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HPC-Tools-for-Quantum-Computing


			
				
					
						
						
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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" class="active"><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"><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/deutsch-jozsa.html#deutsch-jozsa-algorithm" id="deutsch-jozsa-algorithm"><h1>Deutsch-Jozsa Algorithm</h1></a>
<p>This algorithm was the first to show that quantum computers could have a speedup over classical computers.</p>
<p>Consider a function \(f(x)\) which takes an input of an \(n\)-bit string \(x\) and returns 0 or 1.</p>
<p>Suppose that \(f(x)\) is either a <strong>constant</strong> function that has the same value \(c \in {0, 1},  \forall x\), or a <strong>balanced</strong> function, where the value is 0 for half of the inputs, and 1 for the other half.</p>
<p>The Deutsch-Jozsa problem is to find whether \(f\) is <em>constant</em> or <em>balanced</em>, in as few function evaluations as possible.</p>
<p>Using classical computing, in the worst case, this requires \(2^{n-1}+1\) function evaluations.
Using quantum computing, this can be done with just one function evaluation.</p>
<p>The function \(f\), to be used in a quantum computer, must be specified by an oracle circuit \(U_{f}\) such that \(U_{f} \lvert x \rangle = (-1)^{f(x)}\lvert x \rangle\).</p>
<a class="header" href="algorithms/deutsch-jozsa.html#the-algorithm" id="the-algorithm"><h2>The Algorithm</h2></a>
<ol>
<li>Initialize \(n\) qubits in the state \(\lvert 0, \dots, 0\rangle\).</li>
<li>Apply the Hadamard gate \(H\) to each qubit.</li>
<li>Apply the oracle circuit \(U_f\).</li>
<li>Apply the Hadamard gate \(H\) to each qubit.</li>
<li>Measure each qubit. Let \(y = (y_{1}, \dots, y_{n})\) be the list of measurement outcomes.</li>
</ol>
<p>From this procedure, we find that \(f\) is constant if \(y\) is the all zero string.</p>
<pre><pre class="playpen"><code class="language-rust">extern crate qcgpu;

use qcgpu::State;
use qcgpu::gates::h;

fn main() {
    // 3 qubits, f(x) = x_0 NOT x_1 x_2
    // Balanced
    let mut balanced_state = State::new(3, 1);

    balanced_state.apply_all(h());

    // Oracle U_f
    balanced_state.h(2);
    balanced_state.z(0);
    balanced_state.cx(1, 2);
    balanced_state.h(2);

    balanced_state.apply_all(h());

    println!(
        &quot;{}&quot;,
        if balanced_state.measure() == 0 {
            &quot;constant&quot;
        } else {
            &quot;balanced&quot;
        }
    );

    // 3 qubits, f(x) = 0
    // Constant
    let mut constant_state = State::new(3, 1);

    constant_state.apply_all(h());

    // Oracle is equivilant to the identity gate, thus has no effect on the
    // state.

    constant_state.apply_all(h());

    println!(
        &quot;{}&quot;,
        if constant_state.measure() == 0 {
            &quot;constant&quot;
        } else {
            &quot;balanced&quot;
        }
    );
}
</code></pre></pre>

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