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I was working on the Grover's algorithm and the most common example is for a unitary distribution in a quantum database, for example:

$|\psi\rangle = \frac{1}{2}|00\rangle + \frac{1}{2}|01\rangle + \frac{1}{2}|10\rangle + \frac{1}{2}|11\rangle.$

Is there a way to obtain arbitrary distribution (the above one is achieved by applying $H^{\otimes n}$ gates), e.g.

$|\psi\rangle = \frac{1}{3}|00\rangle + \frac{1}{4}|01\rangle + \sqrt{\frac{83}{144}}|10\rangle + \frac{1}{2}|11\rangle$ ? Does the structure of Grover's algorithm differ in such a case?

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According to this paper,

A significant conclusion from this solution is that generically the generalized algorithm also has $O(\sqrt{N/r})$ running time

Where 'r' is the number of marked states. By generalized, the authors meant a distribution with arbitrary complex amplitudes. So it seems to answer your question. That the modified initialization would still perform in the same way as the original one.

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