# Grover's algorithm

## Background

Amplitude amplification is a general purpose quantum algorithm, or subroutine, that can be used to obtain a quadratic speedup over a handful of classical algorithms. Grover’s algorithm was the first to demonstrate this speedup on unstructured search problems. Formulating a Grover's search problem requires an oracle function that marks one or more computational basis states as the states we are interested in finding, and an amplification circuit that increases the amplitude of marked states, consequently suppressing the remaining states.

Here, we demonstrate how to construct Grover oracles and use the

## Setup

Here we import the small number of tools we need for this tutorial.

Output:

Output:

```
'ibmq_mumbai'
```

## Step 1: Map classical inputs to a quantum problem

Grover's algorithm requires an oracle that specifies one or more marked computational basis states, where "marked" means a state with a phase of -1. A controlled-Z gate, or its multi-controlled generalization over $N$ qubits, marks the $2^{N}-1$ state (

Output:

### Specific Grover's instance

Now that we have the oracle function, we can define a specific instance of Grover search. In this example we will mark two computational states out of the eight available in a three-qubit computational space:

Output:

### GroverOperator

The built-in Qiskit

Output:

Repeated applications of this

Output:

### Full Grover circuit

A complete Grover experiment starts with a Hadamard gate on each qubit; creating an even superposition of all computational basis states, followed the Grover operator (

Output:

## Step 2: Optimize problem for quantum execution.

Output:

## Step 3: Execute using Qiskit Primitives.

Amplitude amplification is a sampling problem that is suitable for execution with the

Output:

## Step 4: Post-process, return result in classical format.

Output:

Output:

```
'0.19.1'
```

Output:

```
'1.0.0rc1'
```

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