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‎pr-4271/applications/python/adapt_qaoa.html‎

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parameter</p>
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<p>3- Optimize all parameters currently in the Ansatz <span class="math notranslate nohighlight">\(\beta_m, \gamma_m = 1, 2, ...k\)</span> such that <span class="math notranslate nohighlight">\(\braket{\psi (k)|H_C|\psi(k)}\)</span> is minimized, and return to the second step.</p>
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<p>Below is a schematic representation of the ADAPT-QAOA algorithm explained above.</p>
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<div><p><img alt="ac96507499284e5bb174610da35e2e4a" class="no-scaled-link" src="../../_images/adapt-qaoa.png" style="width: 1000px;" /></p>
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<div><p><img alt="8b228e6d38324e82b35cb569e8ffeb4e" class="no-scaled-link" src="../../_images/adapt-qaoa.png" style="width: 1000px;" /></p>
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</div><div class="nbinput nblast docutils container">
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<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[15]:
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‎pr-4271/applications/python/adapt_qaoa.md‎

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‎pr-4271/applications/python/adapt_vqe.html‎

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<p>7- Perform a VQE experiment to re-optimize all parameters in the ansatz.</p>
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<p>8- go to step 4</p>
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<p>Below is a Schematic depiction of the ADAPT-VQE algorithm</p>
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<div><p><img alt="5a5b5040a6da4306857a297450c0b57d" class="no-scaled-link" src="../../_images/adapt-vqe.png" style="width: 800px;" /></p>
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<div><p><img alt="98e6cd3961cc4814923d0b0af042e663" class="no-scaled-link" src="../../_images/adapt-vqe.png" style="width: 800px;" /></p>
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<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[1]:
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‎pr-4271/applications/python/adapt_vqe.md‎

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‎pr-4271/applications/python/deutsch_algorithm.html‎

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</section>
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<section id="Quantum-oracles">
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<h2>Quantum oracles<a class="headerlink" href="#Quantum-oracles" title="Permalink to this heading"></a></h2>
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<p><img alt="e05dd6f7f7d44af39c5390f2a0370957" class="no-scaled-link" src="../../_images/oracle.png" style="width: 300px; height: 150px;" /></p>
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<p><img alt="6fdef71f34b4410c875fc508533e9ad8" class="no-scaled-link" src="../../_images/oracle.png" style="width: 300px; height: 150px;" /></p>
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<p>Suppose we have <span class="math notranslate nohighlight">\(f(x): \{0,1\} \longrightarrow \{0,1\}\)</span>. We can compute this function on a quantum computer using oracles which we treat as black box functions that yield the output with an appropriate sequence of logical gates.</p>
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<p>Above you see an oracle represented as <span class="math notranslate nohighlight">\(U_f\)</span> which allows us to transform the state <span class="math notranslate nohighlight">\(\ket{x}\ket{y}\)</span> into:</p>
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<h2>Deutsch’s Algorithm:<a class="headerlink" href="#Deutsch's-Algorithm:" title="Permalink to this heading"></a></h2>
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<p>Our aim is to find out if <span class="math notranslate nohighlight">\(f: \{0,1\} \longrightarrow \{0,1\}\)</span> is a constant or a balanced function? If constant, <span class="math notranslate nohighlight">\(f(0) = f(1)\)</span>, and if balanced, <span class="math notranslate nohighlight">\(f(0) \neq f(1)\)</span>.</p>
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<p>We step through the circuit diagram below and follow the math after the application of each gate.</p>
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<p><img alt="916ebb45f3f941a3b9bb03b5914ac8f4" class="no-scaled-link" src="../../_images/deutsch.png" style="width: 500px; height: 210px;" /></p>
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<p><img alt="eb52d30ddebc430daed8549a75f644cc" class="no-scaled-link" src="../../_images/deutsch.png" style="width: 500px; height: 210px;" /></p>
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<div class="math notranslate nohighlight">
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\[\ket{\psi_0} = \ket{01}
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\tag{1}\]</div>

‎pr-4271/applications/python/deutsch_algorithm.md‎

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::: {#Quantum-oracles .section}
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## Quantum oracles[](#Quantum-oracles "Permalink to this heading"){.headerlink}
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Suppose we have [\\(f(x): \\{0,1\\} \\longrightarrow \\{0,1\\}\\)]{.math
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We step through the circuit diagram below and follow the math after the
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‎pr-4271/applications/python/edge_detection.html‎

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<section id="Quantum-Probability-Image-Encoding-(QPIE):">
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<h2>Quantum Probability Image Encoding (QPIE):<a class="headerlink" href="#Quantum-Probability-Image-Encoding-(QPIE):" title="Permalink to this heading"></a></h2>
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<p>Lets take as an example a classical 2x2 image (4 pixels). We can label each pixel with its position</p>
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<div><p><img alt="d07ec0faab584bf789118c16f2a7f2f9" class="no-scaled-link" src="../../_images/pixels-img.png" style="width: 200px;" /></p>
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</div><p>Each pixel will have its own color intensity represented along with its position label as an 8-bit black and white color. To convert the pixel intensity to probability amplitudes of a quantum state</p>
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<div class="math notranslate nohighlight">
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\[c_i = \frac{I_{yx}}{\sqrt(\sum I^2_{yx})}\]</div>

‎pr-4271/applications/python/edge_detection.md‎

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‎pr-4271/applications/python/grovers.html‎

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<p>Let us now examine the geometric picture behind our current discussion. We’ll consider the ambient Hilbert space to be spanned by the standard basis vectors <span class="math notranslate nohighlight">\(|0\rangle, |1\rangle, \dots, |N-1\rangle\)</span>, where the full dimension is <span class="math notranslate nohighlight">\(N = 2^n\)</span>. Since the uniform superposition state <span class="math notranslate nohighlight">\(|\xi\rangle\)</span> can be expressed as a linear combination of the states <span class="math notranslate nohighlight">\(|G\rangle\)</span> and <span class="math notranslate nohighlight">\(|B\rangle\)</span> with real coefficients, all three states <span class="math notranslate nohighlight">\(|\xi\rangle, |G\rangle,\)</span> and <span class="math notranslate nohighlight">\(|B\rangle\)</span>
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reside in a two-dimensional real subspace of the ambient Hilbert space, which we can visualize as a 2D plane as in the image below. Since, <span class="math notranslate nohighlight">\(|G\rangle\)</span> and <span class="math notranslate nohighlight">\(|B\rangle\)</span> are orthogonal, we can imagine them graphed as unit vectors in the positive <span class="math notranslate nohighlight">\(y\)</span> and positive <span class="math notranslate nohighlight">\(x\)</span> directions, respectively. From our previous expression, <span class="math notranslate nohighlight">\(\ket{\xi} = \sin(\theta) |G\rangle + \cos(\theta) |B\rangle,\)</span> we see that the state <span class="math notranslate nohighlight">\(|\xi\rangle\)</span> forms an angle <span class="math notranslate nohighlight">\(\theta\)</span> with
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<span class="math notranslate nohighlight">\(|B\rangle\)</span>.</p>
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</div><p>Given that the number of marked states <span class="math notranslate nohighlight">\(t\)</span> is typically small compared to <span class="math notranslate nohighlight">\(N\)</span>, it follows that <span class="math notranslate nohighlight">\(\theta = \arcsin\left(\sqrt{\frac{t}{N}}\right)\)</span> is a small angle. This assumption is reasonable, as otherwise, a sufficient number of independent queries to the oracle would likely yield a solution through classical search methods.</p>
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\[\mathcal{G} = r_\xi \circ r_B = H^{\otimes n} \big( 2|0^{\otimes n} \rangle \langle 0^{\otimes n}| - \text{Id} \big) H^{\otimes n} \mathcal{O}_f.\]</div>
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<p>The circuit diagram below puts together steps 1 through 3:</p>
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<div style="text-align: center;"><p><img alt="7e28bc462e8541b9a8f30fe95773661d" src="../../_images/grovers-steps1-3.png" /></p>
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</div><p>Running this circuit initializes <span class="math notranslate nohighlight">\(\ket{\xi}\)</span> and performs a rotation by <span class="math notranslate nohighlight">\(2\theta\)</span> (twice the angle between <span class="math notranslate nohighlight">\(|\xi\rangle\)</span> and <span class="math notranslate nohighlight">\(|B\rangle\)</span>) in the direction from <span class="math notranslate nohighlight">\(|B\rangle\)</span> to <span class="math notranslate nohighlight">\(|G\rangle\)</span>.</p>
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<div style="text-align: center;"><p><img alt="93cb397184c142eaafdc124f406db713" src="../../_images/grovers-full-rotation.png" /></p>
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</div><p>Let’s verify that the state resulting from one iteration of Grover’s algorithm brings us closer to the good state, <span class="math notranslate nohighlight">\(\ket{G}\)</span>. In particular, notice that the amplitudes of <code class="docutils literal notranslate"><span class="pre">1001</span></code> and <code class="docutils literal notranslate"><span class="pre">1111</span></code> in the resulting state have been amplified compared to the equal superposition of states.</p>
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‎pr-4271/applications/python/grovers.md‎

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Let's verify that the state resulting from one iteration of Grover's

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