With this computation, the measured result is always | 000 ⟩. Then a second Hadamard gate converts the superposition back into the pure state | 0 ⟩. First a Hadamard gate puts qubit 2 into a superposition of | 0 ⟩ and | 1 ⟩. The measured results should vary randomly between | 000 ⟩ and | 111 ⟩, as they did with the cat state of Eq. Then two CNOTs are used to correlate qubits 3 and 1 with qubit 2. A Hadamard gate is used to put qubit 2 into a superposition of | 0 ⟩ and | 1 ⟩. This is called an entangled state of qubits 2 and 3. The measured values for qubits 2 and 3 are both random, but they are correlated with each other. When the computation is carried out multiple times, the output should vary randomly between | 000 ⟩ and | 011 ⟩. ![]() When this superposition is used to control qubit 3, that qubit will also end up in a superposition. The Hadamard gate puts qubit 2 in a superposition of | 0 ⟩ and | 1 ⟩. In this diagram, there is a Hadamard gate on qubit 2, before qubit 2 is used to control qubit 3. This computation shows that using a phase-shift gate to change the phase of quantum amplitudes can indeed change the results of the calculation. Equations (14) and (15) show why this works. The net effect of the three gates has been to flip qubit 3 from | 0 ⟩ to | 1 ⟩. As in the previous case, the result of the calculation is perfectly definite, but now the result is always | 001 ⟩. But now the phase-shift gate R ̂ θ with θ = π is applied between the two Hadamard gates, shown by a box with π in it. 3(d), again two Hadamard gates are applied to the same qubit. In this case, the second Hadamard undoes the effect of the first one, so the result of the calculation should always be | 000 ⟩. As shown above, the Hadamard gate splits a single quantum amplitude into two but it can also put two quantum amplitudes back together into one. 3(c), two successive Hadamard gates are applied to the same qubit. ![]() Putting the N-qubit register into an equal superposition of all 2 N basis states by applying a Hadamard gate to each qubit is one of the important building blocks of many quantum algorithms. The result of the calculation should vary randomly between all eight possibilities, | 000 ⟩, | 001 ⟩, …, | 111 ⟩. Knowing the state of qubit 1, for example, gives no information about the states of qubits 2 and 3. This leaves each of the three qubits equally likely to be in the states | 0 ⟩ and | 1 ⟩, with no correlations between the qubits. 3(b), Hadamard gates are applied to qubits 1, 2, and 3. Therefore, the result of the calculation should vary randomly between the two possibilities, | 000 ⟩ and | 010 ⟩. (14), this puts qubit 2 in an equal superposition | 0 ⟩ and | 1 ⟩. 3(a), a Hadamard gate is applied to qubit 2.
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