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‎doc/pub/week10/html/week10-bs.html‎

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@@ -121,22 +121,6 @@
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None,
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'more-summary-the-discrete-fourier-transform-dft'),
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('Output vector', 2, None, 'output-vector'),
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('Simple example', 2, None, 'simple-example'),
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('Discrete Fourier Transformations',
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2,
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None,
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'discrete-fourier-transformations'),
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('More details on Discrete Fourier transforms',
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2,
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None,
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'more-details-on-discrete-fourier-transforms'),
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('Inverse DFT', 2, None, 'inverse-dft'),
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('Orthonormality', 2, None, 'orthonormality'),
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('Inverse transform', 2, None, 'inverse-transform'),
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('Fast Fourier transform and polynomial multiplication',
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2,
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None,
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'fast-fourier-transform-and-polynomial-multiplication'),
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('From DFT to QFT', 2, None, 'from-dft-to-qft'),
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('In terms of arbitrary states',
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2,
@@ -277,13 +261,6 @@
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<!-- navigation toc: --> <li><a href="#fast-fourier-transform-fft" style="font-size: 80%;"><b>Fast Fourier transform (FFT)</b></a></li>
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<!-- navigation toc: --> <li><a href="#more-summary-the-discrete-fourier-transform-dft" style="font-size: 80%;"><b>More summary: The discrete Fourier transform (DFT)</b></a></li>
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<!-- navigation toc: --> <li><a href="#output-vector" style="font-size: 80%;"><b>Output vector</b></a></li>
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<!-- navigation toc: --> <li><a href="#simple-example" style="font-size: 80%;"><b>Simple example</b></a></li>
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<!-- navigation toc: --> <li><a href="#discrete-fourier-transformations" style="font-size: 80%;"><b>Discrete Fourier Transformations</b></a></li>
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<!-- navigation toc: --> <li><a href="#more-details-on-discrete-fourier-transforms" style="font-size: 80%;"><b>More details on Discrete Fourier transforms</b></a></li>
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<!-- navigation toc: --> <li><a href="#inverse-dft" style="font-size: 80%;"><b>Inverse DFT</b></a></li>
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<!-- navigation toc: --> <li><a href="#orthonormality" style="font-size: 80%;"><b>Orthonormality</b></a></li>
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<!-- navigation toc: --> <li><a href="#inverse-transform" style="font-size: 80%;"><b>Inverse transform</b></a></li>
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<!-- navigation toc: --> <li><a href="#fast-fourier-transform-and-polynomial-multiplication" style="font-size: 80%;"><b>Fast Fourier transform and polynomial multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="#from-dft-to-qft" style="font-size: 80%;"><b>From DFT to QFT</b></a></li>
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<!-- navigation toc: --> <li><a href="#in-terms-of-arbitrary-states" style="font-size: 80%;"><b>In terms of arbitrary states</b></a></li>
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<!-- navigation toc: --> <li><a href="#unitarity" style="font-size: 80%;"><b>Unitarity</b></a></li>
@@ -375,7 +352,6 @@ <h2 id="plans-for-the-week-of-march-24-28-2025" class="anchor">Plans for the wee
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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> TBA</li>
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<li> Start discussion of discrete Fourier transforms and Quantum Fourier transforms, basic mathematical expressions</li>
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<li> Reading recommendation Hundt, Quantum Computing for Programmers, sections 6.1-6.4 on QFT.
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<!-- o <a href="https://youtu.be/" target="_self">Video of lecture TBA</a> -->
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<!-- !split -->
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<h2 id="possible-paths-for-project-2" class="anchor">Possible paths for project 2 </h2>
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<ul>
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<ol>
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<li> Implement QFTs and study the quantum phase estimation (QPE) algorithm and
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<ol type="a"></li>
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<li> Compare QFTs with Fast Fourier transforms</li>
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<li> Study the solution of quantum mechanical eigenvalue problems with systems from atomic/molecular physics and quantum chemistry using adaptive QPE</li>
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<li> Quantum machine learning projects like quantum Boltzmann machine or quantum neural networks</li>
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<li> Other ideas</li>
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</ul>
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</ol>
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<p>For project 2, in order to be time efficient, you can use software like Qiskit, Pennylane, qBraid and/or other</p>
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<!-- !split -->
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motivations for the DFT. For those of you familiar with signal
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processing, harmonic oscillations, and many other areas of
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applications, Fourier transforms are almost standard kitchen
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items. For those of you who have studied quantum theory, you have
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probably met Fourier transforms when studying Heisenberg's uncertainty
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relation between momentum and position.
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items.
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</p>
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<!-- !split -->
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<!-- !split -->
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<h2 id="continuous-fourier-transforms-and-the-principle-of-superposition" class="anchor">Continuous Fourier transforms and the principle of Superposition </h2>
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<p>It was Fourier's idea to expand a continuous and periodic function \( f(t) \) in terms of sums sinus and cosinus ordered functions (we will use exponentials however)
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<p>It was Fourier's idea to expand a continuous and periodic function \( f(t) \) in terms of sums sine and cosine ordered functions (we will use exponentials however)
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as
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</p>
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$$
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The constant \( c_{0}=c_0^* \) is a real number.
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</p>
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<p>The above sum can also be rewritten in terms of the real part of the exponetials as</p>
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<p>The above sum can also be rewritten in terms of the real part of the exponentials as</p>
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$$
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f(t) =2\mathrm{Re}\left(\sum_{k=0}^{n}c_{k}\exp{(2\pi\imath kt} \right),
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$$
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$$
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<!-- !split -->
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<h2 id="simple-example" class="anchor">Simple example </h2>
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<p>As an example we can have a set of complex numbers
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\( \{x_0,\dots,x_{n-1}\} \) with fixed length \( n \), we can Fourier
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transform this as
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</p>
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$$
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y_k = \frac{1}{\sqrt{n}} \sum_{j=0}^{n-1} x_j \exp{i(2\pi jk)/n},
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$$
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<p>leading to a new set of complex numbers \( \{ y_0,\dots,y_{n-1}\} \). </p>
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<!-- !split -->
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<h2 id="discrete-fourier-transformations" class="anchor">Discrete Fourier Transformations </h2>
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<p>Consider two sets of complex numbers \( x_k \) and \( y_k \) with
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\( k=0,1,\dots,n-1 \) entries. The discrete Fourier transform is defined
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as
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</p>
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$$
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y_k = \frac{1}{\sqrt{n}} \sum_{j=0}^{n-1} \exp{(\frac{2\pi\imath jk}{n})} x_j.
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$$
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<p>As an example, assume \( x_0=1 \) and \( x_1=1 \). We can then use the above expression to find \( y_0 \) and \( y_1 \).</p>
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<p>With the above formula we get then</p>
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$$
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\begin{align*}
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y_0 &= \frac{1}{\sqrt{2}} \left( \exp{(\frac{2\pi\imath 0\times 1}{2})} \times 1+\exp{(\frac{2\pi\imath 0\times 1}{2})}\times 2\right)\\
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& =\frac{1}{\sqrt{2}}(1+2)=\frac{3}{\sqrt{2}},
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\end{align*}
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$$
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<p>and</p>
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$$
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\begin{align*}
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y_1 &= \frac{1}{\sqrt{2}} \left( \exp{(\frac{2\pi\imath 0\times 1}{2})} \times 1+\exp{(\frac{2\pi\imath 1\times 1}{2})}\times 2\right)=\\
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& =\frac{1}{\sqrt{2}}(1+2\exp{(\pi\imath)})=-\frac{1}{\sqrt{2}}.
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\end{align*}
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$$
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<!-- !split -->
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<h2 id="more-details-on-discrete-fourier-transforms" class="anchor">More details on Discrete Fourier transforms </h2>
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<p>Suppose that we have a vector \( f \) of \( n \) complex numbers, \( f_{k}, k
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\in\{0,1, \ldots, n-1\} \). Then the discrete Fourier transform (DFT) is
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a map from these \( n \) complex numbers to \( n \) complex numbers, the
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Fourier transformed coefficients \( \tilde{f}_{j} \), given by
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</p>
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$$
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\begin{equation*}
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\tilde{f}_{j}=\frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} \omega^{-j k} f_{k}
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\end{equation*}
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$$
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<p>where \( \omega=\exp \left(\frac{2 \pi i}{N}\right) \).</p>
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<!-- !split -->
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<h2 id="inverse-dft" class="anchor">Inverse DFT </h2>
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<p>The inverse DFT is given by</p>
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$$
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\begin{equation*}
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f_{j}=\frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} \omega^{j k} \tilde{f}_{k}
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\end{equation*}
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$$
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<p>To see this consider how the basis vectors transform. If \( f_{k}^{l}=\delta_{k, l} \), then</p>
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$$
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\begin{equation*}
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\tilde{f}_{j}^{l}=\frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} \omega^{-j k} \delta_{k, l}=\frac{1}{\sqrt{n}} \omega^{-j l}
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\end{equation*}
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$$
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<!-- !split -->
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<h2 id="orthonormality" class="anchor">Orthonormality </h2>
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<p>These DFT vectors are orthonormal, that is</p>
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$$
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\begin{equation*}
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\sum_{j=0}^{n-1} \tilde{f}^{l}{ }_{j}^{*} \tilde{f}_{j}^{m}=\frac{1}{n} \sum_{j=0}^{n-1} \omega^{j l} \omega^{-j m}=\frac{1}{N} \sum_{j=0}^{n-1} \omega^{j(l-m)}
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\end{equation*}
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$$
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<p>This last sum can be evaluated as a geometric series, but beware of the \( (l-m)=0 \) term, and yields</p>
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$$
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\begin{equation*}
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\sum_{j=0}^{n-1} \tilde{f}^{l}{ }_{j}^{*} \tilde{f}_{j}^{m}=\delta_{l, m}
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\end{equation*}
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$$
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<!-- !split -->
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<h2 id="inverse-transform" class="anchor">Inverse transform </h2>
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<p>From this we can check that the inverse DFT does indeed perform the inverse transform:</p>
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$$
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\begin{equation*}
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f_{j}=\frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} \omega^{j k} \tilde{f}_{k}=\frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} \omega^{j k} \frac{1}{\sqrt{n}} \sum_{l=0}^{n-1} \omega^{-l k} f_{l}=\frac{1}{n} \sum_{k, l=0}^{n-1} \omega^{(j-l) k} f_{l}=\sum_{l=0}^{n-1} \delta_{j, l} f_{l}=f_{j}
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\end{equation*}
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$$
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<!-- !split -->
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<h2 id="fast-fourier-transform-and-polynomial-multiplication" class="anchor">Fast Fourier transform and polynomial multiplication </h2>
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<p>The FFT algorithm is an \( O(n\log{n}) \) divide and conquer algorithm for DFT, used by
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Gauss circa 1805, and popularized by Cooley and Turkey and 1965. Gauss used the
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algorithm to determine periodic asteroid orbits, while Cooley and Turkey used it to
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detect Soviet nuclear tests from o&#64256;shore readings.
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A practical implementation of FFT is FFTW, which was described by Frigo and
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Johnson at MIT. The algorithm is often implemented directly in hardware, for fixed \( n \).
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</p>
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<h2 id="from-dft-to-qft" class="anchor">From DFT to QFT </h2>
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<h2 id="two-qubit-system" class="anchor">Two-qubit system </h2>
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<p>First a two qubit system. See whiteboard notes.</p>
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<p>First a two qubit system. Notes to be added.</p>
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<h2 id="four-qubit-system" class="anchor">Four qubit system </h2>

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