Quantum Walk Search

What is Quantum Walk Search?

Quantum Walk Search is a search algorithm that leverages quantum computing principles, specifically quantum walks, to search through an unstructured database or graph more efficiently than classical algorithms. Quantum walks are the quantum analog of classical random walks and can achieve quadratic speedup in search tasks.

Algorithm for Quantum Walk Search

1. Initialization: Initialize the quantum state representing the search space.
2. Quantum Walk Operator: Apply the quantum walk operator that evolves the quantum state according to the structure of the search space.
3. Marking: Mark the target states by applying a phase shift to the target elements.
4. Amplitude Amplification: Use amplitude amplification to increase the probability of measuring the target state.
5. Measurement: Measure the quantum state to obtain the target element with high probability.

How does Quantum Walk Search work?

• Quantum Walk Search utilizes the principles of superposition and interference to explore the search space efficiently.
• The quantum walk operator spreads the probability amplitude over the search space, while the marking and amplitude amplification steps increase the probability of finding the target element.

Problem Description

Given an unstructured database or graph, implement the Quantum Walk Search algorithm to find the target element. If the element is not present, the algorithm should indicate that as well.

Examples

Example 1: Input: database = [1, 3, 5, 7, 9] target = 5 Output: 2

Example 2: Input: database = [2, 4, 6, 8, 10] target = 7 Output: -1

Your Task

Complete the function quantum_walk_search() which takes a list database and an integer target as input parameters and returns the index of the target element. If the element is not present, return -1.

Expected Time Complexity: $O(\sqrt{n})$ Expected Auxiliary Space: $O(n)$

Constraints

• $1 <= n <= 10^5$
• $1 <= database[i] <= 10^6$
• $1 <= target <= 10^6$

Implementation

Python

import numpy as np

def quantum_walk_search(database, target):
    n = len(database)
    steps = int(np.sqrt(n))
    
    # Initialize quantum state
    state = np.zeros(n)
    state.fill(1 / np.sqrt(n))
    
    # Quantum walk operator
    for step in range(steps):
        for i in range(n):
            if database[i] == target:
                state[i] *= -1
    
    # Amplitude amplification
    for i in range(n):
        state[i] *= -1
    state.fill(2 / n * np.sum(state))
    
    # Measurement
    probabilities = state ** 2
    index = np.argmax(probabilities)
    return index if database[index] == target else -1

# Example usage:
database = [1, 3, 5, 7, 9]
target = 5
print(quantum_walk_search(database, target))  # Output: 2

C++

#include <iostream>
#include <vector>
#include <cmath>
#include <algorithm>

int quantum_walk_search(std::vector<int>& database, int target) {
    int n = database.size();
    int steps = static_cast<int>(std::sqrt(n));
    
    // Initialize quantum state
    std::vector<double> state(n, 1.0 / std::sqrt(n));
    
    // Quantum walk operator
    for (int step = 0; step < steps; ++step) {
        for (int i = 0; i < n; ++i) {
            if (database[i] == target) {
                state[i] *= -1;
            }
        }
    }
    
    // Amplitude amplification
    double sum = std::accumulate(state.begin(), state.end(), 0.0);
    for (int i = 0; i < n; ++i) {
        state[i] = (2.0 / n) * sum - state[i];
    }
    
    // Measurement
    auto max_it = std::max_element(state.begin(), state.end());
    int index = std::distance(state.begin(), max_it);
    return (database[index] == target) ? index : -1;
}

int main() {
    std::vector<int> database = {1, 3, 5, 7, 9};
    int target = 5;
    std::cout << quantum_walk_search(database, target) << std::endl;  // Output: 2
    return 0;
}