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‎doc/pub/week11/html/week11-bs.html‎

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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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('QFT, short review from last week',
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'qft-short-review-from-last-week'),
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('Advantage 1: Exponential Speedup',
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'advantage-1-exponential-speedup'),
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('Advantage 2: Quantum Parallelism',
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'advantage-2-quantum-parallelism'),
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('Advantage 3: Compact Circuit Implementation',
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'advantage-3-compact-circuit-implementation'),
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('Advantage 4: Applications in Quantum Computing',
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'advantage-4-applications-in-quantum-computing'),
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('Advantage 5: Reduced Memory Complexity',
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'advantage-5-reduced-memory-complexity'),
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('Advantage 6: Quantum Signal Processing',
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'advantage-6-quantum-signal-processing'),
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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,
@@ -185,6 +213,13 @@
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<!-- navigation toc: --> <li><a href="#inverse-dft" style="font-size: 80%;">Inverse DFT</a></li>
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<!-- navigation toc: --> <li><a href="#orthonormality" style="font-size: 80%;">Orthonormality</a></li>
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<!-- navigation toc: --> <li><a href="#inverse-transform" style="font-size: 80%;">Inverse transform</a></li>
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<!-- navigation toc: --> <li><a href="#qft-short-review-from-last-week" style="font-size: 80%;">QFT, short review from last week</a></li>
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<!-- navigation toc: --> <li><a href="#advantage-1-exponential-speedup" style="font-size: 80%;">Advantage 1: Exponential Speedup</a></li>
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<!-- navigation toc: --> <li><a href="#advantage-2-quantum-parallelism" style="font-size: 80%;">Advantage 2: Quantum Parallelism</a></li>
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<!-- navigation toc: --> <li><a href="#advantage-3-compact-circuit-implementation" style="font-size: 80%;">Advantage 3: Compact Circuit Implementation</a></li>
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<!-- navigation toc: --> <li><a href="#advantage-4-applications-in-quantum-computing" style="font-size: 80%;">Advantage 4: Applications in Quantum Computing</a></li>
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<!-- navigation toc: --> <li><a href="#advantage-5-reduced-memory-complexity" style="font-size: 80%;">Advantage 5: Reduced Memory Complexity</a></li>
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<!-- navigation toc: --> <li><a href="#advantage-6-quantum-signal-processing" style="font-size: 80%;">Advantage 6: Quantum Signal Processing</a></li>
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<!-- navigation toc: --> <li><a href="#from-dft-to-qft" style="font-size: 80%;">From DFT to 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>
@@ -484,6 +519,62 @@ <h2 id="inverse-transform" class="anchor">Inverse transform </h2>
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$$
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<!-- !split -->
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<h2 id="qft-short-review-from-last-week" class="anchor">QFT, short review from last week </h2>
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<ol>
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<li> QFTs are the quantum analogue of the Discrete Fourier Transform (DFT).</li>
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<li> They play a crucial role in quantum algorithms like Shor&#8217;s Algorithm and the Quantum Phase Estimation (QPE) algorithm</li>
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<li> QFTs provide an <em>exponential speedup</em> over classical Fourier Transform methods.</li>
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</ol>
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<!-- !split -->
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<h2 id="advantage-1-exponential-speedup" class="anchor">Advantage 1: Exponential Speedup </h2>
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<ol>
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<li> Classical Discrete Fourier Transforms (DFTs) require \( O(N^2) \) operations.</li>
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<li> The Fast Fourier Transforms (FFTs) improve this to \( O(N \log N) \) operations.</li>
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<li> QFTs reduce complexity to \( O((\log N)^2) \) using quantum gates.</li>
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</ol>
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<!-- !split -->
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<h2 id="advantage-2-quantum-parallelism" class="anchor">Advantage 2: Quantum Parallelism </h2>
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<p>QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.
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This is crucial in algorithms like:
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</p>
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<ol>
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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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<!-- !split -->
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<h2 id="advantage-3-compact-circuit-implementation" class="anchor">Advantage 3: Compact Circuit Implementation </h2>
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<ol>
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<li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<li> QFTs are highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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<!-- !split -->
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<h2 id="advantage-4-applications-in-quantum-computing" class="anchor">Advantage 4: Applications in Quantum Computing </h2>
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<ol>
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<li> \textbf{Shor&#8217;s Algorithm:} Uses QFT to find periodicity in modular exponentiation.</li>
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<li> \textbf{Quantum Phase Estimation (QPE):} Extracts eigenvalues of unitary matrices.</li>
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<li> \textbf{Quantum Signal Processing:} Enables spectral analysis on quantum data.</li>
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</ol>
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<!-- !split -->
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<h2 id="advantage-5-reduced-memory-complexity" class="anchor">Advantage 5: 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> QFTs 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="advantage-6-quantum-signal-processing" class="anchor">Advantage 6: 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="from-dft-to-qft" class="anchor">From DFT to QFT </h2>
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‎doc/pub/week11/html/week11-reveal.html‎

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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="qft-short-review-from-last-week">QFT, short review from last week </h2>
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<ol>
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<p><li> QFTs are the quantum analogue of the Discrete Fourier Transform (DFT).</li>
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<p><li> They play a crucial role in quantum algorithms like Shor&#8217;s Algorithm and the Quantum Phase Estimation (QPE) algorithm</li>
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<p><li> QFTs provide an <em>exponential speedup</em> over classical Fourier Transform methods.</li>
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</ol>
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</section>
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<section>
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<h2 id="advantage-1-exponential-speedup">Advantage 1: Exponential Speedup </h2>
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<ol>
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<p><li> Classical Discrete Fourier Transforms (DFTs) require \( O(N^2) \) operations.</li>
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<p><li> The Fast Fourier Transforms (FFTs) improve this to \( O(N \log N) \) operations.</li>
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<p><li> QFTs reduce complexity to \( O((\log N)^2) \) using quantum gates.</li>
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</ol>
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</section>
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<section>
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<h2 id="advantage-2-quantum-parallelism">Advantage 2: Quantum Parallelism </h2>
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<p>QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.
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This is crucial in algorithms like:
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</p>
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<ol>
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<p><li> Shor&#8217;s Algorithm for factoring large numbers.</li>
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<p><li> Quantum Phase Estimation (QPE) for eigenvalue extraction.</li>
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</ol>
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</section>
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<section>
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<h2 id="advantage-3-compact-circuit-implementation">Advantage 3: Compact Circuit Implementation </h2>
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<ol>
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<p><li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<p><li> QFTs are highly efficient for <em>quantum hardware</em>.</li>
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</ol>
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</section>
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<section>
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<h2 id="advantage-4-applications-in-quantum-computing">Advantage 4: Applications in Quantum Computing </h2>
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<ol>
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<p><li> \textbf{Shor&#8217;s Algorithm:} Uses QFT to find periodicity in modular exponentiation.</li>
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<p><li> \textbf{Quantum Phase Estimation (QPE):} Extracts eigenvalues of unitary matrices.</li>
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<p><li> \textbf{Quantum Signal Processing:} Enables 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="advantage-5-reduced-memory-complexity">Advantage 5: 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> QFTs 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="advantage-6-quantum-signal-processing">Advantage 6: 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="from-dft-to-qft">From DFT to QFT </h2>
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‎doc/pub/week11/html/week11-solarized.html‎

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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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('QFT, short review from last week',
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2,
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None,
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'qft-short-review-from-last-week'),
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('Advantage 1: Exponential Speedup',
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2,
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None,
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'advantage-1-exponential-speedup'),
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('Advantage 2: Quantum Parallelism',
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2,
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None,
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'advantage-2-quantum-parallelism'),
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('Advantage 3: Compact Circuit Implementation',
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None,
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'advantage-3-compact-circuit-implementation'),
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('Advantage 4: Applications in Quantum Computing',
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2,
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None,
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'advantage-4-applications-in-quantum-computing'),
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('Advantage 5: Reduced Memory Complexity',
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2,
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None,
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'advantage-5-reduced-memory-complexity'),
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('Advantage 6: Quantum Signal Processing',
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2,
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None,
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'advantage-6-quantum-signal-processing'),
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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,
@@ -421,6 +449,62 @@ <h2 id="inverse-transform">Inverse transform </h2>
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$$
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="qft-short-review-from-last-week">QFT, short review from last week </h2>
454+
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<ol>
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<li> QFTs are the quantum analogue of the Discrete Fourier Transform (DFT).</li>
457+
<li> They play a crucial role in quantum algorithms like Shor&#8217;s Algorithm and the Quantum Phase Estimation (QPE) algorithm</li>
458+
<li> QFTs provide an <em>exponential speedup</em> over classical Fourier Transform methods.</li>
459+
</ol>
460+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="advantage-1-exponential-speedup">Advantage 1: Exponential Speedup </h2>
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<ol>
463+
<li> Classical Discrete Fourier Transforms (DFTs) require \( O(N^2) \) operations.</li>
464+
<li> The Fast Fourier Transforms (FFTs) improve this to \( O(N \log N) \) operations.</li>
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<li> QFTs reduce complexity to \( O((\log N)^2) \) using quantum gates.</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="advantage-2-quantum-parallelism">Advantage 2: Quantum Parallelism </h2>
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<p>QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.
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This is crucial in algorithms like:
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</p>
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<ol>
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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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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="advantage-3-compact-circuit-implementation">Advantage 3: Compact Circuit Implementation </h2>
478+
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<ol>
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<li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
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<li> QFTs are highly efficient for <em>quantum hardware</em>.</li>
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</ol>
483+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="advantage-4-applications-in-quantum-computing">Advantage 4: Applications in Quantum Computing </h2>
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<ol>
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<li> \textbf{Shor&#8217;s Algorithm:} Uses QFT to find periodicity in modular exponentiation.</li>
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<li> \textbf{Quantum Phase Estimation (QPE):} Extracts eigenvalues of unitary matrices.</li>
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<li> \textbf{Quantum Signal Processing:} Enables 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="advantage-5-reduced-memory-complexity">Advantage 5: Reduced Memory Complexity </h2>
492+
493+
<ol>
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<li> Classical DFTs require \( O(N) \) space.</li>
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<li> QFTs store Fourier-transformed coefficients <em>implicitly</em> in qubit amplitudes.</li>
496+
<li> Only \( O(n) \) qubits are needed for a size \( N = 2^n \) transformation.</li>
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</ol>
498+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="advantage-6-quantum-signal-processing">Advantage 6: 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>
503+
<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="from-dft-to-qft">From DFT to QFT </h2>
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‎doc/pub/week11/html/week11.html‎

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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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('QFT, short review from last week',
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2,
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None,
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'qft-short-review-from-last-week'),
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('Advantage 1: Exponential Speedup',
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2,
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None,
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'advantage-1-exponential-speedup'),
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('Advantage 2: Quantum Parallelism',
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2,
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None,
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'advantage-2-quantum-parallelism'),
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('Advantage 3: Compact Circuit Implementation',
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2,
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None,
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'advantage-3-compact-circuit-implementation'),
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('Advantage 4: Applications in Quantum Computing',
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2,
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None,
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'advantage-4-applications-in-quantum-computing'),
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('Advantage 5: Reduced Memory Complexity',
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2,
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None,
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'advantage-5-reduced-memory-complexity'),
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('Advantage 6: Quantum Signal Processing',
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2,
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None,
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'advantage-6-quantum-signal-processing'),
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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,
@@ -498,6 +526,62 @@ <h2 id="inverse-transform">Inverse transform </h2>
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$$
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="qft-short-review-from-last-week">QFT, short review from last week </h2>
531+
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<ol>
533+
<li> QFTs are the quantum analogue of the Discrete Fourier Transform (DFT).</li>
534+
<li> They play a crucial role in quantum algorithms like Shor&#8217;s Algorithm and the Quantum Phase Estimation (QPE) algorithm</li>
535+
<li> QFTs provide an <em>exponential speedup</em> over classical Fourier Transform methods.</li>
536+
</ol>
537+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
538+
<h2 id="advantage-1-exponential-speedup">Advantage 1: Exponential Speedup </h2>
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<ol>
540+
<li> Classical Discrete Fourier Transforms (DFTs) require \( O(N^2) \) operations.</li>
541+
<li> The Fast Fourier Transforms (FFTs) improve this to \( O(N \log N) \) operations.</li>
542+
<li> QFTs reduce complexity to \( O((\log N)^2) \) using quantum gates.</li>
543+
</ol>
544+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
545+
<h2 id="advantage-2-quantum-parallelism">Advantage 2: Quantum Parallelism </h2>
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<p>QFTs act on a <em>superposition</em> of states, processing all inputs simultaneously.
547+
This is crucial in algorithms like:
548+
</p>
549+
<ol>
550+
<li> Shor&#8217;s Algorithm for factoring large numbers.</li>
551+
<li> Quantum Phase Estimation (QPE) for eigenvalue extraction.</li>
552+
</ol>
553+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
554+
<h2 id="advantage-3-compact-circuit-implementation">Advantage 3: Compact Circuit Implementation </h2>
555+
556+
<ol>
557+
<li> QFTs require only <em>Hadamard gates and controlled-phase gates</em>.</li>
558+
<li> QFTs are highly efficient for <em>quantum hardware</em>.</li>
559+
</ol>
560+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="advantage-4-applications-in-quantum-computing">Advantage 4: Applications in Quantum Computing </h2>
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<ol>
563+
<li> \textbf{Shor&#8217;s Algorithm:} Uses QFT to find periodicity in modular exponentiation.</li>
564+
<li> \textbf{Quantum Phase Estimation (QPE):} Extracts eigenvalues of unitary matrices.</li>
565+
<li> \textbf{Quantum Signal Processing:} Enables spectral analysis on quantum data.</li>
566+
</ol>
567+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
568+
<h2 id="advantage-5-reduced-memory-complexity">Advantage 5: Reduced Memory Complexity </h2>
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<ol>
571+
<li> Classical DFTs require \( O(N) \) space.</li>
572+
<li> QFTs store Fourier-transformed coefficients <em>implicitly</em> in qubit amplitudes.</li>
573+
<li> Only \( O(n) \) qubits are needed for a size \( N = 2^n \) transformation.</li>
574+
</ol>
575+
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="advantage-6-quantum-signal-processing">Advantage 6: 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="from-dft-to-qft">From DFT to QFT </h2>
503587

‎doc/pub/week11/ipynb/ipynb-week11-src.tar.gz‎

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