RONCC
/
HPC-Tools-for-Quantum-Computing


			
				
					
						
						
							123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257
							<!DOCTYPE HTML>
<html lang="en" class="sidebar-visible no-js">
    <head>
        <!-- Book generated using mdBook -->
        <meta charset="UTF-8">
        <title>Bernstein-Vazirani - QCGPU User Guide</title>
        <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
        <meta name="description" content="The user guide / prose documentation for QCGPU">
        <meta name="viewport" content="width=device-width, initial-scale=1">
        <meta name="theme-color" content="#ffffff" />

        <base href="../">

        <link rel="stylesheet" href="book.css">
        <link href="https://fonts.googleapis.com/css?family=Open+Sans:300italic,400italic,600italic,700italic,800italic,400,300,600,700,800" rel="stylesheet" type="text/css">
        <link href="https://fonts.googleapis.com/css?family=Source+Code+Pro:500" rel="stylesheet" type="text/css">

        <link rel="shortcut icon" href="favicon.png">

        <!-- Font Awesome -->
        <link rel="stylesheet" href="_FontAwesome/css/font-awesome.css">

        <link rel="stylesheet" href="highlight.css">
        <link rel="stylesheet" href="tomorrow-night.css">
        <link rel="stylesheet" href="ayu-highlight.css">

        <!-- Custom theme stylesheets -->
        

        
        <!-- MathJax -->
        <script async type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
        

    </head>
    <body class="light">
        <!-- Work around some values being stored in localStorage wrapped in quotes -->
        <script type="text/javascript">
            try {
                var theme = localStorage.getItem('mdbook-theme');
                var sidebar = localStorage.getItem('mdbook-sidebar');

                if (theme.startsWith('"') && theme.endsWith('"')) {
                    localStorage.setItem('mdbook-theme', theme.slice(1, theme.length - 1));
                }

                if (sidebar.startsWith('"') && sidebar.endsWith('"')) {
                    localStorage.setItem('mdbook-sidebar', sidebar.slice(1, sidebar.length - 1));
                }
            } catch (e) { }
        </script>

        <!-- Set the theme before any content is loaded, prevents flash -->
        <script type="text/javascript">
            var theme;
            try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { } 
            if (theme === null || theme === undefined) { theme = 'light'; }
            document.body.className = theme;
            document.querySelector('html').className = theme + ' js';
        </script>

        <!-- Hide / unhide sidebar before it is displayed -->
        <script type="text/javascript">
            var html = document.querySelector('html');
            var sidebar = 'hidden';
            if (document.body.clientWidth >= 1080) {
                try { sidebar = localStorage.getItem('mdbook-sidebar'); } catch(e) { }
                sidebar = sidebar || 'visible';
            }
            html.classList.remove('sidebar-visible');
            html.classList.add("sidebar-" + sidebar);
        </script>

        <nav id="sidebar" class="sidebar" aria-label="Table of contents">
            <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>
        </nav>

        <div id="page-wrapper" class="page-wrapper">

            <div class="page">
                
                <div id="menu-bar" class="menu-bar">
                    <div id="menu-bar-sticky-container">
                        <div class="left-buttons">
                            <button id="sidebar-toggle" class="icon-button" type="button" title="Toggle Table of Contents" aria-label="Toggle Table of Contents" aria-controls="sidebar">
                                <i class="fa fa-bars"></i>
                            </button>
                            <button id="theme-toggle" class="icon-button" type="button" title="Change theme" aria-label="Change theme" aria-haspopup="true" aria-expanded="false" aria-controls="theme-list">
                                <i class="fa fa-paint-brush"></i>
                            </button>
                            <ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
                                <li role="none"><button role="menuitem" class="theme" id="light">Light <span class="default">(default)</span></button></li>
                                <li role="none"><button role="menuitem" class="theme" id="rust">Rust</button></li>
                                <li role="none"><button role="menuitem" class="theme" id="coal">Coal</button></li>
                                <li role="none"><button role="menuitem" class="theme" id="navy">Navy</button></li>
                                <li role="none"><button role="menuitem" class="theme" id="ayu">Ayu</button></li>
                            </ul>
                            
                            <button id="search-toggle" class="icon-button" type="button" title="Search. (Shortkey: s)" aria-label="Toggle Searchbar" aria-expanded="false" aria-keyshortcuts="S" aria-controls="searchbar">
                                <i class="fa fa-search"></i>
                            </button>
                            
                        </div>

                        <h1 class="menu-title">QCGPU User Guide</h1> 

                        <div class="right-buttons">
                            <a href="print.html" title="Print this book" aria-label="Print this book">
                                <i id="print-button" class="fa fa-print"></i>
                            </a>
                        </div>
                    </div>
                </div>

                
                <div id="search-wrapper" class="hidden">
                    <form id="searchbar-outer" class="searchbar-outer">
                        <input type="search" name="search" id="searchbar" name="searchbar" placeholder="Search this book ..." aria-controls="searchresults-outer" aria-describedby="searchresults-header">
                    </form>
                    <div id="searchresults-outer" class="searchresults-outer hidden">
                        <div id="searchresults-header" class="searchresults-header"></div>
                        <ul id="searchresults">
                        </ul>
                    </div>
                </div>
                

                <!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
                <script type="text/javascript">
                    document.getElementById('sidebar-toggle').setAttribute('aria-expanded', sidebar === 'visible');
                    document.getElementById('sidebar').setAttribute('aria-hidden', sidebar !== 'visible');
                    Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
                        link.setAttribute('tabIndex', sidebar === 'visible' ? 0 : -1);
                    });
                </script>

                <div id="content" class="content">
                    <main>
                        <a class="header" href="algorithms/bernstein-vazirani.html#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="algorithms/bernstein-vazirani.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 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>

                    </main>

                    <nav class="nav-wrapper" aria-label="Page navigation">
                        <!-- Mobile navigation buttons -->
                        
                            <a rel="prev" href="algorithms/algorithms.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
                                <i class="fa fa-angle-left"></i>
                            </a>
                        

                        
                            <a rel="next" href="algorithms/deutsch-jozsa.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
                                <i class="fa fa-angle-right"></i>
                            </a>
                        

                        <div style="clear: both"></div>
                    </nav>
                </div>
            </div>

            <nav class="nav-wide-wrapper" aria-label="Page navigation">
                
                    <a href="algorithms/algorithms.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
                        <i class="fa fa-angle-left"></i>
                    </a>
                

                
                    <a href="algorithms/deutsch-jozsa.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
                        <i class="fa fa-angle-right"></i>
                    </a>
                
            </nav>

        </div>

        

        

        

        

        
        <script src="searchindex.js" type="text/javascript" charset="utf-8"></script>
        
        
        <script src="elasticlunr.min.js" type="text/javascript" charset="utf-8"></script>
        <script src="mark.min.js" type="text/javascript" charset="utf-8"></script>
        <script src="searcher.js" type="text/javascript" charset="utf-8"></script>
        

        <script src="clipboard.min.js" type="text/javascript" charset="utf-8"></script>
        <script src="highlight.js" type="text/javascript" charset="utf-8"></script>
        <script src="book.js" type="text/javascript" charset="utf-8"></script>

        <!-- Custom JS scripts -->
        

    </body>
</html>