Grover's Algorithm

We want to find $x\in A$, given an Unitary Operator $U_f:=(-1{)}^f(x)|x⟩$, where $f(x):=1_A$. Grover’s algorithm solves this when $A$ has a single element by:

1. Getting our system into the superposition $|{\psi }_0⟩=H^{\otimes n}{|0⟩}^n$.
2. Apply the loop
   1. Apply our oracle
   2. Invert amplitudes around the mean through the diffusion operator $U_d:=2|\psi ⟩⟨\psi |-1$, where $|\psi ⟩:=\frac{1}{\sqrt{N}}{\sum }_x|x⟩$.
3. After $t$ iterations we moved our Quantum state in the direction of the solution by an angle of $2(t+1)\frac{\theta }{2}$. We want to choose $t$ to make this as close as possible to $\frac{\pi }{2}$, and then make a measurement to find the solution. I still need to write this out.
4. Read the result, where the amplitude now is mostly centred in the set around $f(x)=0$

• Do the above when the set $A$ has a known number $n>1$ of solutions, and an unknown number of solutions.