# Why Grover's search return even distribution with half solution

The image below is the result of finding 4 correct answers from 8 candidates using qiskit. The 4 correct answers I set were 000, 001, 010, 011, but the graph shows the probability is incorrect (the result gives even distribution). Then, I tried on cases with more qubits, and they are all confused on half solution condition. I have no idea about why it occurs.

• You need to give us more details of what you were doing. Nov 5, 2021 at 9:36

If half the possible strings are solutions, then we start with $$|s\rangle$$ exactly halfway between the $$|\text{solution}\rangle$$ vector and the $$|\text{not-solution}\rangle$$ vector. When we then do the two reflections, we end up with a vector that is $$U_s U_w |s\rangle = \tfrac{1}{\sqrt{2}}(|\text{solution}\rangle - |\text{not-solution}\rangle)$$