The Grover Circuit

Let’s reiterate the steps of the algorithm that we have covered. Then we’ll convert the steps into the language of quantum gates and circuits.

1. Equal superposition

Mostly the first step is to create our superposition state $|E\rangle$. We need this to recreate our whole search space in the state space of our qubits. You’ve probably figured this out by now that we need to apply the Hadamard gate $H$ on each qubit. After application of $H$, the state of our system looks like this:

$$|\psi_1\rangle = H^{\otimes n}|\psi_0\rangle$$

$$|\psi_1\rangle = \frac{1} {\sqrt{2^{n}}}\sum_{x=0}^{2^n-1} |x\rangle$$