# Slight modification of Grover's algorithm

Let $$S$$ be a set $$\{| j ⟩~|~j = 0, 1, · · · ,N − 1\}$$ with assuming $$⟨j_i|j_j⟩ = δ_{ij}$$. Thus, total number of quantum states in $$S$$ is $$N$$. Usual Grover’s algorithm is to find a state $$|w⟩ \in S$$ with $$\sqrt{N}$$ queries. This means that the oracle and diffuser should be repeated approximately $$\sqrt{N}$$ times.

Now, I want to change a situation slightly. Let us assume that the number of $$|j⟩$$ in $$S$$ is $$α_j$$. Thus, total number of quantum states in $$S$$ is $$\tilde{N} = \sum_{j=0}^{N-1} α_j$$. Of course, the number of the state $$|w⟩ \in S$$ is $$α_w$$. I want to find the value of $$α_w$$ and all $$|w⟩$$ quantum state efficiently by modifying the Grover’s algorithm. Is it possible? We assume that the total number of states $$\tilde{N}$$ is known, but $$α_j$$ is unknown.

First, your notations and terminologies suggest you are finding a 'state' among possible 'quantum states' $$\newcommand{\ket}[1]{\left|{#1}\right\rangle}\ket{0}, \dots, \ket{N-1}$$, but with possible duplication. Of course, if two quantum states are identical, then they are, well, identical: there cannot be any duplicates.
It is better to imagine Grover's algorithm as working with a classical logical predicate $$P$$: given some index $$i$$, $$P(i)=1$$ or $$P(i)=0$$. What Grover's algorithm is doing is to find some $$i$$ satisfying $$P(i)=1$$, when $$P$$ is given as an oracle.
Of course, you can define/implement this predicate in many different ways and hence use Grover's algorithm in different contexts. One way would be 'database search': you are given a database of $$N$$ items: $$D(i)$$ is the entry stored at the $$i$$th position. Now, you want to find a particular $$y$$. Rather, you want to find an index $$i$$ satisfying $$D(i)=y$$. This can be done by defining the predicate $$P$$ as $$P(i)=[D(i)=y]$$; in other words, $$P(i)=1$$ iff $$D(i)=y$$, and $$P(i)=0$$ iff $$D(i)\neq y$$.
In this scenario, Grover's algorithm does not say anything about uniqueness of items $$D(0), \dots, D(N-1)$$: it works without any restriction on uniqueness. In fact, Grover's algorithm only makes oracle queries to the predicate $$P$$; it does not care how this $$P$$ is implemented, or whether there is a database behind the implementation of $$P$$, with duplicates or not.