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8

You won't find the barrier in quantum computing textbooks because it isn't a standard primitive of quantum information theory like unitary gates and quantum circuits. The barrier as a directive for circuit compilation to separate pieces of a circuit so that any optimizations or re-writes are constrained to only act between barriers (and if there a no ...

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Look at the picture of layer #9. It tells you explicitly how to group the qubits. There's a pair (q0,q5), another pair (q3,q8) and the rest of the qubits (q1,q2,q4,q6,q7,q9,q10,11). To see the relevance, look back to the circuit. Start with qubits 0 and 5. You can see that there's a two-qubit gate between them, but there are no two-qubit gates going from one ...

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Measurements cannot produce an imaginary result. So if you want to measure an imaginary part, you need a suitable transformation before you measure. I haven't looked into the details of the mentioned operations but I'm sure that is what they do. On the last part of your question: the operation $R_x (\pi /2)$ can be visualized on the Bloch sphere by a ...

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Here's one decomposition: It was made by decomposing a controlled S (which is easier to think about because it only phases; it's diagonal) and then converting the basis of the target by conjugating with Hadamards. It generalizes in-place to any $\text{CNOT}^{2t}$:

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