# Time Complexity Analysis - Grover

Let’s learn why Grover's Algorithm runs in O(sqrt(N)) time.

## We'll cover the following

This is going to be a geometric proof to show that the number of **Grover iterations** we must do to achieve a high probability of measuring the **winner** state is bounded by $O(\sqrt{N})$.

## Equal superposition

We will carry forward what we’ve learned from the past lessons and start with the situation where some arbitrary state $|x\rangle$ could be measured from the superposition, and it lies in between $|w\rangle$ and $|E\rangle$ in the state space. The $|x'\rangle$ state is **orthogonal** to $|w\rangle$ and the angle between $|x'\rangle$ and $|E\rangle$ is $\Delta$

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