HPC-Tools-for-Quantum-Computing/book/src/algorithms/shor.md

Shors Algorithm

This algorithm finds the prime factors (\)u\) and \(v\)) of an odd, composite integer \(n\), that is not a prime power.

The Algorithm

(pseudo code)

Repeat
    Randomly choose \\(a \in \{ 2, \dots, n - 1 \}\\)
    Compute \\(d = gcd(a, n)\\)
    If \\(d \geq 2\\) then
        Return \\(u = d\\) and \\(v = n/d\\)
    Else // We know \\(a \in \mathbb{Z}^*_N\\)
        Let \\(r\\) be the order of \\(a\\) in \\(\mathbb{Z}^*_N\\) // Order finding algorithm
        If \\(r\\) is even then
            Compute \\(x = a^{r/2} - 1 (\mod n)\\)
            Compute \\(d = \gcd(x, n)\\)
            If \\(d \geq 2\\) then
                Return \\(u = d\\) and \\(v = n/d\\)
Until a value is returned.

See https://cs.uwaterloo.ca/~watrous/LectureNotes.html

extern crate qcgpu;
extern crate rand;

use rand::{thread_rng, Rng};
use qcgpu::{gcd, State};
use qcgpu::gates::h;


fn main() {
    let n = 15; // Number to factor
    println!("Factoring {}.", n);

    // Here we should check if the number is even, or if it is a power factor

    let mut rng = thread_rng();
    loop {
        let mut a = rng.gen_range(2, n); // Randomly choose \\(a \in \{ 2, \dots, n - 1 \}\\)

        let mut d = gcd(a, n);
        if d >= 2 {
            // Found the factors; No Quantum needed
            println!(
                "Factors are {} and {} (No quantum algorithm used)",
                d,
                n / d
            );
            break;
        } else {
            // We know \\(a \in \mathbb{Z}^*_N\\)
            let r = find_order(a, n);
            if r % 2 == 0 {
                let x = (a.pow((r as f32 / 2 as f32) as u32) - 1) % n;
                d = gcd(x, n);
                if d >= 2 {
                    println!("Factors are {} and {}", d, n / d);
                    break;
                }
            } else {
                println!("Period is odd");
            }
        }
    }
}

Order Finding

Given a positive integer \(n \geq 2\) and an element \(a \in \mathbb{Z}_n^* \), Find the order of \(a\) in \(\mathbb{Z}_n^*\).

Order finding is the only quantum part in shors algorithm.

\(\mathbb{Z}_n\) is the set of integers from \({0,\dots, n - 1}\) or \(\mathbb{z} \mod n\). The set \(\mathbb{Z}_n^* = {a \in \mathbb{Z}_n : \gcd(a,n) = 1}\)

The set \( \mathbb{Z}_n^* \) forms a group wth the multiplication modulo \(n\) operation. Thus, for \( a \in \mathbb{Z}_n^* \) , \(\exists b \in \mathbb{Z}_n^*\) that uniquely satisfies

\[ ab \equiv 1 \mod n \]

The order of a given element \(a \in \mathbb{Z}_n^*\) is the smallest positive integer \(r\) such that

\[ a^r \equiv 1 \mod n \]

```rust fn find_order(a: i32, n: i32) -> i32 {

unimplemented!()

}