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The histogram of a circuit is the result of running the circuit (with measurement) many times, right?

Does this correspond to the squared amplitudes of the final state-vector? If not, why?

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Yes, probabilities shown in a histogram are squares of absolute values of coefficients in a state vector (i.e. probability amplitudes).

Note that the amplitudes are generally complex numbers, hence we square the absolute values and not the amplitudes themselves.

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