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<p>Return a random spin operator with the given number of terms (<code class="code docutils literal notranslate"><span class="pre">term_count</span></code>) where each term acts on all targets in the open range [0, qubit_count). An optional seed value may also be provided.</p>
Copy file name to clipboardExpand all lines: pr-3301/applications/python/adapt_qaoa.html
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parameter</p>
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<p>3- Optimize all parameters currently in the Ansatz <spanclass="math notranslate nohighlight">\(\beta_m, \gamma_m = 1, 2, ...k\)</span> such that <spanclass="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>
<p>Suppose we have <spanclass="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 <spanclass="math notranslate nohighlight">\(U_f\)</span> which allows us to transform the state <spanclass="math notranslate nohighlight">\(\ket{x}\ket{y}\)</span> into:</p>
<h2>Deutsch’s Algorithm:<aclass="headerlink" href="#Deutsch's-Algorithm:" title="Permalink to this heading">¶</a></h2>
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<p>Our aim is to find out if <spanclass="math notranslate nohighlight">\(f: \{0,1\} \longrightarrow \{0,1\}\)</span> is a constant or a balanced function? If constant, <spanclass="math notranslate nohighlight">\(f(0) = f(1)\)</span>, and if balanced, <spanclass="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>
<h2>Quantum Probability Image Encoding (QPIE):<aclass="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>
</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>
Copy file name to clipboardExpand all lines: pr-3301/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 <spanclass="math notranslate nohighlight">\(|0\rangle, |1\rangle, \dots, |N-1\rangle\)</span>, where the full dimension is <spanclass="math notranslate nohighlight">\(N = 2^n\)</span>. Since the uniform superposition state <spanclass="math notranslate nohighlight">\(|\xi\rangle\)</span> can be expressed as a linear combination of the states <spanclass="math notranslate nohighlight">\(|G\rangle\)</span> and <spanclass="math notranslate nohighlight">\(|B\rangle\)</span> with real coefficients, all three states <spanclass="math notranslate nohighlight">\(|\xi\rangle, |G\rangle,\)</span> and <spanclass="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, <spanclass="math notranslate nohighlight">\(|G\rangle\)</span> and <spanclass="math notranslate nohighlight">\(|B\rangle\)</span> are orthogonal, we can imagine them graphed as unit vectors in the positive <spanclass="math notranslate nohighlight">\(y\)</span> and positive <spanclass="math notranslate nohighlight">\(x\)</span> directions, respectively. From our previous expression, <spanclass="math notranslate nohighlight">\(\ket{\xi} = \sin(\theta) |G\rangle + \cos(\theta) |B\rangle,\)</span> we see that the state <spanclass="math notranslate nohighlight">\(|\xi\rangle\)</span> forms an angle <spanclass="math notranslate nohighlight">\(\theta\)</span> with
</div><p>Given that the number of marked states <spanclass="math notranslate nohighlight">\(t\)</span> is typically small compared to <spanclass="math notranslate nohighlight">\(N\)</span>, it follows that <spanclass="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>
</div><p>Running this circuit initializes <spanclass="math notranslate nohighlight">\(\ket{\xi}\)</span> and performs a rotation by <spanclass="math notranslate nohighlight">\(2\theta\)</span> (twice the angle between <spanclass="math notranslate nohighlight">\(|\xi\rangle\)</span> and <spanclass="math notranslate nohighlight">\(|B\rangle\)</span>) in the direction from <spanclass="math notranslate nohighlight">\(|B\rangle\)</span> to <spanclass="math notranslate nohighlight">\(|G\rangle\)</span>.</p>
</div><p>Let’s verify that the state resulting from one iteration of Grover’s algorithm brings us closer to the good state, <spanclass="math notranslate nohighlight">\(\ket{G}\)</span>. In particular, notice that the amplitudes of <codeclass="docutils literal notranslate"><spanclass="pre">1001</span></code> and <codeclass="docutils literal notranslate"><spanclass="pre">1111</span></code> in the resulting state have been amplified compared to the equal superposition of states.</p>
<h3>Step 1: Initialize (Classical pre-processing)<aclass="headerlink" href="#Step-1:-Initialize-(Classical-pre-processing)" title="Permalink to this heading">¶</a></h3>
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