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Suppose after some computation we get a superposition like $\frac{1}{\sqrt{4}}(\left|00\right>+\left|00\right>+\left|10\right>+\left|11\right>)$. How do we merge the same term in this superposition. Simply adding two coefficients does not work here as the superposition should equal to $\frac{1}{\sqrt{2}}\left|00\right>+\frac{1}{\sqrt{4}}\left|10\right>+\frac{1}{\sqrt{4}}\left|11\right>$. For the general case, is there a formula to merge two same term in a superposition?

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Note that $ \dfrac{|00\rangle + |00\rangle + |10\rangle + |11\rangle}{\sqrt{4}} = \dfrac{2|00\rangle+ |10\rangle + |11\rangle }{2} = \begin{pmatrix}1 \\ 0 \\ 1/2 \\ 1/2 \end{pmatrix}$ is not a quantum state as it is not normalized.

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As stated by @KAJ226, a quantum computation cannot get the superposition you gave as an example because that superposition isn't a quantum state (because it's not normalized). But if it were normalized, then indeed all you do is add the coefficients as expected. You could modify your initial state to get something like.

$\frac{1}{\sqrt{4}}\left(\frac{\sqrt{2}}{2}|00\rangle+\frac{\sqrt{2}}{2}|00\rangle+|10\rangle + |11\rangle\right)$.

This is a normalized state and indeed you can just add the two coefficients. This is totally acceptable but a bit contrived. A much more common example is

\begin{align} H\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right) &= \frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle\right) \\ &= \frac{1}{\sqrt{2}}\left(\frac{2}{\sqrt{2}}|0\rangle+\frac{0}{\sqrt{2}}|1\rangle\right) \\ &=\frac{2}{\sqrt{4}}|0\rangle = |0\rangle \end{align} where going from the first line to the second line was obtained by just adding the coefficients

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