Shor's Factoring Algorithm

Shor's factoring algorithm is a quantum algorithm for integer factorization that runs much faster than classical algorithms. It was developed by the mathematician Peter Shor in 1994. The algorithm takes as input an integer N and outputs its prime factors. It has important implications for cryptography, which relies on the difficulty of factoring large numbers.

How it works

The algorithm works by finding the period of a certain function, called the modular exponential function, using a quantum computer. The period of the function can be used to factor the input integer.

The quantum part of the algorithm involves creating a superposition of possible solutions to the problem, and then applying a quantum Fourier transform to extract the period. The period can then be used to factor the input integer using classical algorithms.

Code example

Here is an example implementation of Shor's factoring algorithm in Python:

from math import gcd
from random import randrange
from qiskit import QuantumCircuit, Aer, execute

def shor_factor(n):
    while True:
        a = randrange(2, n)
        d = gcd(a, n)
        if d > 1:
            return d
        qc = QuantumCircuit(2 * qubits, qubits)
        for qubit in range(qubits):
            qc.h(qubit)
            qc.measure(qubit, qubit)
        backend = Aer.get_backend('qasm_simulator')
        job = execute(qc, backend=backend, shots=1)
        result = list(job.result().get_counts().keys())[0]
        r = int(result, 2)
        if r % 2 != 0:
            continue
        x = (a**(r/2)) % n
        if (x + 1) % n == 0:
            continue
        d = gcd(x + 1, n)
        if d > 1:
            return d

This implementation uses the quantum simulator provided by the Qiskit library to simulate the quantum computations, since a real quantum computer with enough qubits is not yet available.

Conclusion

Shor's factoring algorithm is a groundbreaking advance in quantum computing and has important implications for cryptography. While the algorithm is primarily of theoretical interest at present, continued improvements in quantum hardware may eventually make it a practical tool for breaking certain cryptographic systems.