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            <ol class="chapter"><li><a href="qcgpu.html"><strong aria-hidden="true">1.</strong> QCGPU</a></li><li><ol class="section"><li><a href="getting-started.html"><strong aria-hidden="true">1.1.</strong> Getting Started</a></li></ol></li><li><a href="user-guide/user-guide.html"><strong aria-hidden="true">2.</strong> User Guide</a></li><li><ol class="section"><li><a href="user-guide/registers.html"><strong aria-hidden="true">2.1.</strong> Quantum Registers</a></li><li><a href="user-guide/gates.html"><strong aria-hidden="true">2.2.</strong> Quantum Gates</a></li><li><a href="user-guide/operations.html"><strong aria-hidden="true">2.3.</strong> Quantum Operations</a></li><li><a href="user-guide/examples.html"><strong aria-hidden="true">2.4.</strong> Examples</a></li><li><a href="user-guide/decoherence.html"><strong aria-hidden="true">2.5.</strong> Decoherence</a></li></ol></li><li><a href="algorithms/algorithms.html"><strong aria-hidden="true">3.</strong> Algorithms</a></li><li><ol class="section"><li><a href="algorithms/bernstein-vazirani.html"><strong aria-hidden="true">3.1.</strong> Bernstein-Vazirani</a></li><li><a href="algorithms/deutsch-jozsa.html"><strong aria-hidden="true">3.2.</strong> Deutsch-Jozsa</a></li><li><a href="algorithms/grover.html"><strong aria-hidden="true">3.3.</strong> Grovers</a></li><li><a href="algorithms/shor.html" class="active"><strong aria-hidden="true">3.4.</strong> Shors</a></li><li><a href="algorithms/super-dense.html"><strong aria-hidden="true">3.5.</strong> Super Dense Coding</a></li></ol></li></ol>
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                        <a class="header" href="algorithms/shor.html#shors-algorithm" id="shors-algorithm"><h1>Shors Algorithm</h1></a>
<p>This algorithm finds the prime factors (\)u\) and \(v\)) of an odd, composite integer \(n\),
that is not a prime power.</p>
<a class="header" href="algorithms/shor.html#the-algorithm" id="the-algorithm"><h2>The Algorithm</h2></a>
<p>(pseudo code)</p>
<pre><code class="language-pseudo">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.
</code></pre>
<p>See <a href="https://cs.uwaterloo.ca/%7Ewatrous/LectureNotes.html">https://cs.uwaterloo.ca/~watrous/LectureNotes.html</a></p>
<pre><pre class="playpen"><code class="language-rust">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!(&quot;Factoring {}.&quot;, 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 &gt;= 2 {
            // Found the factors; No Quantum needed
            println!(
                &quot;Factors are {} and {} (No quantum algorithm used)&quot;,
                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 &gt;= 2 {
                    println!(&quot;Factors are {} and {}&quot;, d, n / d);
                    break;
                }
            } else {
                println!(&quot;Period is odd&quot;);
            }
        }
    }
}
</code></pre></pre>
<a class="header" href="algorithms/shor.html#order-finding" id="order-finding"><h2>Order Finding</h2></a>
<p>Given a positive integer \(n \geq 2\) and an element \(a \in \mathbb{Z}_n^* \),
Find the order of \(a\) in \(\mathbb{Z}_n^*\).</p>
<p>Order finding is the only quantum part in shors algorithm.</p>
<p>\(\mathbb{Z}_n\) is the set of integers from \({0,\dots, n - 1}\) or \(\mathbb{z} \mod n\).
The set \(\mathbb{Z}_n^* = {a \in \mathbb{Z}_n : \gcd(a,n) = 1}\)</p>
<p>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</p>
<p>\[
ab \equiv 1 \mod n
\]</p>
<p>The order of a given element \(a \in \mathbb{Z}_n^*\) is the smallest positive integer \(r\) such that</p>
<p>\[
a^r \equiv 1 \mod n
\]</p>
<pre><pre class="playpen"><code class="language-rust">
# #![allow(unused_variables)]
#fn main() {
fn find_order(a: i32, n: i32) -&gt; i32 {
    unimplemented!()
}
#}</code></pre></pre>

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