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‎doc/pub/week10/html/week10-bs.html‎

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@@ -57,23 +57,25 @@
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@@ -268,10 +270,10 @@
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transforms-qfts" style="font-size: 80%;"><b>Quantum Fourier Transforms (QFTs)</b></a></li>
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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="#quantum-fourier-transforms-and-implementations" style="font-size: 80%;"><b>Quantum Fourier Transforms and implementations</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="#quantum-fourier-transforms-and-reduced-memory-complexity-not-covered" style="font-size: 80%;"><b>Quantum Fourier Transforms and Reduced Memory Complexity (not covered)</b></a></li>
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<!-- navigation toc: --> <li><a href="#quantum-fourier-transforms-and-quantum-signal-processing-not-covered-in-this-course" style="font-size: 80%;"><b>Quantum Fourier Transforms and Quantum Signal Processing (not covered in this course)</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>
@@ -457,39 +459,39 @@ <h2 id="why-quantum-fourier-transforms-and-exponential-speedup" class="anchor">W
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<h2 id="quantum-fourier-transforms-and-quantum-parallelism" class="anchor">Quantum Fourier Transforms and quantum parallelism </h2>
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<ol>
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<li> QFTs acts on a <em>superposition</em> of states, processing all inputs simultaneously.</li>
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<li> QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.</li>
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<li> This is crucial in algorithms like:
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<ol type="a"></li>
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<li> Shor&#8217;s Algorithm for factoring large numbers.</li>
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<li> Quantum Phase Estimation (QPE) for eigenvalue extraction.</li>
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</ol>
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</ol>
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<!-- !split -->
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<h2 id="quantum-fourier-transforms-and-implementation" class="anchor">Quantum Fourier Transforms and implementation </h2>
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<h2 id="quantum-fourier-transforms-and-implementations" class="anchor">Quantum Fourier Transforms and implementations </h2>
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<ol>
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<li> QFT requires only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<li> A 3-qubit QFT circuit uses only \( O(n^2) \).</li>
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<li> This makes QFT highly efficient for <em>quantum hardware</em>.</li>
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<li> This makes QFTs highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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<!-- !split -->
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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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<li> \textbf{Quantum Signal Processing:} QFTs enable spectral analysis on quantum data. Important for AI applications.</li>
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</ol>
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<!-- !split -->
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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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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity-not-covered" class="anchor">Quantum Fourier Transforms and Reduced Memory Complexity (not covered) </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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<h2 id="quantum-fourier-transforms-and-quantum-signal-processing-not-covered-in-this-course" class="anchor">Quantum Fourier Transforms and Quantum Signal Processing (not covered in this course) </h2>
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<p>QFTs enable applications such as:</p>
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<ol>
@@ -503,9 +505,9 @@ <h2 id="quantum-fourier-transforms-quantum-signal-processing" class="anchor">Qua
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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> Quantum Fourier Transforms provide <em>exponential speedup</em></li>
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<li> They enable key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
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<li> QFTs are 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 -->

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

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<h2 id="quantum-fourier-transforms-and-quantum-parallelism">Quantum Fourier Transforms and quantum parallelism </h2>
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<p><li> QFTs acts on a <em>superposition</em> of states, processing all inputs simultaneously.</li>
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<p><li> QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.</li>
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<p><li> This is crucial in algorithms like:
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<p><li> Shor&#8217;s Algorithm for factoring large numbers.</li>
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<h2 id="quantum-fourier-transforms-and-implementation">Quantum Fourier Transforms and implementation </h2>
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<h2 id="quantum-fourier-transforms-and-implementations">Quantum Fourier Transforms and implementations </h2>
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<p><li> QFT requires only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<p><li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<p><li> A 3-qubit QFT circuit uses only \( O(n^2) \).</li>
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<p><li> This makes QFT highly efficient for <em>quantum hardware</em>.</li>
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<p><li> This makes QFTs highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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</section>
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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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<p><li> \textbf{Quantum Signal Processing:} QFTs enable spectral analysis on quantum data. Important for AI applications.</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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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity-not-covered">Quantum Fourier Transforms and Reduced Memory Complexity (not covered) </h2>
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<ol>
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<p><li> Classical DFTs require \( O(N) \) space.</li>
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</section>
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<section>
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<h2 id="quantum-fourier-transforms-quantum-signal-processing">Quantum Fourier Transforms Quantum Signal Processing </h2>
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<h2 id="quantum-fourier-transforms-and-quantum-signal-processing-not-covered-in-this-course">Quantum Fourier Transforms and Quantum Signal Processing (not covered in this course) </h2>
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<p>QFTs enable applications such as:</p>
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<ol>
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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> Quantum Fourier Transforms provide <em>exponential speedup</em></li>
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<p><li> They enable key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
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<p><li> QFTs are 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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</section>

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

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<h2 id="quantum-fourier-transforms-and-quantum-parallelism">Quantum Fourier Transforms and quantum parallelism </h2>
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<li> QFTs acts on a <em>superposition</em> of states, processing all inputs simultaneously.</li>
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<li> QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.</li>
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<li> This is crucial in algorithms like:
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<li> Shor&#8217;s Algorithm for factoring large numbers.</li>
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<li> Quantum Phase Estimation (QPE) for eigenvalue extraction.</li>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="quantum-fourier-transforms-and-implementation">Quantum Fourier Transforms and implementation </h2>
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<h2 id="quantum-fourier-transforms-and-implementations">Quantum Fourier Transforms and implementations </h2>
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<li> QFT requires only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<li> A 3-qubit QFT circuit uses only \( O(n^2) \).</li>
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<li> This makes QFT highly efficient for <em>quantum hardware</em>.</li>
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<li> This makes QFTs highly efficient for <em>quantum hardware</em>.</li>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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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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<li> \textbf{Quantum Signal Processing:} QFTs enable spectral analysis on quantum data. Important for AI applications.</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-and-reduced-memory-complexity">Quantum Fourier Transforms and Reduced Memory Complexity </h2>
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<h2 id="quantum-fourier-transforms-and-reduced-memory-complexity-not-covered">Quantum Fourier Transforms and Reduced Memory Complexity (not covered) </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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<h2 id="quantum-fourier-transforms-and-quantum-signal-processing-not-covered-in-this-course">Quantum Fourier Transforms and Quantum Signal Processing (not covered in this course) </h2>
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<p>QFTs enable applications such as:</p>
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<ol>
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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> Quantum Fourier Transforms provide <em>exponential speedup</em></li>
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<li> They enable key quantum algorithms like <em>Shor&#8217;s Algorithm</em> and <em>Quantum Phase Estimation</em>.</li>
415+
<li> QFTs are 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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