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‎doc/pub/week13/html/week13-bs.html‎

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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Plans for the week of April 15-19',
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'sections': [('Plans for the week of April 21-25',
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2,
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None,
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'plans-for-the-week-of-april-15-19'),
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('QFT', 2, None, 'qft'),
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('In terms of arbitrary states',
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2,
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None,
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'in-terms-of-arbitrary-states'),
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('Unitarity', 2, None, 'unitarity'),
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('Binary representation', 2, None, 'binary-representation'),
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('Rewriting the QFT', 2, None, 'rewriting-the-qft'),
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('Short hand notation', 2, None, 'short-hand-notation'),
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('Four qubit system', 2, None, 'four-qubit-system'),
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('QFT acts as follows', 2, None, 'qft-acts-as-follows'),
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('Quantum circuits', 2, None, 'quantum-circuits'),
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('Rotation gates', 2, None, 'rotation-gates'),
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('Using the Hadamard gate', 2, None, 'using-the-hadamard-gate'),
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('Algorithm', 2, None, 'algorithm'),
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('Binary fraction', 2, None, 'binary-fraction'),
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('Controlled rotation gate', 2, None, 'controlled-rotation-gate'),
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('Applying to all qubits', 2, None, 'applying-to-all-qubits'),
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('Plans for the week of April 22-26',
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2,
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None,
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'plans-for-the-week-of-april-22-26')]}
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'plans-for-the-week-of-april-21-25')]}
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end of tocinfo -->
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<body>
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<script type="text/x-mathjax-config">
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MathJax.Hub.Config({
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TeX: {
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equationNumbers: { autoNumber: "AMS" },
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extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
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}
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});
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</script>
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<script type="text/javascript" async
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src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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<!-- Bootstrap navigation bar -->
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<div class="navbar navbar-default navbar-fixed-top">
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<div class="navbar-header">
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-april-15-19" style="font-size: 80%;">Plans for the week of April 15-19</a></li>
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<!-- navigation toc: --> <li><a href="#qft" style="font-size: 80%;">QFT</a></li>
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<!-- navigation toc: --> <li><a href="#in-terms-of-arbitrary-states" style="font-size: 80%;">In terms of arbitrary states</a></li>
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<!-- navigation toc: --> <li><a href="#unitarity" style="font-size: 80%;">Unitarity</a></li>
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<!-- navigation toc: --> <li><a href="#binary-representation" style="font-size: 80%;">Binary representation</a></li>
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<!-- navigation toc: --> <li><a href="#rewriting-the-qft" style="font-size: 80%;">Rewriting the QFT</a></li>
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<!-- navigation toc: --> <li><a href="#short-hand-notation" style="font-size: 80%;">Short hand notation</a></li>
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<!-- navigation toc: --> <li><a href="#four-qubit-system" style="font-size: 80%;">Four qubit system</a></li>
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<!-- navigation toc: --> <li><a href="#qft-acts-as-follows" style="font-size: 80%;">QFT acts as follows</a></li>
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<!-- navigation toc: --> <li><a href="#quantum-circuits" style="font-size: 80%;">Quantum circuits</a></li>
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<!-- navigation toc: --> <li><a href="#rotation-gates" style="font-size: 80%;">Rotation gates</a></li>
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<!-- navigation toc: --> <li><a href="#using-the-hadamard-gate" style="font-size: 80%;">Using the Hadamard gate</a></li>
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<!-- navigation toc: --> <li><a href="#algorithm" style="font-size: 80%;">Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="#binary-fraction" style="font-size: 80%;">Binary fraction</a></li>
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<!-- navigation toc: --> <li><a href="#controlled-rotation-gate" style="font-size: 80%;">Controlled rotation gate</a></li>
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<!-- navigation toc: --> <li><a href="#applying-to-all-qubits" style="font-size: 80%;">Applying to all qubits</a></li>
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<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-april-22-26" style="font-size: 80%;">Plans for the week of April 22-26</a></li>
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<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-april-21-25" style="font-size: 80%;">Plans for the week of April 21-25</a></li>
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</ul>
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</li>
@@ -130,281 +76,36 @@ <h1>Quantum Computing, Quantum Machine Learning and Quantum Information Theories
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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<b>Morten Hjorth-Jensen</b>
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</center>
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<!-- institution(s) -->
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<!-- institution -->
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<center>
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[1] <b>Department of Physics, University of Oslo</b>
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</center>
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<center>
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[2] <b>Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University</b>
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<b>Department of Physics, University of Oslo, Norway</b>
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</center>
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<br>
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<center>
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<h4>April 17</h4>
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<h4>April 23</h4>
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</center> <!-- date -->
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<br>
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<!-- potential-jumbotron-button -->
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</div> <!-- end jumbotron -->
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<!-- !split -->
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<h2 id="plans-for-the-week-of-april-15-19" class="anchor">Plans for the week of April 15-19 </h2>
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<h2 id="plans-for-the-week-of-april-21-25" class="anchor">Plans for the week of April 21-25 </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<ol>
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<li> Quantum Fourier transforms (QFT), reminder from last week</li>
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<li> Implementing QFTs</li>
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<li> Quantum phase estimation (QPE) and computation of eigenvalues</li>
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<li> Reading recommendations: Hundt's text <b>Quantum Computing for Programmers</b>, sections 6.1-6.4</li>
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<li> <a href="https://youtu.be/gNKJ_sBrPuE" target="_self">Video of lecture at</a></li>
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<li> <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesApril17.pdf" target="_self">Whiteboard notes</a></li>
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<li> Discussion of project 2</li>
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<li> TBA</li>
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<li> <a href="https://youtu.be/" target="_self">Video of lecture TBA</a>
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<!-- o <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesApril17.pdf" target="_self">Whiteboard notes</a> --></li>
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</ol>
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</div>
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</div>
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<!-- !split -->
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<h2 id="qft" class="anchor">QFT </h2>
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<p>The Quantum Fourier Transform (QFT) has mathematically the same equation as starting point
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but the notation is generally different. We generally compute the
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quantum Fourier transform on a set of orthonormal basis state vectors
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\(|0\rangle, |1\rangle, ..., |N-1\rangle\). The linear operator defining
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the transform is given by the action on basis states
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</p>
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$$
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|j\rangle \mapsto \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} e^{2\pi ijk/N}|k\rangle.
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$$
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<!-- !split -->
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<h2 id="in-terms-of-arbitrary-states" class="anchor">In terms of arbitrary states </h2>
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<p>This can be written on arbitrary states,</p>
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$$
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\sum_{j=0}^{N-1}x_j|j\rangle \mapsto \sum_{k=0}^{N-1} y_k|k\rangle
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$$
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<p>where each amplitude \(y_k\) is the discrete Fourier transform of
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\(x_j\).
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</p>
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<!-- !split -->
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<h2 id="unitarity" class="anchor">Unitarity </h2>
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<p>The quantum Fourier transform is unitary. Taking \(N = 2^n\),
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for \(n\) qubits gives us the orthonormal (computational) basis
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</p>
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$$
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|0\rangle, |1\rangle, ..., |2^{n}-1\rangle.
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$$
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<!-- !split -->
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<h2 id="binary-representation" class="anchor">Binary representation </h2>
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<p>Each of the computational basis states can be represented in binary</p>
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$$
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j = j_1j_2 \cdots j_n
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$$
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<p>where each \(j_k\) is either \(0\) or \(1\), and the corresponding
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binary vector is \(|j_1j_2 \cdots j_n\rangle\).
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</p>
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<!-- !split -->
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<h2 id="rewriting-the-qft" class="anchor">Rewriting the QFT </h2>
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<p>The quantum Fourier
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transform on one of these \(n\)-qubit vectors can be written as,
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</p>
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$$
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\begin{align*}
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|j_1j_2 \cdots j_n \rangle = \frac{\left(|0\rangle +e^{2\pi i0.j_n}|1\rangle\right) \otimes \left(|0\rangle +e^{2\pi i0.j_{n-1}j_n}|1\rangle\right) \otimes \cdots \otimes \left(|0\rangle +e^{2\pi i0.j_1j_2 \cdots j_n}|1\rangle\right)}{2^{n/2}}
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\end{align*}
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$$
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<!-- !split -->
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<h2 id="short-hand-notation" class="anchor">Short hand notation </h2>
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<p>In the above, we use the notation</p>
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$$
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0.j_lj_{l+1} \cdots j_n = \frac{j+l}{2} + \frac{j_{l+1}}{2^2} + \cdots + \frac{j_n}{2^{m-l+1}}
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$$
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<!-- !split -->
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<h2 id="four-qubit-system" class="anchor">Four qubit system </h2>
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<p>The basis states are</p>
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$$
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|j_1j_2j_3j_4 \rangle
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$$
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<p>where \(j_k\) is either \(0\) or \(1\). We have</p>
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$$
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\begin{align*}
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0.j_3 &= \frac{j_3}{2} \\
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0.j_2j_3 &= \frac{j_2}{2} + \frac{j_3}{4} \\
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0.j_1j_2j_3 &= \frac{j_1}{2} + \frac{j_2}{4} + \frac{j_3}{8} \\
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0.j_0j_1j_2j_3 &= \frac{j_0}{2} + \frac{j_1}{4} + \frac{j_2}{8} + \frac{j_3}{16} \\
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\end{align*}
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$$
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<!-- !split -->
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<h2 id="qft-acts-as-follows" class="anchor">QFT acts as follows </h2>
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<p>The quantum Fourier transform acts as follows:</p>
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$$
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\begin{align*}
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|j_1j_2j_3j_4 \rangle \mapsto \frac{1}{\sqrt{2^{4/2}}}
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\left(|0\rangle + e^{2 \pi i 0.j4}|1\rangle \right) \otimes
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\left(|0\rangle + e^{2 \pi i 0.j_3j4}|1\rangle \right) \otimes
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\left(|0\rangle + e^{2 \pi i 0.j_2j_3j4}|1\rangle \right) \otimes
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\left(|0\rangle + e^{2 \pi i 0.j_1j_2j_3j4}|1\rangle \right)
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\end{align*}
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$$
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<!-- !split -->
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<h2 id="quantum-circuits" class="anchor">Quantum circuits </h2>
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<p>To compose a quantum circuit that calculates the quantum Fourier
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transform we use the operators
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</p>
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$$
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\begin{align*}
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R_k =
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\begin{bmatrix}
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1 & 0 \\
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0 & e^{2\pi i/2^k}
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\end{bmatrix}.
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\end{align*}
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$$
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<!-- !split -->
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<h2 id="rotation-gates" class="anchor">Rotation gates </h2>
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<p>In this example, the \(R_k\) gates are:</p>
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$$
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\begin{align*}
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R_1 = \begin{bmatrix}
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1 & 0 \\
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0 & e^{2 \pi i /2^0}
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\end{bmatrix} =
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\begin{bmatrix}
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1 & 0 \\
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0 & 1
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\end{bmatrix}, \quad
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R_2 = \begin{bmatrix}
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1 & 0 \\
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0 & e^{2 \pi i /2^2}
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\end{bmatrix}, \quad
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R_3 = \begin{bmatrix}
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1 & 0 \\
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0 & e^{2 \pi i /2^3}
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\end{bmatrix}, \quad
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R_4 = \begin{bmatrix}
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1 & 0 \\
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0 & e^{2 \pi i /2^4}
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\end{bmatrix}.
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\end{align*}
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$$
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<!-- !split -->
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<h2 id="using-the-hadamard-gate" class="anchor">Using the Hadamard gate </h2>
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<p>The Hadamard
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gate on a single qubit creates an equal superposition of its basis
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states, assuming it is not already in a superposition, such that
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</p>
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$$
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H\vert 0 \rangle = \frac{1}{\sqrt{2}} \left(\vert 0 \rangle + \vert 1\rangle\right), \quad H\vert 1\rangle = \frac{1}{\sqrt{2}} \left(\vert 0 \rangle - \vert 1\rangle\right)
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$$
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<p>The \( R_k \)&#160;gate simply adds a phase if the qubit it acts on is in the state \( \vert 1\rangle \)</p>
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$$
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R_k\vert 0 \rangle = \vert 0 \rangle, \quad R_k\vert 1\rangle = e^{2\pi i/2^{k}}\vert 1\rangle
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$$
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<p>Since all this gates are unitary, the quantum Fourier transfrom is also unitary.</p>
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<!-- !split -->
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<h2 id="algorithm" class="anchor">Algorithm </h2>
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<p>Assume we have a quantum register of \( n \) qubits in the state \( \vert j_1 j_2 \dots j_n\rangle \).
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Applying the Hadamard gate to the first qubit
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produces the state
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</p>
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$$
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H\vert j_1 j_2 \dots j_n\rangle = \frac{\left(\vert 0 \rangle + e^{2\pi i 0.j_1}\vert 1\rangle\right)}{2^{1/2}} \vert j_2 \dots j_n\rangle.
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$$
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<!-- !split -->
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<h2 id="binary-fraction" class="anchor">Binary fraction </h2>
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<p>Here we have made use of the binary fraction to represent the action of the Hadamard gate </p>
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$$
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\exp{2\pi i 0.j_1} = -1,
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$$
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<p>if \( j_1 = 1 \) and \( +1 \) if \( j_1 = 0 \).</p>
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<!-- !split -->
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<h2 id="controlled-rotation-gate" class="anchor">Controlled rotation gate </h2>
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<p>Furthermore we can apply the controlled-\( R_k \)&#160;gate, with all the other qubits \( j_k \) for \( k>1 \) as control qubits to produce the state</p>
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$$
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\frac{\left(\vert 0 \rangle + e^{2\pi i 0.j_1j_2\dots j_n}\vert 1\rangle\right)}{2^{1/2}} \vert j_2 \dots j_n\rangle
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$$
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<p>Next we do the same procedure on qubit \( 2 \)&#160;producing the state</p>
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$$
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\frac{\left(\vert 0 \rangle + e^{2\pi i 0.j_1j_2\dots j_n}\vert 1\rangle\right)\left(\vert 0 \rangle + e^{2\pi i 0.j_2\dots j_n}\vert 1\rangle\right)}{2^{2/2}} \vert j_2 \dots j_n\rangle
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$$
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<!-- !split -->
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<h2 id="applying-to-all-qubits" class="anchor">Applying to all qubits </h2>
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<p>Doing this for all \( n \) qubits yields state</p>
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$$
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\frac{\left(\vert 0 \rangle + e^{2\pi i 0.j_1j_2\dots j_n}\vert 1\rangle\right)\left(\vert 0 \rangle + e^{2\pi i 0.j_2\dots j_n}\vert 1\rangle\right)\dots \left(\vert 0 \rangle + e^{2\pi i 0.j_n}\vert 1\rangle\right)}{2^{n/2}} \vert j_2 \dots j_n\rangle
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$$
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<p>At the end we use swap gates to reverse the order of the qubits</p>
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$$
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\frac{\left(\vert 0 \rangle + e^{2\pi i 0.j_n}\vert 1\rangle\right)\left(\vert 0 \rangle + e^{2\pi i 0.j_{n-1}j_n}\vert 1\rangle\right)\dots\left(\vert 0 \rangle + e^{2\pi i 0.j_1j_2\dots j_n}\vert 1\rangle\right) }{2^{n/2}} \vert j_2 \dots j_n\rangle
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$$
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<p>This is just the product representation from earlier, obviously our desired output.</p>
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<!-- !split -->
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<h2 id="plans-for-the-week-of-april-22-26" class="anchor">Plans for the week of April 22-26 </h2>
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<ol>
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<li> Summary of algorithms and discussion of project 2</li>
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<li> Discussion of physical realizations of quantum computers</li>
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</ol>
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