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  <div class="section" id="quantum-gates">
<h1>Quantum Gates<a class="headerlink" href="#quantum-gates" title="Permalink to this headline">¶</a></h1>
<p>In quantum computing, gates are used to manipulate quantum registers and
to implement quantum algorithms.</p>
<div class="section" id="built-in-gates">
<h2>Built-In Gates<a class="headerlink" href="#built-in-gates" title="Permalink to this headline">¶</a></h2>
<p>There are a number of gates built into QCGPU. They can all be applied
the same way:</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">qcgpu</span>

<span class="n">register</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">State</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>

<span class="n">state</span><span class="o">.</span><span class="n">h</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c1"># Applies the Hadamard (H) gate to the first qubit.</span>
<span class="n">state</span><span class="o">.</span><span class="n">x</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># Applies a pauli-x (NOT) gate to the second qubit.</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">h</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span></code> can be replaced with any of the following:</p>
<ul class="simple">
<li>The Hadamard gate: <strong>h</strong> - <code class="docutils literal notranslate"><span class="pre">state.h(0)</span></code></li>
<li>The S gate: <strong>s</strong> - <code class="docutils literal notranslate"><span class="pre">state.s(0)</span></code></li>
<li>The T gate: <strong>t</strong> - <code class="docutils literal notranslate"><span class="pre">state.t(0)</span></code></li>
<li>The Pauli-X / NOT gate: <strong>x</strong> - <code class="docutils literal notranslate"><span class="pre">state.x(0)</span></code></li>
<li>The Pauli-Y gate: <strong>y</strong> - <code class="docutils literal notranslate"><span class="pre">state.y(0)</span></code></li>
<li>The Pauli-Z gate: <strong>z</strong> - <code class="docutils literal notranslate"><span class="pre">state.z(0)</span></code></li>
<li>The CNOT gate: <strong>cx</strong> -
<code class="docutils literal notranslate"><span class="pre">state.cx(0,</span> <span class="pre">1)</span> <span class="pre">#</span> <span class="pre">CNOT</span> <span class="pre">with</span> <span class="pre">control</span> <span class="pre">=</span> <span class="pre">0,</span> <span class="pre">target</span> <span class="pre">=</span> <span class="pre">1</span></code></li>
<li>The SWAP gate: <strong>swap</strong> -
<code class="docutils literal notranslate"><span class="pre">state.swap(0,1)</span> <span class="pre">#</span> <span class="pre">Swaps</span> <span class="pre">the</span> <span class="pre">0th</span> <span class="pre">and</span> <span class="pre">1st</span> <span class="pre">qubit</span></code></li>
<li>The Toffoli gate: <strong>toffoli</strong> -
<code class="docutils literal notranslate"><span class="pre">state.toffoli(0,</span> <span class="pre">1,</span> <span class="pre">2)</span> <span class="pre">#</span> <span class="pre">Toffoli</span> <span class="pre">with</span> <span class="pre">controls</span> <span class="pre">=</span> <span class="pre">(0,</span> <span class="pre">1),</span> <span class="pre">target</span> <span class="pre">=</span> <span class="pre">2</span></code></li>
</ul>
<p>These are all shorthand methods for the application of arbitrary gates.
For example, the application of a Hadamard gate above is shorthand for</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">qcgpu</span>

<span class="n">h</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">h</span><span class="p">()</span>

<span class="n">register</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">State</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">register</span><span class="o">.</span><span class="n">apply_gate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
</pre></div>
</div>
<p>You can also use any of the gates as controlled gates. For example, the
application of the CNOT gate above is shorthand for</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">qcgpu</span>

<span class="n">x</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">x</span><span class="p">()</span>

<span class="n">register</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">State</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">register</span><span class="o">.</span><span class="n">apply_controlled_gate</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="applying-a-gate-to-a-whole-register">
<h2>Applying A Gate To A Whole Register<a class="headerlink" href="#applying-a-gate-to-a-whole-register" title="Permalink to this headline">¶</a></h2>
<p>There is a convenience method to apply a gate to every qubit in the register.
The following applies a Hadamard gate to the whole register,</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">qcgpu</span>

<span class="n">h</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">h</span><span class="p">()</span>

<span class="n">register</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">State</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">register</span><span class="o">.</span><span class="n">apply_all</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="user-defined-gates">
<h2>User Defined Gates<a class="headerlink" href="#user-defined-gates" title="Permalink to this headline">¶</a></h2>
<p>Gates in QCGPU are represented by the <code class="docutils literal notranslate"><span class="pre">qcgpu.Gate</span></code> class.</p>
<p>The only gates that can be defined by the user are single qubit gates.</p>
<p>The process of creating a gate is</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">qcgpu</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>

<span class="n">gate_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
    <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">4</span><span class="p">)]</span>
<span class="p">])</span>

<span class="n">gate</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">Gate</span><span class="p">(</span><span class="n">gate_matrix</span><span class="p">)</span>
</pre></div>
</div>
<p>The input to the <code class="docutils literal notranslate"><span class="pre">Gate</span></code> constructor is checked to be a 2x2 unitary
matrix.</p>
<p>This newly created gate can then be applied the long hand way,</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">qcgpu</span>

<span class="n">register</span> <span class="o">=</span> <span class="n">qcgpu</span><span class="o">.</span><span class="n">State</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">register</span><span class="o">.</span><span class="n">apply_gate</span><span class="p">(</span><span class="n">gate</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
</pre></div>
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