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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" class="active"><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"><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="#bernstein-vazirani-algorithm" id="bernstein-vazirani-algorithm"><h1>Bernstein-Vazirani Algorithm</h1></a>
<p>This algorithm finds a hidden integer \(a \in { 0, 1}^n\)from
an oracle \(f_a\)which returns a bit \(a \cdot x \equiv \sum_i a_i x_i \mod 2\)
for an input \(x \in {0,1}^n\).</p>
<p>A classical oracle returns \(f_a(x) = a \dot x \mod 2\), while the quantum oracle
must be queried with superpositions of input \(x\)'s.</p>
<p>To solve this problem classically, the hidden integer can be found by checking the
oracle with the inputs \(x = 1,2,\dots,2^i,2^{n-1}\), where each
query reveals the \(i\)th bit of \(a\) (\(a_i\)).
This is the optimal classical solution, and is O(n). Using a quantum oracle and the
Bernstein-Vazirani algorithm, \(a\) can be found with just one query to the oracle.</p>
<a class="header" href="#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 inner product oracle.</li>
<li>Apply the Hadamard gate \(H\) to each qubit.</li>
<li>Measure the register</li>
</ol>
<p>From this procedure, we find that the registers measured value is equal to that of
the original hidden integer.</p>
<pre><pre class="playpen"><code class="language-rust">extern crate qcgpu;

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

fn main() {
    let num_qubits = 16; // Number of qubits to use
    let a = 101; // Hidden integer, bitstring is 1100101

    // You should also make sure that a is representable with n qubits,
    // by settings a as a mod 2^n.

    // Bernstein-Vazirani algorithm
    let mut state = State::new(num_qubits, 1); // New quantum register, using the GPU.

    // Apply a hadamard gate to each qubit
    state.apply_all(h());

    // Apply the inner products oracle
    for i in 0..num_qubits {
        if a &amp; (1 &lt;&lt; i) != 0 {
            state.z(i as i32);
        }
        // Otherwise should apply identity gate, but computationally this doens't change the state.
    }

    // Apply hadamard gates before measuring
    state.apply_all(h());

    println!(&quot;Measurement Results: {:?}&quot;, state.measure_many(1000));
    // Measurement Results: {&quot;0000000001100101&quot;: 1000}
}
</code></pre></pre>

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