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What is the need to apply the Hadamard gate as the first step while designing the diffuser circuit in the implementation of Grover's algorithm? I know what the gate does but I cannot understand what specific purpose it serves in this case.

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In our Grover’s algorithm, we:

  1. Add the initial Hadamard gates ie put our circuit in equal superposition |s>
  2. Apply the oracle
  3. Apply reflection about |s>

For the reflection (diffuser), we need to implement U = 2|s><s| - 1 As we want to add negative phase to every state orthogonal to |s>, we implement as follows:

Transform |s> to |0> by applying Hadamard gates (this is why we use it!) Apply a circuit to add negative phase to states orthogonal to |0> Transform |0> back to |s> using Hadamard gates

Without the Hadamards, we cannot add the negative phase to states orthogonal to |0> as it needs to be converted first.

This link may help: https://qiskit.org/textbook/ch-algorithms/grover.html#2.-Example:-2-Qubits-

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The H gate is used to create an equal superposition of all computational states, each of which represents an item in the search region. Then the following circuits are applied to rotate the vector to the target.

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The Hadamard gate is used (in most descriptions) as a simplification to generate a superposition of the database.

Say we want to find a marked element in a database $D$. The simplification is based on the assumption that the database has a number of entries that's a power of $2$. Formally, one would use a operator $U_{D}$ that generates a superposition of all the elements in the database $D$.

For Example: Say the database consists the integers from $\{0, 1, ..., 7\}$. Then the Hadamard transform $H^{\otimes 3}$ performs the map $U_D: |0\rangle \mapsto \sum_{i = 0}^{7} |i\rangle$. Circuit with 0,1, ..., 7

If the database would instead be the integers $\{0, 1, ..., 6\}$, then one would use a different operator $U_D: |0\rangle \mapsto \sum_{i = 0}^{6} |i\rangle$. Circuit with 0,1, ..., 6, where $U_D$ is called "{0,...,6}".

(You can verify that the operators generate the databases by looking at the probability distribution after the operator)

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