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How to see that both are adder circuits in terms of conventional classical circuit logic ?

Strange, the first circuit uses QFT , while the second circuit is using ripple-carry architecture

adders

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The first one uses Fourier basis to represent numbers, so that addition and subtraction can just be translated into rotations of respectively sized angles. Look at Qiskit Quantum Fourier Transform tutorial, it gives an intuitive view on the number representations.

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