Thirty Years Later, a Speed Boost for Quantum Factoring

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scale1 mha'> <div class='scale0'> <div class='mb1 pb025 mt0 h6t post__title__kicker '> <a class="kicker theme__accent theme__text-hover uppercase" href="/tag/quantum-computing/">quantum computing</a> </div> <h1 class='post__title__title mv025 noe theme__text' >Thirty Years Later, a Speed Boost for Quantum Factoring</h1> <div class='post__title__author-date h5 mt1'> <div class='byline relative flex flex-items-start merriweather mv025 mr1 gray3 h6t theme__text-hover'> <em class='byline__by gray4 mr075'>By </em> <a href='https://www.quantamagazine.org/authors/brubaker_ben/'> <span class='byline__author uppercase kern light small'>Ben Brubaker</span> </a> </div> <p class="h6 mv1 pv025 o6"><em>October 17, 2023</em></p> </div> <div class='post__title__meta flex flex-wrap flex-items-start'> <div class='post__title__excerpt wysiwyg p italic mb1 mt025 pr2 o4 theme__text ' > Shor’s algorithm will enable future quantum computers to factor large numbers quickly, undermining many online security protocols. Now a researcher has shown how to do it even faster. </div> <div class='post__title__actions h5 mt025 flex-auto flex flex-items-center'> <div class='comments-button mr05 flex flex-items-center theme__accent-hover h5 theme__accent' data-name='comment-bubble' > <a href='#comments' class='flex flex-items-center pangram'> <svg viewbox='0 0 50 50' enable-background='new 0 0 50 50' xmlns='http://www.w3.org/2000/svg' class=''><title>Comment</title><path fill='currentColor' d='M9.4 4.2h31.2c8.6 0 9.4 7 9.4 15.6s-.7 15.6-9.4 15.6h-2.2l-.9 9.4-18.8-9.4H9.4c-8.6 0-9.4-7-9.4-15.6S.7 4.2 9.4 4.2z'></path></svg> <div> <span class='comments-button__count ml075 small mbold disqus-comment-count' data-disqus-identifier='131364 https://www.quantamagazine.org/?p=131364'></span> </div> </a> </div> <span class='o-divider mh075 relative'></span> <div class='relative'> <button class='bookmark-button mh05 flex flex-items-center theme__accent-hover ' data-toggle-hover='tooltip' data-id='131364' data-active-class='' aria-expanded='false'> <svg class="icon" viewbox="0 0 50 50" enable-background="new 0 0 50 50" xmlns="http://www.w3.org/2000/svg"><title>Save Article</title><path fill="currentColor" d="M2.1 0h45.8v50L25.5 37.5 2.1 50V0z"></path></svg> </button> <div class='q-tooltip hidden force-mobile-placement w-32 ml-n16 -top-10'> <div class='q-tooltip-content'> <div class='q-tooltip-arrow -bottom-3'></div> <div class='q-tooltip-inner py-1 pl-1' role='tooltip'> <div class='h6t relative z1 uppercase kern--w mx025'> <span class='small no-wrap'>Read Later</span> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </section> </section> <figure class='acf--code '> <div class=''> <div class=''> <div class='acf-media acf-media--large'> <div class='mha '> <figure class='acf-media__inner mb2 relative '> <div class='mha container--ml'> <div class=''> <video src="/wp-content/uploads/2023/10/BetterShor-byDVDP-Lede-1.mp4" autoplay="autoplay" muted="true" loop="true" playsinline="true" width="100%" poster="/wp-content/uploads/2023/10/BetterShor-byDVDP-Lede-Still-scaled.webp"></video> </div> </div> <figcaption class='acf-media__content mt1 relative fill-h full-caption'> <section class='outer acf-media__content__inner outer--content'> <div class='relative mha container--ml'> <div class='image__meta'> <div class='image__meta__caption mha container--xs'> <div class='caption wysiwyg h5 theme__anchors--solid fill-h acf-media__caption'> <p>A new paper extends Shor’s algorithm to multiple dimensions.</p> </div> </div> <div class='image__meta__attribution mha container--xs'> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h acf-media__attr'> <p>DVDP for <em>Quanta Magazine</em></p> </div> </div> </div> </div> </section> </figcaption> </figure> </div> </div> </div> </div> </figure> <div class='acf-content scale1 mt2'> <h2 class='screen-reader-text'>Introduction</h2> <div class='post__wrapper scale0 show-dropcap'> <div class='mha container--m'> <div class='post__content relative flex flex-items-start flex-justify-between'> <aside class='post__sidebar hide flex flex-justify-center '> <div class='post__sidebar__content'> <div class='post__sidebar__content__inner'> <div class='align-c mb075'> <div class='sidebar__author'> <a class='theme__accent-hover transition--color ' href='https://www.quantamagazine.org/authors/brubaker_ben/'> <div class='sidebar__author__avatar mha mb1'> <div class='image mx0 relative image--circle '> <div class='image__inner absolute fit-x is-loaded'> <img width="1500" height="1500" src="https://www.quantamagazine.org/wp-content/uploads/2022/10/Ben_Brubaker.jpg" class="absolute fit-x fill-h fill-v mxa" alt="" decoding="async" fetchpriority="high" srcset="https://www.quantamagazine.org/wp-content/uploads/2022/10/Ben_Brubaker.jpg 1500w, https://www.quantamagazine.org/wp-content/uploads/2022/10/Ben_Brubaker-520x520.jpg 520w, https://www.quantamagazine.org/wp-content/uploads/2022/10/Ben_Brubaker-768x768.jpg 768w, https://www.quantamagazine.org/wp-content/uploads/2022/10/Ben_Brubaker-160x160.jpg 160w" sizes="(max-width: 1500px) 100vw, 1500px" /> </div> </div> </div> <div class='h3t mv05'> <span class='screen-reader-text'>By </span>Ben Brubaker </div> <p class="o8 theme__text h5 mt05 mb1"><em>Staff Writer</em></p> </a> </div> <hr class='mb075 o1'> <p class="h6 mv1 pv025 o6"><em>October 17, 2023</em></p> </div> <form action='https://www.quantamagazine.org/thirty-years-later-a-speed-boost-for-quantum-factoring-20231017' method='get' class='sidebar__actions'> <hr class='mt075 mb1 o1'> <input type='hidden' id='print' name='print' value='1' /> <button type="submit" class='h6 pangram uppercase mv05 pb1 bold kern flex flex-items-center flex-justify-between sidebar__print transition--color theme__accent-hover'> <small>View PDF/Print Mode</small> <svg class="icon-l theme__accent ml05" viewbox="0 0 50 50" 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data-role='selectable'> <div class='post__content wysiwyg p theme__anchors--underline '> <p><a href="https://math.mit.edu/~shor/">Peter Shor</a> didn’t set out to break the internet. But an algorithm he developed in the mid-1990s threatened to do just that. In a <a href="https://ieeexplore.ieee.org/document/365700">landmark paper</a>, Shor showed how a hypothetical computer that exploited the quirks of quantum physics could break large numbers into their prime factors far faster than any ordinary classical machine.</p> <p>The result had implications far beyond mathematics. At the time, a vital component of internet security called <a href="https://www.quantamagazine.org/quantum-secure-cryptography-crosses-red-line-20150908/">public-key cryptography</a> relied on the assumption that factoring large numbers is so computationally difficult as to be effectively impossible. That assumption still underpins some critical protocols today. Shor’s algorithm showed that it would fail spectacularly in a world with powerful <a href="https://www.quantamagazine.org/tag/quantum-computing/">quantum computers</a>.</p> <p>In the past 30 years, computer scientists have streamlined Shor’s algorithm in preparation for the day that quantum technology matures enough to run it. But <a href="https://arxiv.org/abs/2308.06572">a new variant</a>, from the New York University computer scientist <a href="https://cims.nyu.edu/~regev/">Oded Regev</a>, is faster in a fundamentally new sense. It’s the first to improve the relationship between the size of the number being factored and the number of quantum operations required to factor it.</p> <p>“It’s really remarkable that somebody has apparently been able to improve the complexity of this result many, many years later,” said <a href="https://people.maths.bris.ac.uk/~csxam/">Ashley Montanaro</a>, a quantum computing researcher at the University of Bristol. “This is really exciting.”</p> <p><a href="https://github.com/ekera">Martin Ekerå</a>, a cryptographer at the Swedish National Communications Security Authority, agreed that Regev’s paper is interesting but cautioned that beating the state of the art in practice will require further optimization. “Shor’s original algorithms are already surprisingly efficient, so it is not trivial to make major improvements,” he wrote in an email.</p> <p>Regev developed his new algorithm by augmenting Shor’s algorithm with techniques from a branch of cryptography dealing with high-dimensional geometry.</p> <p>“I would have thought that any algorithm that worked with this basic outline would be doomed,” said Shor, an applied mathematician now at the Massachusetts Institute of Technology. “But I was wrong.”</p> </div> </div> </div> </section> <aside class='post__sidebar hide flex flex-justify-center theme__anchors--solid '> <div class='post__sidebar__content'> <div class='post__sidebar__content__inner'> <div class='h3t mt0'> Share this article </div> <div data-id='social-links'> <div class='social-links social-links--share flex flex-justify-between'> <a href='http://www.facebook.com/sharer.php?u=https://www.quantamagazine.org/thirty-years-later-a-speed-boost-for-quantum-factoring-20231017/' class='social-links__link flex flex-items-center relative' target="_blank" > <div class='icon-button inline-block relative pointer theme__accent-hover-child'> <div class='absolute fit-x mxa o2 hover--opaque theme__text'> </div> <svg class="absolute fit-x mxa theme__text" 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items-start justify-center flex-col max-w-img-regular py-0 px-0 l:px-16 l:flex-row '> <div class='w-full flex justify-center max-w-mod-regular m:max-w-unset image--module s:flex-col s:m-auto'> <div class='w-auto mr-2 last:mr-0 s:m-0 s:flex s:flex-col flex-1'> <img width="2560" height="2050" src="https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-scaled.webp" class="mb-6 w-full s:mb-4" alt="Peter Shor in a green shirt sitting at a messy desk" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-scaled.webp 2560w, https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-1720x1377.webp 1720w, https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-520x416.webp 520w, https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-768x615.webp 768w, https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-1536x1230.webp 1536w, https://www.quantamagazine.org/wp-content/uploads/2023/10/PeterShor-byChristopherHarting-2048x1640.webp 2048w" sizes="(max-width: 2560px) 100vw, 2560px" /> </div> </div> <figcaption class='block my-0 mx-auto w-full l:max-w-mod-inline m:w-full l:w-72'> <section class='p-0 m:py-0 m:px-16'> <div class='w-full my-0 mx-auto flex flex-col s:mx-5 s:w-full-minus-25 s:max-w-mod-inline m:max-w-mod-inline l:m-0 l:p-0 max-w-mod-inline l:pl-8 l:max-w-75 s:mb-4'> <div class='caption wysiwyg h5 theme__anchors--solid fill-h '> <p>Internet security depends on the computational difficulty of breaking large numbers into their prime factors. In 1994, Peter Shor discovered a quantum algorithm that can factor large numbers quickly.</p> </div> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h '> <p>Christopher Harting</p> </div> </div> </section> </figcaption> </figure> <div class='acf-content scale1 mt2'> <section class='outer mha js-router-anchors outer--content'> <div class='flex-auto mha container--xs '> <div class='post__content__section wysiwyg p theme__anchors--underline' data-role='selectable'> <div class='post__content wysiwyg p theme__anchors--underline '> <h2><strong>Finding Factors<br /> </strong></h2> <p>Quantum computers derive their power from the peculiar way they process information. Classical computers use bits, each of which must always be in one of two states, labeled 0 and 1. Quantum bits, or “qubits,” can additionally be in combinations of their 0 and 1 states — a phenomenon called superposition. It’s also possible to coax multiple qubits into a collective superposition state: A two-qubit superposition has four components that can perform different computations simultaneously, and the number of such components grows exponentially as the number of qubits increases. That allows quantum computers to effectively perform exponentially many different computations in parallel.</p> <p>But <a href="https://www.quantamagazine.org/why-is-quantum-computing-so-hard-to-explain-20210608/">there’s a catch</a>: Reading the result of a computation performed in superposition only reveals the answer to the part computed by one random component. To reap the benefits of computing in superposition, you must somehow map the end result onto a simpler state where it’s safe to read the result. That’s not possible in most cases, and at first nobody knew how to make it work for any problem. “There were very few people who even had the courage to think about quantum computations,” Regev said.</p> <p>Then in 1994, Shor read <a href="https://ieeexplore.ieee.org/document/365701">a paper</a> by the computer scientist Daniel Simon that showed how to exploit quantum superposition to solve a contrived problem. Shor figured out how to extend Simon’s result to a more general and practical problem called period finding. A mathematical function is said to be periodic when its output cycles repeatedly through the same values as the input increases; the length of a single cycle is known as the function’s period.</p> <aside class="post__aside mb2 relative post__aside--right post__aside--medium hide-on-print"> <div class='card clearfix mv05 pv1 theme__anchors--none is-grid-variant is-shortcode border-grey-o25'> <div class='card__image mr1 mb1' data-toggle-hover='card'> <a href='https://www.quantamagazine.org/why-is-quantum-computing-so-hard-to-explain-20210608/' class='relative'> <span class='screen-reader-text'>What Makes Quantum Computing So Hard to Explain?</span> <div class='image mx0 relative image--standard'> <div class='image__inner absolute fit-x is-loaded'> <img width="520" height="292" src="https://www.quantamagazine.org/wp-content/uploads/2021/06/0607_QComputer_520x292_F.jpg" class="absolute fit-x fill-h fill-v mxa" alt="" decoding="async" /> </div> </div> </a> </div> <div class='card__content '> <div class='mb1 pb025 mt0 h6t card__kicker'> <a class="kicker theme__accent theme__text-hover uppercase" href="/tag/quantized/">Quantized Columns</a> </div> <a href='https://www.quantamagazine.org/why-is-quantum-computing-so-hard-to-explain-20210608/' data-toggle-hover='card'> <h3 class='card__title noe mv0 theme__accent-hover transition--color h2t'> What Makes Quantum Computing So Hard to Explain? </h3> </a> <div class='card__meta mb0 mt025 flex flex-items-center flex-wrap'> <div class='card-date h6 flex flex-items-center mv05'> <div class='uppercase mr1 kern small gray3'>June 8, 2021</div> <span class='o-divider o-divider--vertical mr1 relative'></span> </div> <div class='card__actions h6 flex flex-items-center mv05'> <div class='comments-button mr05 flex flex-items-center theme__accent-hover ' data-name='comment-bubble' > <a href='https://www.quantamagazine.org/why-is-quantum-computing-so-hard-to-explain-20210608/#comments' class='flex flex-items-center pangram'> <svg viewbox='0 0 50 50' enable-background='new 0 0 50 50' xmlns='http://www.w3.org/2000/svg' class='o2'><title>Comment</title><path fill='currentColor' d='M9.4 4.2h31.2c8.6 0 9.4 7 9.4 15.6s-.7 15.6-9.4 15.6h-2.2l-.9 9.4-18.8-9.4H9.4c-8.6 0-9.4-7-9.4-15.6S.7 4.2 9.4 4.2z'></path></svg> </a> </div> <span class='o-divider mh075 relative'></span> <div class='relative'> <button class='bookmark-button mh05 flex flex-items-center theme__accent-hover ' data-toggle-hover='tooltip' data-id='105347' data-active-class='' aria-expanded='false'> <svg class="icon" viewbox="0 0 50 50" enable-background="new 0 0 50 50" xmlns="http://www.w3.org/2000/svg"><title>Save Article</title><path fill="currentColor" d="M2.1 0h45.8v50L25.5 37.5 2.1 50V0z"></path></svg> </button> <div class='q-tooltip hidden force-mobile-placement w-32 ml-n16 -top-10'> <div class='q-tooltip-content'> <div class='q-tooltip-arrow -bottom-3'></div> <div class='q-tooltip-inner py-1 pl-1' role='tooltip'> <div class='h6t relative z1 uppercase kern--w mx025'> <span class='small no-wrap'>Read Later</span> </div> </div> </div> </div> </div> </div> </div> <div class='card__excerpt h5 pangram o6 mb4 mt025'> </div> </div> </div> </aside> <p>To find the period of a given function using a quantum computer, start by setting up a very large superposition in which each component computes the function’s output for a different input. Then use Shor’s method to convert that large superposition into a simpler state and read the result. At that point, a classical computer can take over and finish the calculation quickly. Overall, Shor’s period-finding algorithm runs exponentially faster than any classical alternative because it computes different outputs of the periodic function simultaneously using superposition.</p> <p>As Shor looked for applications for his quantum period-finding algorithm, he rediscovered a previously known but obscure mathematical theorem: For every number, there exists a periodic function whose periods are related to the number’s prime factors. So if there’s a number you want to factor, you can compute the corresponding function and then solve the problem using period finding — “exactly what quantum computers are so good at,” Regev said.</p> <p>On a classical computer, this would be an agonizingly slow way to factor a large number — slower even than trying every possible factor. But Shor’s method speeds up the process exponentially, making period finding an ideal way to construct a fast quantum factoring algorithm.</p> <p>Shor’s algorithm was one of a few key early results that transformed quantum computing from an obscure subfield of theoretical computer science to the juggernaut it is today. But putting the algorithm into practice is a daunting task, because quantum computers are notoriously susceptible to errors: In addition to the qubits required to perform their computations, they need many others doing <a href="https://www.quantamagazine.org/how-quantum-computers-will-correct-their-errors-20211116/">extra work</a> to keep them from failing. A <a href="https://quantum-journal.org/papers/q-2021-04-15-433/">recent paper</a> by Ekerå and the Google researcher <a href="https://algassert.com/about.html">Craig Gidney</a> estimates that using Shor’s algorithm to factor a security-standard 2,048-bit number (about 600 digits long) would require a quantum computer with 20 million qubits. Today’s state-of-the-art machines have at most a few hundred.</p> <p>That’s why some critical internet protocols still rely on how hard it is to factor large numbers, but researchers don’t want to get too complacent. <a href="https://www.quantamagazine.org/new-codes-could-make-quantum-computing-10-times-more-efficient-20230825/">Theoretical</a> and technological innovations could bring the required qubit count down further, and there’s no proof that Shor’s algorithm is optimal — there might be a better quantum factoring algorithm out there that nobody’s discovered.</p> <p>If so, Regev said, “we should know as early as possible, before it’s too late.”</p> <h2><strong>Lost in the Trees </strong></h2> <p>Regev began his academic career in the late 1990s, when cryptographers were searching for a new form of public-key cryptography that wasn’t vulnerable to Shor’s algorithm. The most promising approach, called <a href="https://www.quantamagazine.org/cryptographys-future-will-be-quantum-safe-heres-how-it-will-work-20221109/">lattice-based cryptography</a>, relies on the apparent difficulty of computational problems involving high-dimensional arrays of points, or lattices. One such problem is akin to the task of locating the tree closest to a random point in a forest.</p> <aside class="post__aside mb2 relative post__aside--right post__aside--wide"><figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img width="1300" height="897" src="https://www.quantamagazine.org/wp-content/uploads/2023/10/OdedRegev-CourtesyofOdedRegev.webp" class="block fit-x fill-h fill-v is-loaded mxa" alt="Oded Regev in a white shirt in front of a green background" decoding="async" srcset="https://www.quantamagazine.org/wp-content/uploads/2023/10/OdedRegev-CourtesyofOdedRegev.webp 1300w, https://www.quantamagazine.org/wp-content/uploads/2023/10/OdedRegev-CourtesyofOdedRegev-520x359.webp 520w, https://www.quantamagazine.org/wp-content/uploads/2023/10/OdedRegev-CourtesyofOdedRegev-768x530.webp 768w" sizes="(max-width: 1300px) 100vw, 1300px" /> </div> <figcaption class='image__meta mt1'> <div class='caption wysiwyg h5 theme__anchors--solid fill-h post__aside__caption post__aside__caption--shortcode'> <p>Oded Regev developed a multidimensional version of Shor’s algorithm that runs even faster.</p> </div> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h post__aside__attribution'> <p>Courtesy of Oded Regev</p> </div> </figcaption> </figure> </aside> <p>“If it’s a hundred-dimensional forest, then that’s much more complicated than if it’s a two-dimensional forest,” said <a href="https://www.math.ucdavis.edu/~greg/">Greg Kuperberg</a>, a mathematician at the University of California, Davis.</p> <p>Regev began studying lattice-based cryptography as a postdoc, initially as an attacker — he wanted to stress-test the new approach by finding weaknesses that a quantum computer could exploit. But he couldn’t make any progress, and he soon wondered if there was a deeper reason for that. In 2005, he found a way to parlay those failed attacks into a <a href="https://dl.acm.org/doi/10.1145/1060590.1060603">form of lattice-based cryptography</a> superior to all other variants.</p> <p>“Oded is absolutely brilliant with lattices,” Kuperberg said.</p> <p>Over the years, as Regev taught Shor’s algorithm to successive generations of students, he found himself wondering whether the techniques he’d used for attacking lattice-based cryptography might actually prove useful in factoring algorithms. That’s because one step in the final, classical stage of Shor’s algorithm amounts to finding the nearest point in a one-dimensional lattice. That one-dimensional problem is trivially easy, but the resemblance to the analogous problem in hundreds of dimensions whose hardness underpins lattice-based cryptography was unmistakable.</p> <p>“If you’re someone that does lattices like me, you think, ‘OK, there’s some lattice going on here,’” Regev said. “But it wasn’t clear to me how to make use of that.” For years he toyed with other ideas for new quantum factoring algorithms, but he never got anywhere. Then last winter he returned to the problem and resolved to pin down that tantalizing connection between factoring and lattice-based cryptography. This time, he found success.</p> <h2><strong>Extra Dimensions</strong></h2> <p>Regev knew he needed to start by generalizing the periodic function at the heart of Shor’s algorithm from one dimension to many dimensions. In Shor’s algorithm, that function involves repeatedly multiplying a random number, dubbed <em>g</em>, with itself. But the period of this function — the number of times you must multiply by <em>g</em> before the output of the function starts repeating — can be very large, and that means that a quantum computer must multiply large numbers in some components of the superposition it uses to compute the periodic function. Those large multiplications are the most computationally costly part of Shor’s algorithm.</p> <aside class="post__aside mb2 relative mha post__aside--right hide-on-print"><div class='post__aside__pullquote relative'> <div class='pullquote theme__text mb2 align-c'> <div class='mb1'> <p>I would have thought that any algorithm that worked with this basic outline would be doomed. But I was wrong.</p> </div> </div> <cite class='o6 block align-c'> <p>Peter Shor</p> </cite> </div> </aside> <p>The analogous two-dimensional function instead uses a pair of numbers, <em>g</em><sub>1</sub> and <em>g</em><sub>2</sub>. It involves multiplying <em>g</em><sub>1</sub> with itself many times and then repeatedly multiplying by <em>g</em><sub>2</sub>. The period of this function is also two-dimensional — it’s defined by the number of <em>g</em><sub>1</sub> multiplications and <em>g</em><sub>2</sub> multiplications that together make the function’s output start repeating. There are many different combinations of <em>g</em><sub>1</sub> and <em>g</em><sub>2</sub> multiplications that will do the trick.</p> <p>Regev worked through the technical details to generalize the algorithm to an arbitrary number of dimensions, not just two, but his initial results weren’t encouraging. To compute the periodic function in many dimensions, the quantum computer would still have to multiply many numbers together. Each number wouldn’t need to get multiplied as many times as in the one-dimensional case, but there were more distinct numbers to multiply. The whole thing seemed to be a wash.</p> <p>“You think, ‘Great, I just did everything in high dimensions, and it’s exactly the same running time as Shor’s,’” Regev said. “I was stuck with that for a while.” Then he realized he could get around the problem by changing the order of the multiplications. Instead of repeatedly tacking numbers onto a single product that would grow progressively larger over the course of the quantum computation, he started with pairs of small numbers, multiplied the resulting products together, and proceeded upward. The total number of multiplications didn’t change much, but now nearly all of them involve relatively small numbers, making the calculation faster.</p> <p>“That makes all the difference in the world,” said <a href="http://people.csail.mit.edu/vinodv/">Vinod Vaikuntanathan</a>, a cryptographer at MIT.</p> <figure class='mb2 mt1 image--shortcode s:mt-0'> <div class='relative image mx0'> <img src="https://www.quantamagazine.org/wp-content/uploads/2023/10/SeekingCyclesbyMerrillSherman-v3_Desktop.svg" class="block fit-x fill-h fill-v is-loaded mxa large-print-img s:hidden m:hidden" alt="" decoding="async" /><img src="https://www.quantamagazine.org/wp-content/uploads/2023/10/SeekingCyclesbyMerrillSherman-v3_Mobile.svg" class="block fit-x fill-h fill-v is-loaded mxa large-print-img l:hidden" alt="" decoding="async" /> </div> <figcaption class='image__meta mt1'> <div class='attribution theme__anchors--solid wysiwyg pangram h6 mb1 fill-h post__aside__attribution'> <p>Merrill Sherman/<i data-stringify-type="italic">Quanta Magazine</i>; source: Oded Regev</p> </div> </figcaption> </figure> <p>At first, it looked as though Regev had just replaced one problem with another. He’d sped up the quantum computation of the periodic function by increasing the number of dimensions, but the subsequent classical computation required to extract the period was now similar to locating the nearest lattice point in a high-dimensional space — a task widely believed to be hard. The analogy to lattice-based cryptography that motivated his new approach seemed to doom it to failure.</p> <p>One cold morning in March before a trip to a seminar at Princeton University, Regev found himself waiting for the colleague he was carpooling with. “I arrived early, and he was late to pick up the car,” he said. While he was sitting around waiting, the last piece of the puzzle suddenly came to him. “That’s the moment when things fell into place, but it was baking for a while.”</p> <p>It all came down to the right number of dimensions. When the lattice dimension was too low, his algorithm couldn’t take full advantage of the speedup from multiplying smaller numbers. When it was too high, the quantum computation was fast, but the classical part required solving a prohibitively hard lattice problem. Regev had known from the beginning that to have any hope of success, he would have to work somewhere in between, but it wasn’t clear whether a sweet spot existed. That morning in March, he realized how he could tweak the details of the algorithm to make it run quickly in a few dozen dimensions.</p> <h2><strong>Writing in the Sand</strong></h2> <p>The improvement was profound. The number of elementary logical steps in the quantum part of Regev’s algorithm is proportional to <em>n</em><sup>1.5</sup> when factoring an <em>n</em>-bit number, rather than <em>n</em><sup>2</sup> as in Shor’s algorithm. The algorithm repeats that quantum part a few dozen times and combines the results to map out a high-dimensional lattice, from which it can deduce the period and factor the number. So the algorithm as a whole may not run faster, but speeding up the quantum part by reducing the number of required steps could make it easier to put it into practice.</p> <p>Of course, the time it takes to run a quantum algorithm is just one of several considerations. Equally important is the number of qubits required, which is analogous to the memory required to store intermediate values during an ordinary classical computation. The number of qubits that Shor’s algorithm requires to factor an <em>n</em>-bit number is proportional to <em>n</em>, while Regev’s algorithm in its original form requires a number of qubits proportional to <em>n</em><sup>1.5</sup> — a big difference for 2,048-bit numbers.</p> <p>In classical computing, speed is usually a more important consideration than memory, because classical bits are extremely robust: You can save a file on your computer and not worry about it randomly changing when you open it again later. Quantum computing researchers aren’t always so lucky.</p> <p>“Our qubits are constantly trying to fall apart, and we’re trying to stop them from falling apart,” Gidney said. “It’s like you’re trying to write in the sand and the wind is blowing it away.” That means the extra qubits required by Regev’s algorithm could be a major drawback.</p> <div class="related-list"> <aside class="post__aside mb2 relative hide-on-print post__aside--right"> <h2 class="mv0 h5 bold uppercase kern">Related:</h2> <hr class="mt1 o2"> <ol class="post__aside__list theme__anchors--none pt05"> <li class="mv075 flex"> <h3 class="h5t mt0 mb025 medium"> <a class="theme__accent theme__text-hover" href="https://www.quantamagazine.org/quantum-secure-cryptography-crosses-red-line-20150908/"> <span class="' . $span_class . '">A Tricky Path to Quantum-Safe Encryption</span> </a> </h3> </li> <li class="mv075 flex"> <h3 class="h5t mt0 mb025 medium"> <a class="theme__accent theme__text-hover" href="https://www.quantamagazine.org/a-new-approach-to-multiplication-opens-the-door-to-better-quantum-computers-20190424/"> <span class="' . $span_class . '">A New Approach to Multiplication Opens the Door to Better Quantum Computers</span> </a> </h3> </li> <li class="mv075 flex"> <h3 class="h5t mt0 mb025 medium"> <a class="theme__accent theme__text-hover" href="https://www.quantamagazine.org/post-quantum-cryptography-scheme-is-cracked-on-a-laptop-20220824/"> <span class="' . $span_class . '">‘Post-Quantum’ Cryptography Scheme Is Cracked on a Laptop</span> </a> </h3> </li> </ol> </aside> </div> <p>But Regev’s paper isn’t the end of the story. Two weeks ago, Vaikuntanathan and his graduate student Seyoon Ragavan found a way to reduce the algorithm’s memory use. <a href="https://arxiv.org/abs/2310.00899">Their variant</a> of Regev’s algorithm, like Shor’s original algorithm, requires a number of qubits proportional to <em>n</em> rather than <em>n</em><sup>1.5</sup>. Ekerå wrote in an email that the work “brings us a lot closer to an implementation that would be more efficient in practice.”</p> <p>The broader lesson of Regev’s new algorithm, beyond the implications for factoring, is that quantum computing researchers should always be open to surprises, even in problems that have been studied for decades.</p> <p>“This variant of my algorithm was undiscovered for 30 years and came out of the blue,” Shor said. “There’s still probably lots of other quantum algorithms to be found.”</p> <p><em>Editor’s note: Oded Regev receives funding from the </em><a href="https://www.simonsfoundation.org/"><em>Simons Foundation</em></a><em>, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage. More details are </em><a href="https://www.quantamagazine.org/about/"><em>available here</em></a><em>.</em></p> </div> </div> </div> </section> </div> <div class='pv2'></div> <div class='post__footer'> <div class='mha container--m'> <div class='flex flex-wrap flex-items-stretch'> <div class='post__footer__col fill-h pv2 flex flex-items-center flex-justify-center'> <section class='outer'> <aside class='post__sidebar hide flex flex-justify-center post__sidebar--footer'> <div class='post__sidebar__content'> <div class='post__sidebar__content__inner'> <div class='align-c mb075'> <div class='sidebar__author'> <a class='theme__accent-hover transition--color ' href='https://www.quantamagazine.org/authors/brubaker_ben/'> <div class='sidebar__author__avatar mha mb1'> <div class='image mx0 relative image--circle '> <div class='image__inner absolute fit-x is-loaded'> <img width="1500" height="1500" src="https://www.quantamagazine.org/wp-content/uploads/2022/10/Ben_Brubaker.jpg" class="absolute 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