How to obtain arbitrary distribution in quantum database


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?


Posted 2018-05-30T10:03:51.980

Reputation: 929



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.

Hasan Iqbal

Posted 2018-05-30T10:03:51.980

Reputation: 1 270