 architecture is shown in Fig. 19.10. From the figure, we observe the initialization of the register state R) = s) |-> from n |0) ancilla qubits, one |1) ancilla qubit, and n+1 Hadamard gates H. The circuit shown in the dashed-line box is the iterative Grover operator G = UsUw, with Us = 2/s) (s - I, which now acts on the register state |R) = s). It is only necessary to cascade the Grover circuit G shown in Fig. 19.10 kmax times to obtain the full GQDS circuit shown in Fig. 19.11. As the figure shows, the algorithm implementation is complete with a measurement (eigenvalue 2w) of the observable S with success probability p(w) 2 1. This description of the GQDS algorithm hopefully conveys the elegance of parallel computing through quantum circuits. Yet there are many questions remaining on the possible or even optimal circuit architectures for the "oracle" and Grover operators, Quantum database search algorithms let aside the practical physical implementation of the circuit at very large scales (N 2 103 -10°), - where the GQDS algorithm brings superior value. This reservation being stated, in a rising twenty-first century dominated by Internet communications, with its billions of host addresses, web pages, references, and databases, the physical possibility of implementing GQDS algorithms through practical quantum computers would, beyond any doubt, represent a most significant breakthrough. It would mean the possibility of elaborate search algorithms over the Internet with relatively fast access times and, in some applications, near-instantaneous access to practical information, however complex or remote! Exercises 19.1 (M): Show that for any n-qubit /x) = /X1X2...Xn), Xi E {0, 1}, and Hadamard gate H, we have: where . X1Z1 + X2Z2 + XnZn is a scalar product modulo 2. 19.2 (M): Given the orthonormal basis {/xx)}k=0...N-1, show that the transformation QFT, as defined by is unitary. 19.3 (B): Calculate the quantum Fourier transform of the qubit: 19.4 (B): Calculate the quantum Fourier transform of four EPR-Bell states 