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update week 12
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‎doc/pub/week12/html/week12-bs.html‎

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@@ -39,10 +39,71 @@
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'sections': [('Plans for the week of April 7-11, 2025',
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2,
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None,
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'plans-for-the-week-of-april-7-11-2025')]}
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'plans-for-the-week-of-april-7-11-2025'),
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('Reminder on the Quantum Fourier transform',
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2,
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None,
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'reminder-on-the-quantum-fourier-transform'),
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('Quantum phase estimation algorithm',
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2,
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None,
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'quantum-phase-estimation-algorithm'),
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("Brief overview of Shor's algorithm",
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2,
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None,
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'brief-overview-of-shor-s-algorithm'),
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('Some history', 2, None, 'some-history'),
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('Problem Statement', 2, None, 'problem-statement'),
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('Reduction to Order-Finding',
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2,
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None,
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'reduction-to-order-finding'),
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('Continued Fraction Expansion',
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2,
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None,
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'continued-fraction-expansion'),
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('Basics of number theory', 2, None, 'basics-of-number-theory'),
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('QFTs again', 2, None, 'qfts-again'),
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('Quantum Period Finding', 2, None, 'quantum-period-finding'),
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('Applying Quantum Fourier Transform (QFT)',
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2,
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None,
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'applying-quantum-fourier-transform-qft'),
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('Classical Post-Processing',
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2,
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None,
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'classical-post-processing'),
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('Complexity and Practical Considerations',
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2,
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None,
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'complexity-and-practical-considerations'),
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('Error Analysis and Success Probability',
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2,
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None,
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'error-analysis-and-success-probability'),
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("Summarizing Shor's algorithm",
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2,
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None,
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'summarizing-shor-s-algorithm')]}
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end of tocinfo -->
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<body>
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<script type="text/x-mathjax-config">
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MathJax.Hub.Config({
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TeX: {
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equationNumbers: { autoNumber: "AMS" },
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extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
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}
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});
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</script>
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<script type="text/javascript" async
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src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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<!-- Bootstrap navigation bar -->
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<div class="navbar navbar-default navbar-fixed-top">
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<div class="navbar-header">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-april-7-11-2025" style="font-size: 80%;">Plans for the week of April 7-11, 2025</a></li>
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<!-- navigation toc: --> <li><a href="#reminder-on-the-quantum-fourier-transform" style="font-size: 80%;">Reminder on the Quantum Fourier transform</a></li>
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<!-- navigation toc: --> <li><a href="#quantum-phase-estimation-algorithm" style="font-size: 80%;">Quantum phase estimation algorithm</a></li>
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<!-- navigation toc: --> <li><a href="#brief-overview-of-shor-s-algorithm" style="font-size: 80%;">Brief overview of Shor's algorithm</a></li>
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<!-- navigation toc: --> <li><a href="#some-history" style="font-size: 80%;">Some history</a></li>
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<!-- navigation toc: --> <li><a href="#problem-statement" style="font-size: 80%;">Problem Statement</a></li>
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<!-- navigation toc: --> <li><a href="#reduction-to-order-finding" style="font-size: 80%;">Reduction to Order-Finding</a></li>
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<!-- navigation toc: --> <li><a href="#continued-fraction-expansion" style="font-size: 80%;">Continued Fraction Expansion</a></li>
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<!-- navigation toc: --> <li><a href="#basics-of-number-theory" style="font-size: 80%;">Basics of number theory</a></li>
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<!-- navigation toc: --> <li><a href="#qfts-again" style="font-size: 80%;">QFTs again</a></li>
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<!-- navigation toc: --> <li><a href="#quantum-period-finding" style="font-size: 80%;">Quantum Period Finding</a></li>
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<!-- navigation toc: --> <li><a href="#applying-quantum-fourier-transform-qft" style="font-size: 80%;">Applying Quantum Fourier Transform (QFT)</a></li>
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<!-- navigation toc: --> <li><a href="#classical-post-processing" style="font-size: 80%;">Classical Post-Processing</a></li>
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<!-- navigation toc: --> <li><a href="#complexity-and-practical-considerations" style="font-size: 80%;">Complexity and Practical Considerations</a></li>
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<!-- navigation toc: --> <li><a href="#error-analysis-and-success-probability" style="font-size: 80%;">Error Analysis and Success Probability</a></li>
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<!-- navigation toc: --> <li><a href="#summarizing-shor-s-algorithm" style="font-size: 80%;">Summarizing Shor's algorithm</a></li>
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</ul>
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</li>
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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> <a href="https://youtu.be/" target="_self">Video of lecture TBA</a>
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<li> Quantum phase estimation algorithm, final derivation and discussions</li>
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<li> Brief discussion of Shor's algorithm</li>
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<li> Reading recommendation: Hundt section 6.4 for the QPE and section 6.5 for Shor's algorithm
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<!-- o <a href="https://youtu.be/" target="_self">Video of lecture TBA</a> -->
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<!-- o <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesApril10.pdf" target="_self">Whiteboard notes</a> --></li>
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</ol>
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</div>
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</div>
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<!-- !split -->
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<h2 id="reminder-on-the-quantum-fourier-transform" class="anchor">Reminder on the Quantum Fourier transform </h2>
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<!-- !split -->
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<h2 id="quantum-phase-estimation-algorithm" class="anchor">Quantum phase estimation algorithm </h2>
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<!-- !split -->
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<h2 id="brief-overview-of-shor-s-algorithm" class="anchor">Brief overview of Shor's algorithm </h2>
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<ol>
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<li> Shor's algorithm is a quantum algorithm for integer factorization.</li>
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<li> It efficiently factors large integers by leveraging quantum period finding.</li>
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<li> Exponential speedup over the best-known classical algorithms.</li>
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<li> Breaks RSA encryption, which relies on the difficulty of factoring.</li>
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</ol>
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<!-- !split -->
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<h2 id="some-history" class="anchor">Some history </h2>
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<p>Shor's algorithm, introduced by Peter Shor in 1994, revolutionized the
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field of cryptography. It demonstrated that a quantum computer could
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solve integer factorization and the discrete logarithm problem
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efficiently, posing a significant threat to classical cryptosystems
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such as RSA.
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</p>
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<!-- !split -->
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<h2 id="problem-statement" class="anchor">Problem Statement </h2>
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<p>The objective of Shor's algorithm is to factorize a large composite
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integer \( N \) into its prime factors. Given \( N = p \times q \),
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where \( p \) and \( q \) are unknown distinct primes, the problem of
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factorization can be reduced to finding an integer \( a \) such that
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\( 1 < a < N \) and \( \gcd(a, N) = 1 \).
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</p>
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<!-- !split -->
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<h2 id="reduction-to-order-finding" class="anchor">Reduction to Order-Finding </h2>
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<p>The key reduction in Shor's algorithm lies in transforming
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factorization into an order-finding problem:
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</p>
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<ol>
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<li> Choose a random integer \( a \) such that \( 1 < a < N \) and \( \gcd(a, N) = 1 \).</li>
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<li> Determine the smallest positive integer \( r \) such that \( a^r \equiv 1 \pmod{N} \).</li>
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</ol>
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<p>The integer \( r \) is known as the \textit{order} of \( a \) modulo
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\( N \), and it helps in discovering the factors of \( N \).
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</p>
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<!-- !split -->
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<h2 id="continued-fraction-expansion" class="anchor">Continued Fraction Expansion </h2>
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<p>After obtaining the rational approximation from the quantum algorithm,
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classical computation using continued fraction expansion aids in
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deducing the correct order \( r \).
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</p>
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<!-- !split -->
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<h2 id="basics-of-number-theory" class="anchor">Basics of number theory </h2>
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<ol>
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<li> The problem: Factor an integer \( N \) into its prime factors.</li>
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<li> Reduce factoring to period finding \( f(x) = a^x \mod N \), where \( a \) is randomly chosen such that \( \gcd(a, N) = 1 \).</li>
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<li> The period \( r \) satisfies \( a^r \equiv 1 \mod N \).</li>
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<li> Once \( r \) is known, factors are given by \( \gcd(a^{r/2} \pm 1, N) \).</li>
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</ol>
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<!-- !split -->
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<h2 id="qfts-again" class="anchor">QFTs again </h2>
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<p>The Quantum Fourier Transform is crucial to Shor's algorithm. It
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leverages quantum parallelism and interference to efficiently estimate
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the periodicity of a function.
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</p>
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$$
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\text{QFT}: \vert x\rangle \mapsto \frac{1}{\sqrt{2^n}} \sum_{y=0}^{2^n-1} e^{2\pi ixy/2^n} \vert y\rangle
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$$
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<p>The quantum part of Shor's algorithm is designed to find the period \(
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r \) of the modular exponentiation function \( f(x) = a^x \bmod N \):
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</p>
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<!-- !split -->
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<h2 id="quantum-period-finding" class="anchor">Quantum Period Finding </h2>
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<p>Quantum Period Finding: Key to Shor's Algorithm</p>
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<ol>
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<li> Quantum subroutine for finding the period \( r \) of \( f(x) \).</li>
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<li> Similar to Quantum Phase Estimation (QPE).</li>
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<li> Uses two quantum registers:</li>
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<ul>
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<li> First register: Superposition of all possible inputs \( x \).</li>
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<li> Second register: Computes \( f(x) = a^x \mod N \).</li>
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</ul>
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</ol>
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<!-- !split -->
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<h2 id="applying-quantum-fourier-transform-qft" class="anchor">Applying Quantum Fourier Transform (QFT) </h2>
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<ol>
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<li> The QFT is applied to the first register after the controlled unitary operations.</li>
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<li> Peaks in the measurement correspond to multiples of \( 1/r \).</li>
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<li> Post-processing classically determines \( r \) by continued fractions.</li>
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</ol>
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<!-- !split -->
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<h2 id="classical-post-processing" class="anchor">Classical Post-Processing </h2>
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<ol>
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<li> Once \( r \) is found, factors of \( N \) are computed by \( \gcd(a^{r/2} \pm 1, N) \).</li>
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<li> If \( r \) is odd or \( a^{r/2} \equiv -1 \mod N \), try a different \( a \).</li>
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<li> With high probability, this yields a non-trivial factor of \( N \).</li>
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</ol>
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<!-- !split -->
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<h2 id="complexity-and-practical-considerations" class="anchor">Complexity and Practical Considerations </h2>
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<ol>
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<li> Quantum Complexity: Polynomial in \( \log N \).</li>
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<li> Classical Best Known: Sub-exponential (Number Field Sieve).</li>
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<li> Quantum Advantage: Exponential speedup over classical algorithms.</li>
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<li> Practical Challenges: Quantum error correction, large qubit counts.</li>
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</ol>
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<!-- !split -->
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<h2 id="error-analysis-and-success-probability" class="anchor">Error Analysis and Success Probability </h2>
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<p>The probability that a randomly chosen \( a \) leads to successful
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factoring is generally greater than 50\%. The quantum component
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requires carefully managed resources for high precision.
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</p>
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<p>Error correction is crucial, as any errors in operations or QFT can
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significantly impact the accuracy of period estimation and the
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eventual factorization.
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</p>
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<!-- !split -->
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<h2 id="summarizing-shor-s-algorithm" class="anchor">Summarizing Shor's algorithm </h2>
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<p>Shor's algorithm is a central quantum algorithm demonstrating
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exponential speedup over classical counterparts. It brought both
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excitement for potential quantum advancements and a reconsideration of
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current cryptographic standards.
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</p>
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<ol>
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<li> Shor's algorithm is a groundbreaking quantum algorithm for factoring.</li>
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<li> Combines quantum period finding with classical number theory.</li>
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<li> Highlights the potential of quantum computing to break RSA.</li>
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</ol>
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<!-- ------------------- end of main content --------------- -->
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</div> <!-- end container -->
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<!-- include javascript, jQuery *first* -->
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</footer>
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-->
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<center style="font-size:80%">
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<!-- copyright --> &copy; 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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<!-- copyright --> &copy; 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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</body>
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</html>

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