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‎doc/pub/week10/html/week10-bs.html‎

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@@ -256,9 +269,11 @@
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<!-- navigation toc: --> <li><a href="#why-quantum-fourier-transforms-and-exponential-speedup" style="font-size: 80%;"><b>Why Quantum Fourier Transforms and exponential speedup</b></a></li>
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transforms-and-quantum-parallelism" style="font-size: 80%;"><b>Quantum Fourier Transforms and quantum parallelism</b></a></li>
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transforms-and-implementation" style="font-size: 80%;"><b>Quantum Fourier Transforms and implementation</b></a></li>
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<!-- navigation toc: --> <li><a href="#why-quantum-fourier-transforms-vi" style="font-size: 80%;"><b>Why Quantum Fourier Transforms? VI</b></a></li>
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<!-- navigation toc: --> <li><a href="#why-quantum-fourier-transforms-i" style="font-size: 80%;"><b>Why Quantum Fourier Transforms? I</b></a></li>
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<!-- navigation toc: --> <li><a href="#a-familiar-case" style="font-size: 80%;"><b>A familiar case</b></a></li>
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transform-and-applications-in-quantum-computing" style="font-size: 80%;"><b>Quantum Fourier Transform and Applications in Quantum Computing</b></a></li>
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transforms-and-reduced-memory-complexity" style="font-size: 80%;"><b>Quantum Fourier Transforms and Reduced Memory Complexity</b></a></li>
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transforms-quantum-signal-processing" style="font-size: 80%;"><b>Quantum Fourier Transforms Quantum Signal Processing</b></a></li>
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<!-- navigation toc: --> <li><a href="#why-quantum-fourier-transforms-brief-summary" style="font-size: 80%;"><b>Why Quantum Fourier Transforms? Brief summary</b></a></li>
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<!-- navigation toc: --> <li><a href="#now-technicalities-reminder-on-fourier-theory-a-familiar-case-first" style="font-size: 80%;"><b>Now technicalities: Reminder on Fourier theory, a familiar case first</b></a></li>
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<!-- navigation toc: --> <li><a href="#several-driving-forces" style="font-size: 80%;"><b>Several driving forces</b></a></li>
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<!-- navigation toc: --> <li><a href="#periodicity" style="font-size: 80%;"><b>Periodicity</b></a></li>
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<!-- navigation toc: --> <li><a href="#simple-code-example" style="font-size: 80%;"><b>Simple Code Example</b></a></li>
@@ -456,13 +471,43 @@ <h2 id="quantum-fourier-transforms-and-implementation" class="anchor">Quantum Fo
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<li> This makes QFT highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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<!-- !split -->
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<h2 id="why-quantum-fourier-transforms-vi" class="anchor">Why Quantum Fourier Transforms? VI </h2>
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<h2 id="quantum-fourier-transform-and-applications-in-quantum-computing" class="anchor">Quantum Fourier Transform and Applications in Quantum Computing </h2>
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<ol>
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<li> \textbf{Shor&#8217;s Algorithm:} Uses QFTs to find periodicity in modular exponentiation.</li>
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<li> \textbf{Quantum Phase Estimation (QPE):} QFTs extract eigenvalues of unitary matrices.</li>
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<li> \textbf{Quantum Signal Processing:} QFTs Enable spectral analysis on quantum data.</li>
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</ol>
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<!-- !split -->
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<h2 id="why-quantum-fourier-transforms-i" class="anchor">Why Quantum Fourier Transforms? I </h2>
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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity" class="anchor">Quantum Fourier Transforms and Reduced Memory Complexity </h2>
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<ol>
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<li> Classical DFTs require \( O(N) \) space.</li>
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<li> QFT store Fourier-transformed coefficients <em>implicitly</em> in qubit amplitudes.</li>
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<li> Only \( O(n) \) qubits are needed for a size \( N = 2^n \) transformation.</li>
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</ol>
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<!-- !split -->
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<h2 id="quantum-fourier-transforms-quantum-signal-processing" class="anchor">Quantum Fourier Transforms Quantum Signal Processing </h2>
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<p>QFTs enable applications such as:</p>
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<ol>
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<li> Quantum spectral analysis.</li>
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<li> Quantum image processing.</li>
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<li> Quantum filtering and denoising.</li>
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</ol>
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<p>These can enhance AI, cryptography, and data processing.</p>
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<!-- !split -->
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<h2 id="why-quantum-fourier-transforms-brief-summary" class="anchor">Why Quantum Fourier Transforms? Brief summary </h2>
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<ol>
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<li> Quantum Fourier Transform provides <em>exponential speedup</em></li>
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<li> It enables key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
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<li> QFT is more efficient in <em>memory usage and circuit complexity</em> than classical methods.</li>
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<li> Future quantum applications in <em>signal processing, AI, and cryptography</em> will heavily rely on QFT.</li>
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</ol>
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<!-- !split -->
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<h2 id="a-familiar-case" class="anchor">A familiar case </h2>
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<h2 id="now-technicalities-reminder-on-fourier-theory-a-familiar-case-first" class="anchor">Now technicalities: Reminder on Fourier theory, a familiar case first </h2>
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<p>For problems with so-called harmonic oscillations, given by for example the following differential equation</p>
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$$

‎doc/pub/week10/html/week10-reveal.html‎

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</section>
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<section>
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<h2 id="why-quantum-fourier-transforms-vi">Why Quantum Fourier Transforms? VI </h2>
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<h2 id="quantum-fourier-transform-and-applications-in-quantum-computing">Quantum Fourier Transform and Applications in Quantum Computing </h2>
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<ol>
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<p><li> \textbf{Shor&#8217;s Algorithm:} Uses QFTs to find periodicity in modular exponentiation.</li>
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<p><li> \textbf{Quantum Phase Estimation (QPE):} QFTs extract eigenvalues of unitary matrices.</li>
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<p><li> \textbf{Quantum Signal Processing:} QFTs Enable spectral analysis on quantum data.</li>
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</ol>
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</section>
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<section>
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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity">Quantum Fourier Transforms and Reduced Memory Complexity </h2>
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<ol>
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<p><li> Classical DFTs require \( O(N) \) space.</li>
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<p><li> QFT store Fourier-transformed coefficients <em>implicitly</em> in qubit amplitudes.</li>
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<p><li> Only \( O(n) \) qubits are needed for a size \( N = 2^n \) transformation.</li>
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</ol>
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</section>
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<section>
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<h2 id="why-quantum-fourier-transforms-i">Why Quantum Fourier Transforms? I </h2>
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<h2 id="quantum-fourier-transforms-quantum-signal-processing">Quantum Fourier Transforms Quantum Signal Processing </h2>
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<p>QFTs enable applications such as:</p>
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<ol>
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<p><li> Quantum spectral analysis.</li>
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<p><li> Quantum image processing.</li>
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<p><li> Quantum filtering and denoising.</li>
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</ol>
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<p>
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<p>These can enhance AI, cryptography, and data processing.</p>
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</section>
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<section>
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<h2 id="why-quantum-fourier-transforms-brief-summary">Why Quantum Fourier Transforms? Brief summary </h2>
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<ol>
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<p><li> Quantum Fourier Transform provides <em>exponential speedup</em></li>
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<p><li> It enables key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
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<p><li> QFT is more efficient in <em>memory usage and circuit complexity</em> than classical methods.</li>
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<p><li> Future quantum applications in <em>signal processing, AI, and cryptography</em> will heavily rely on QFT.</li>
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</ol>
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</section>
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<section>
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<h2 id="a-familiar-case">A familiar case </h2>
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<h2 id="now-technicalities-reminder-on-fourier-theory-a-familiar-case-first">Now technicalities: Reminder on Fourier theory, a familiar case first </h2>
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<p>For problems with so-called harmonic oscillations, given by for example the following differential equation</p>
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<p>&nbsp;<br>

‎doc/pub/week10/html/week10-solarized.html‎

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('Simple Code Example', 2, None, 'simple-code-example'),
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<li> This makes QFT highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-quantum-fourier-transforms-vi">Why Quantum Fourier Transforms? VI </h2>
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<h2 id="quantum-fourier-transform-and-applications-in-quantum-computing">Quantum Fourier Transform and Applications in Quantum Computing </h2>
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<ol>
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<li> \textbf{Shor&#8217;s Algorithm:} Uses QFTs to find periodicity in modular exponentiation.</li>
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<li> \textbf{Quantum Phase Estimation (QPE):} QFTs extract eigenvalues of unitary matrices.</li>
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<li> \textbf{Quantum Signal Processing:} QFTs Enable spectral analysis on quantum data.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-quantum-fourier-transforms-i">Why Quantum Fourier Transforms? I </h2>
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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity">Quantum Fourier Transforms and Reduced Memory Complexity </h2>
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<ol>
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<li> Classical DFTs require \( O(N) \) space.</li>
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<li> QFT store Fourier-transformed coefficients <em>implicitly</em> in qubit amplitudes.</li>
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<li> Only \( O(n) \) qubits are needed for a size \( N = 2^n \) transformation.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="quantum-fourier-transforms-quantum-signal-processing">Quantum Fourier Transforms Quantum Signal Processing </h2>
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<p>QFTs enable applications such as:</p>
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<ol>
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<li> Quantum spectral analysis.</li>
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<li> Quantum image processing.</li>
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<li> Quantum filtering and denoising.</li>
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</ol>
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<p>These can enhance AI, cryptography, and data processing.</p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-quantum-fourier-transforms-brief-summary">Why Quantum Fourier Transforms? Brief summary </h2>
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<ol>
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<li> Quantum Fourier Transform provides <em>exponential speedup</em></li>
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<li> It enables key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
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<li> QFT is more efficient in <em>memory usage and circuit complexity</em> than classical methods.</li>
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<li> Future quantum applications in <em>signal processing, AI, and cryptography</em> will heavily rely on QFT.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="a-familiar-case">A familiar case </h2>
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<h2 id="now-technicalities-reminder-on-fourier-theory-a-familiar-case-first">Now technicalities: Reminder on Fourier theory, a familiar case first </h2>
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<p>For problems with so-called harmonic oscillations, given by for example the following differential equation</p>
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$$

‎doc/pub/week10/html/week10.html‎

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('Periodicity', 2, None, 'periodicity'),
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('Simple Code Example', 2, None, 'simple-code-example'),
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<li> This makes QFT highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-quantum-fourier-transforms-vi">Why Quantum Fourier Transforms? VI </h2>
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<h2 id="quantum-fourier-transform-and-applications-in-quantum-computing">Quantum Fourier Transform and Applications in Quantum Computing </h2>
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<ol>
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<li> \textbf{Shor&#8217;s Algorithm:} Uses QFTs to find periodicity in modular exponentiation.</li>
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<li> \textbf{Quantum Phase Estimation (QPE):} QFTs extract eigenvalues of unitary matrices.</li>
461+
<li> \textbf{Quantum Signal Processing:} QFTs Enable spectral analysis on quantum data.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-quantum-fourier-transforms-i">Why Quantum Fourier Transforms? I </h2>
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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity">Quantum Fourier Transforms and Reduced Memory Complexity </h2>
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<ol>
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<li> Classical DFTs require \( O(N) \) space.</li>
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<li> QFT store Fourier-transformed coefficients <em>implicitly</em> in qubit amplitudes.</li>
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<li> Only \( O(n) \) qubits are needed for a size \( N = 2^n \) transformation.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="quantum-fourier-transforms-quantum-signal-processing">Quantum Fourier Transforms Quantum Signal Processing </h2>
473+
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<p>QFTs enable applications such as:</p>
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<ol>
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<li> Quantum spectral analysis.</li>
477+
<li> Quantum image processing.</li>
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<li> Quantum filtering and denoising.</li>
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</ol>
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<p>These can enhance AI, cryptography, and data processing.</p>
481+
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-quantum-fourier-transforms-brief-summary">Why Quantum Fourier Transforms? Brief summary </h2>
484+
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<ol>
486+
<li> Quantum Fourier Transform provides <em>exponential speedup</em></li>
487+
<li> It enables key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
488+
<li> QFT is more efficient in <em>memory usage and circuit complexity</em> than classical methods.</li>
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<li> Future quantum applications in <em>signal processing, AI, and cryptography</em> will heavily rely on QFT.</li>
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</ol>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="a-familiar-case">A familiar case </h2>
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<h2 id="now-technicalities-reminder-on-fourier-theory-a-familiar-case-first">Now technicalities: Reminder on Fourier theory, a familiar case first </h2>
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<p>For problems with so-called harmonic oscillations, given by for example the following differential equation</p>
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$$

‎doc/pub/week10/ipynb/ipynb-week10-src.tar.gz‎

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