5.10 Exercises 5.10.1 Justification of the circuits of Fig. 5.4 1. Justify the upper circuit of Fig. 5.4. Show that the action of the cNOT gate on the tensor product 1 (|0) + 1))|0) gives an entangled state. 2. Let us assume that qubits are measured immediately after a cU gate. Show that the probabilities of finding the target qubit in the state (0) or 1) and its final states are the same as if the control bit were measured before the gate and the target bit were transformed or not according to whether the control bit was in the state (0) or 1). This observation allows the two-qubit gate cU to be replaced by a one- qubit gate acting on the target bit, which is an enormous technical simplification. However, it works only at the end of a calculation, not for an intermediate cu gate! 3. Show that the lower circuit of Fig. 5.4 constructs the Toffoli gate, with U = √σx.