successive measurements X123456 = = X1 X2X3X4X5X X456789 X4X5X6X7X8X9, where X is the Pauli matrix, x ( i l ) . 24.3 (M): Assuming a 3-qubit entangled block with the second qubit being flipped, show that the syndrome measurement Z13 of the last two qubits yields the expec- tation value (Z13) = =1. 24.4 (T): Establish that the set C of 16 elements shown in Table 24.7 (underscore "_ being for reading clarity), together with bitwise addition +, form a group (C,O). Then determine the differ