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‎doc/pub/week1/html/week1-bs.html‎

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‎doc/pub/week1/html/week1-reveal.html‎

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@@ -356,6 +356,322 @@ <h2 id="online-material-of-possible-interest">Online material of possible intere
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</ol>
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</section>
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<section>
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<h2 id="what-is-quantum-computing">What is quantum computing? </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> It is computation (and more generally, information processing) based on the principles of quantum mechanics, rather than classical physics.</li>
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<p><li> Quantum mechanics is a description of nature at its most fundamental level.</li>
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</ol>
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</div>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Formulated in the early 20th century to explain the behavior of subatomic particles.</li>
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<p><li> QM, and its specializations (quantum field theory, quantum chromodynamics, many-body physics etc.), have been spectacularly successful in explaining microscopic physical phenomena.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="the-exponentiality-of-qm">The exponentiality of QM </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> After nearly a century of study, the best (classical) methods for predicting the behavior of general quantum systems require exponential resources.</li>
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<p><li> The state of \( N \) particles requires at least \( 2^{N} \) numbers to describe.</li>
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<p><li> For \( N \sim 300 \) (the number of particles in a single uranium atom), \[
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<p> 2^{300} \gg \approx 10^{82}\quad (\text{\# of atoms in the observable universe}).
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\]
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</p></li>
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<p><li> Nature seems to be doing extravagant amounts of computation <em>behind the scenes</em>!</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="the-exponentiality-of-qm-i">The exponentiality of QM I </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>How do physicists actually do (quantum) physics?</b>
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<p>
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<ol>
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<p><li> Answer 1: 100+ years of extremely clever approximations and shortcuts for calculations.</li>
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<ul>
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<p><li> Examples: Bethe ansatz, Feynman diagrams, perturbation theory, mean field approximations, \dots</li>
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</ul>
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<p>
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<p><li> Answer 2: computer simulations</li>
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<ul>
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<p><li> Examples: Density functional theory, Quantum Markov Chain Monte Carlo</li>
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</ul>
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<p>
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<p><li> Answer 3: studying systems that allow for Answers 1 and 2 above</li>
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<ul>
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<p><li> Systems that are mostly classical</li>
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<p><li> Systems with non-strongly interacting particles</li>
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<p><li> Lucky that most things we&#8217;ve studied so far fall in this category!</li>
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</ul>
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<p>
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</ol>
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</div>
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<p>But simulating general quantum physics problems is still a hard task.</p>
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</section>
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<section>
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<h2 id="the-exponentiality-of-qm-ii">The exponentiality of QM II </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<p>In 1981, in his talk titled \emph{&#8220;Simulating Physics with Computers&#8221;}, Richard Feynman asked the following question:
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<b>Can probabilistic computers simulate quantum mechanics?</b>.
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</p>
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</div>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<p>His conclusion, after a lengthy exploration was that quantum mechanics [doesn&#8217;t] seem to be imitable by a local classical computer.
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Instead we need a computer built of quantum mechanical elements which obey quantum mechanical laws.
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Feynman, along with David Deutsch, Paul Benioff and Yuri Manin, are credited with the idea of computing based on quantum mechanical principles.
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</p>
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</div>
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</section>
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<section>
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<h2 id="the-early-days">The early days </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Feynman&#8217;s motivation for quantum computers is the most obvious one: simulating quantum systems.</li>
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<p><li> To physicists in the '80s, this idea might have seemed obvious. To a computer scientist in the '80s, QCs would have seemed like a wild crackpot idea.</li>
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<p><li> Also in the '80s, David Deutsch contemplated other applications of quantum computers. He gave an example of a (classical) problem where the &#8220;parallel universes&#8221; of QM seem to speed up (by a constant factor).</li>
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<p><li> 1992: Dan Simon came up with an example of a problem that could be solved exponentially faster on a QC.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="shor-s-breakthrough">Shor&#8217;s breakthrough </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> 1993: Peter Shor realized that Simon&#8217;s idea could be extended to efficiently solve the following problem:</li>
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<p><li> <b>Integer Factorization:</b> Given integer \( N>0 \), find integer \( M>1 \) such that \( M \) divides \( N \).</li>
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<p><li> The security of the RSA cryptosystem (the most widely deployed public-key encryption on the internet) crucially relies on the assumption that factoring integers is hard!</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="crossroads">Crossroads </h2>
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<p>Since Shor&#8217;s algorithm, physicists and computer scientists have been faced with three options:</p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Quantum mechanics is wrong.</li>
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<p><li> There is a fast classical algorithm for factoring.</li>
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<p><li> Quantum computers are more powerful than classical computers.</li>
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</ol>
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</div>
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<p>At least one of these must be true! Which do you think is most likely to be true?</p>
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</section>
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<section>
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<h2 id="which-is-wrong">Which is wrong? </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Option 1: QM is wrong</li>
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<ul>
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<p><li> Perhaps the most successful theory of nature to date.</li>
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<p><li> In all its domains of applicability, we have never found experimental disagreement.</li>
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</ul>
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<p>
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<p><li> Option 2: Polynomial-time classical factoring algorithm.</li>
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<ul>
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<p><li> RSA has been out for nearly 50 years. There is enormous economic pressure to discover weaknesses, if it exists.</li>
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<p><li> The <b>Gauss argument</b>: if the Prince of Mathematics couldn&#8217;t find it, then it probably doesn&#8217;t exist.</li>
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</ul>
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<p>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="more">More </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Option 3: Quantum computers are more powerful than classical computers.</li>
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<ul>
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<p><li> This would refute the Extended Church&ndash;Turing Thesis.</li>
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<p><li> Church&ndash;Turing Thesis: A Turing machine can simulate any effective computation process.</li>
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<p><li> Extended Church&ndash;Turing Thesis: A probabilistic Turing machine can efficiently simulate any effective computation process.</li>
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<p><li> Option \#3 would violate the ECT (assuming factoring is hard for classical computers)!</li>
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</ul>
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<p>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="the-options">The options </h2>
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<p>Since Shor&#8217;s algorithm, physicists and computer scientists have been faced with three options:</p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Quantum mechanics is wrong.</li>
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<p><li> There is a fast classical algorithm for factoring.</li>
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<p><li> Quantum computers are more powerful than classical computers.</li>
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</ol>
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</div>
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<p>At least one of these must be true!</p>
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</section>
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<section>
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<h2 id="present-day">Present day </h2>
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<p>Quantum computing is extremely active and exciting:</p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Engineering: primitive quantum computers are being built.</li>
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<p><li> Theoretical developments: new algorithms, new cryptographic protocols, new insights into how quantum computing compares to classical computing.</li>
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<p><li> Deep connections to other sciences: condensed matter physics, quantum gravity, materials science, chemistry, computer science, pure mathematics.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="emerging-quantum-computers">Emerging quantum computers </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Many players racing to build (scalable) quantum computers.</li>
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<p><li> Different labs/companies are betting on different horses.</li>
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<p><li> Superconducting qubits, ion traps, photonic systems, topological qubits, \dots</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="emerging-quantum-computers">Emerging quantum computers </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Until 2017 or so, most quantum computing devices had \( \sim 10 \) qubits.</li>
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<p><li> Suddenly, there was a flood of resources devoted to engineering efforts, and now we are seeing programmable quantum computers with \( 50+ \) qubits.</li>
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</ol>
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</div>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Major problem: devices are insanely noisy! We are in the &#8220;NISQ&#8221; era:</b>
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<p>
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<ol>
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<p><li> Noisy Intermediate-Scale Quantum</li>
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<p><li> Noisy devices with \( \sim 100 \) qubits.</li>
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<p><li> Not capable of running Shor&#8217;s algorithm, but should still be capable of solving interesting, hard problems.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="quantum-supremacy">Quantum supremacy </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> In Fall 2019, Google announced that they had achieved &#8220;Quantum Supremacy&#8221; using their 53-qubit Sycamore quantum processor.</li>
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<p><li> <b>Quantum supremacy:</b> a moment when there is a convincing real-world demonstration of a quantum computing task that cannot feasibly be performed by a classical computer.</li>
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<p><li> Sweet spot: 53 qubits just beyond the reach of Google&#8217;s compute clusters to simulate in a reasonable time. (\( \sim 2^{53} \) ops)</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="summary-of-hardware-efforts">Summary of hardware efforts </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> We are in the NISQ era, but we have compelling evidence that we can already perform super-classical tasks using these machines.</li>
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<p><li> There is a <b>qubit race</b> going on, but qubit count is not everything!</li>
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<p><li> Scaling up (i.e. more, higher-quality qubits) is a tough engineering challenge, but no fundamental obstacles to building a QC with \( 10^{6} \) noiseless qubits.</li>
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<p><li> <a href="https://qunorth.com/" target="_blank">QUNorth</a> in Denmark and <a href="https://lumi-supercomputer.eu/europe-takes-a-quantum-leap-lumi-q-consortium-signs-contract/" target="_blank">LUMIQ</a> as a European project</li>
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<p><li> Still, large-scale fault-tolerant QCs look like they&#8217;re many years away.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="quantum-algorithms">Quantum algorithms </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ol>
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<p><li> Exponential speedups for structured, algebraic problems</li>
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<p><li> Factoring (Shor&#8217;s algorithm)</li>
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<p><li> Polynomial speedups for unstructured search problems</li>
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<p><li> Grover search</li>
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<p><li> Exponential speedups for quantum simulation</li>
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<ul>
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<p><li> Simulating quantum physics and chemistry (Feynman&#8217;s dream and mine as well). Probably the most important application of quantum computers so far!</li>
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</ul>
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<p>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="quantum-algorithms">Quantum algorithms </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Algorithms to be run on near-term QCs:</b>
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<p>
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<ol>
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<p><li> Variational eigensolvers</li>
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<p><li> Classical&ndash;quantum hybrid algorithms</li>
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<p><li> Quantum Machine Learning</li>
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<p><li> Linear system solvers / SDPs / Convex Optimization</li>
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<p><li> Quantum neural nets</li>
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<p><li> Solving differential equations</li>
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<p><li> more</li>
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</ol>
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</div>
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<p>Connections with fundamental physics: Perhaps key to a theory of Quantum Gravity could be quantum entanglement, and quantum error correcting codes.</p>
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</section>
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<section>
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<h2 id="notations-and-definitions">Notations and definitions </h2>
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