How to Implement Grover’s Algorithm on Dynex

This guide walks you through the implementation of Grover’s algorithm for integer factorization using quantum gates. Grover’s algorithm is a quantum search algorithm that offers a quadratic speedup for unstructured search problems, making it highly effective for finding solutions in large datasets, including integer factorization.

Full source code

1. Introduction to Grover’s Algorithm

Grover’s algorithm is designed to search through an unsorted database of NNN elements in O(N)O(\sqrt{N})O(N​) time, offering significant speedup over classical algorithms. In this context, Grover’s algorithm is adapted for integer factorization, identifying the factors of a number by searching through possible factor pairs. This quantum algorithm uses amplitude amplification, which iteratively increases the probability of the correct solution, enabling it to be identified with high probability.

Why Grover’s Algorithm for Factorization? Integer factorization is a problem of great importance in cryptography, particularly in the security of RSA encryption. Classical algorithms for factoring large integers are computationally expensive and time-consuming. By applying Grover’s algorithm to this problem, quantum computers can potentially factorize large integers exponentially faster than classical computers, posing a threat to current cryptographic systems.

2. Setting Up the Quantum Circuit

The quantum circuit is initialized with qubits corresponding to the input number and the solution space. The qubits are first prepared in a superposition state to explore all possible factor pairs simultaneously.

• Initialize Qubits: The qubits are initialized based on the bit-length required for the factorization.
• Superposition: Apply Hadamard gates to create a superposition of all possible states, enabling the quantum parallelism that Grover’s algorithm exploits.

import pennylane as qml
from pennylane import numpy as np

n = 6  # Number to factorize
nimWiresPQ = max(2, math.ceil(math.log2(n + 1) / 2))
numWiresSOL = math.ceil(math.log2(n))
wires_p = list(range(nimWiresPQ))
wires_q = list(range(nimWiresPQ, 2 * nimWiresPQ))
wires_solution = list(range(2 * nimWiresPQ, 2 * nimWiresPQ + numWiresSOL))
params = (n, wires_p, wires_q, wires_solution)
wires = len(wires_p) + len(wires_q) + len(wires_solution)
def FactorizationCircuit(params):
    n, wires_p, wires_q, wires_solution = params
    for wire in wires_p + wires_q:
        qml.Hadamard(wires=wire)

3. Oracle and Diffusion Operator

The oracle is the heart of Grover’s algorithm, marking the correct solutions by flipping their amplitude. The diffusion operator (also known as the Grover operator) then amplifies the marked solutions, making them more likely to be measured.

• Oracle: The multiplication and FlipSign functions act as the oracle, identifying factor pairs that multiply to the target number.
• Diffusion: The Grover operator enhances the probability of these factor pairs by inverting all amplitudes around their mean.

multiplication(wires_p, wires_q, wires_solution)
    qml.FlipSign(n, wires=wires_solution)
    qml.adjoint(multiplication)(wires_p, wires_q, wires_solution)
    qml.GroverOperator(wires=wires_p + wires_q)
    return qml.probs(wires=wires_p + wires_q)

4. Implementing the Multiplication Function

This function simulates the multiplication of the factors and checks if the product matches the target number. If it does, the oracle marks this state.

def multiplication(wires_p, wires_q, wires_solution):
    qml.QFT(wires=wires_solution)
    for i in range(len(wires_q)):
        for j in range(len(wires_p)):
            coeff = 2 ** (len(wires_p) + len(wires_q) - i - j - 2)
            qml.ctrl(Kfourier, control=[wires_q[i], wires_p[j]])(coeff, wires=wires_solution)
    qml.adjoint(qml.QFT)(wires=wires_solution)

Example: Grover’s algorithm circuit

5. Executing Grover’s Algorithm

Run the algorithm by initializing the circuit with the number to factorize. The output will show the probabilities of different factor pairs, with the correct factors having the highest probability.

result = FactorizationCircuit(params)
print("Grover's Algorithm Factorization Result:", result)

Conclusion

This Grover’s algorithm quantum circuit efficiently factors integers by leveraging quantum parallelism and amplitude amplification. The application of Grover’s algorithm to integer factorization is particularly significant in cryptography, as it provides a pathway to breaking encryption methods that rely on the difficulty of factorizing large numbers.

Why Grover’s Algorithm on Dynex Matters: Running Grover’s algorithm on the Dynex neuromorphic quantum computing cloud with its support for quantum circuits allows for scalable and efficient factorization of integers. This is critical in cryptography and other fields requiring fast and accurate solutions to complex search problems, showcasing the potential of quantum computing in solving tasks that are computationally prohibitive for classical computers.

About Dynex

Dynex is the world’s only accessible neuromorphic quantum computing cloud for solving real-world problems, at scale. The company began as an informal project in September 2020 in collaboration amongst a community of extraordinary minds and quickly evolved into a technological leader ready to scale into global markets. The Dynex n.quantum computing cloud performs quantum computing based algorithms without limitation, executing calculations with unparalleled speed and efficiency, surpassing usual quantum computing constraints. Dynex is dedicated to pushing the boundaries of technology to create sustainable, secure, and innovative solutions that address complex challenges and drive progress. For more information, visit dynex.co.