Grover’s Algorithm

Grover (1996) devised a quantum algorithm for searching through an unsorted database with N entries in O(N1/2) time and using O(logN) storage space.

This particular algorithm cannot be solved in fewer steps than O(N) hit-and-trial classical methods of calculation.(because, in the worst case, the N-th member of the domain might be the correct member). It was later proved by Bennett, Bernstein, Brassard, and Vazirani that no quantum solution to the problem can evaluate the function fewer than O(N1/2) times. This makes Grover’s algorithm asymptotically optimal.

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Python Version of Grover’s Algorithm

import math

# Define the function to search for (returns True if the element is the desired one, False otherwise)
def search_func(element, desired_element):
    return element == desired_element

# Define the Grover's Algorithm function
def grovers_algorithm(database, desired_element):
    n = len(database)
    num_iterations = int(math.sqrt(n))
    
    # Initialize the amplitudes of all states to 1/sqrt(n)
    amplitudes = [1 / math.sqrt(n) for i in range(n)]
    
    # Perform the iterations
    for i in range(num_iterations):
        # Phase inversion
        for j in range(n):
            if search_func(database[j], desired_element):
                amplitudes[j] = -amplitudes[j]
        
        # Amplitude amplification
        mean = sum(amplitudes) / n
        amplitudes = [(2 * mean - amplitude) for amplitude in amplitudes]
    
    # Measure the state with the highest probability and return the corresponding index
    max_prob = max(amplitudes)
    index = amplitudes.index(max_prob)
    
    return database[index]

This implementation takes in a database (as a list of elements) and a desired element to search for. It first initializes the amplitudes of all states to 1/sqrt(n), where n is the size of the database. It then performs the iterations of the algorithm, which consists of a phase inversion step and an amplitude amplification step. Finally, it measures the state with the highest probability and returns the corresponding element in the database.

Note that this is a simple implementation and may not be optimized for large databases.