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For a superconducting qubit, we start with an LC circuit and "quantize" it, mapping the variables analogously to the variables for the harmonic oscillator.

In general, when are we allowed to quantize our Hamiltonian? How does a superconducting quantized circuit differ from the classical superconducting circuit?

It seems like the appropriate approximation is to take everything as quantum and then say that the quantum energy eigenstates are so close that they become classical, but the usual presentation I see is the other way around.

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