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What Makes Quantum Computing So Hard to Explain?



Quantum computers, you might have heard, are magical uber-machines that will soon cure cancer and global warming by trying all possible answers in different parallel universes. For 15 years, on my blog and elsewhere, I’ve railed against this cartoonish vision, trying to explain what I see as the subtler but ironically even more fascinating truth. I approach this as a public service and almost my moral duty as a quantum computing researcher. Alas, the work feels Sisyphean: The cringeworthy hype about quantum computers has only increased over the years, as corporations and governments have invested billions, and as the technology has progressed to programmable 50-qubit devices that (on certain contrived benchmarks) really can give the world’s biggest supercomputers a run for their money. And just as in cryptocurrency, machine learning and other trendy fields, with money have come hucksters.

In reflective moments, though, I get it. The reality is that even if you removed all the bad incentives and the greed, quantum computing would still be hard to explain briefly and honestly without math. As the quantum computing pioneer Richard Feynman once said about the quantum electrodynamics work that won him the Nobel Prize, if it were possible to describe it in a few sentences, it wouldn’t have been worth a Nobel Prize.

Not that that’s stopped people from trying. Ever since Peter Shor discovered in 1994 that a quantum computer could break most of the encryption that protects transactions on the internet, excitement about the technology has been driven by more than just intellectual curiosity. Indeed, developments in the field typically get covered as business or technology stories rather than as science ones.

That would be fine if a business or technology reporter could truthfully tell readers, “Look, there’s all this deep quantum stuff under the hood, but all you need to understand is the bottom line: Physicists are on the verge of building faster computers that will revolutionize everything.”

The trouble is that quantum computers will not revolutionize everything.

Yes, they might someday solve a few specific problems in minutes that (we think) would take longer than the age of the universe on classical computers. But there are many other important problems for which most experts think quantum computers will help only modestly, if at all. Also, while Google and others recently made credible claims that they had achieved contrived quantum speedups, this was only for specific, esoteric benchmarks (ones that I helped develop). A quantum computer that’s big and reliable enough to outperform classical computers at practical applications like breaking cryptographic codes and simulating chemistry is likely still a long way off.

But how could a programmable computer be faster for only some problems? Do we know which ones? And what does a “big and reliable” quantum computer even mean in this context? To answer these questions we have to get into the deep stuff.

Let’s start with quantum mechanics. (What could be deeper?) The concept of superposition is infamously hard to render in everyday words. So, not surprisingly, many writers opt for an easy way out: They say that superposition means “both at once,” so that a quantum bit, or qubit, is just a bit that can be “both 0 and 1 at the same time,” while a classical bit can be only one or the other. They go on to say that a quantum computer would achieve its speed by using qubits to try all possible solutions in superposition — that is, at the same time, or in parallel.

This is what I’ve come to think of as the fundamental misstep of quantum computing popularization, the one that leads to all the rest. From here it’s just a short hop to quantum computers quickly solving something like the traveling salesperson problem by trying all possible answers at once — something almost all experts believe they won’t be able to do.

The thing is, for a computer to be useful, at some point you need to look at it and read an output. But if you look at an equal superposition of all possible answers, the rules of quantum mechanics say you’ll just see and read a random answer. And if that’s all you wanted, you could’ve picked one yourself.

What superposition really means is “complex linear combination.” Here, we mean “complex” not in the sense of “complicated” but in the sense of a real plus an imaginary number, while “linear combination” means we add together different multiples of states. So a qubit is a bit that has a complex number called an amplitude attached to the possibility that it’s 0, and a different amplitude attached to the possibility that it’s 1. These amplitudes are closely related to probabilities, in that the further some outcome’s amplitude is from zero, the larger the chance of seeing that outcome; more precisely, the probability equals the distance squared.

But amplitudes are not probabilities. They follow different rules. For example, if some contributions to an amplitude are positive and others are negative, then the contributions can interfere destructively and cancel each other out, so that the amplitude is zero and the corresponding outcome is never observed; likewise, they can interfere constructively and increase the likelihood of a given outcome. The goal in devising an algorithm for a quantum computer is to choreograph a pattern of constructive and destructive interference so that for each wrong answer the contributions to its amplitude cancel each other out, whereas for the right answer the contributions reinforce each other. If, and only if, you can arrange that, you’ll see the right answer with a large probability when you look. The tricky part is to do this without knowing the answer in advance, and faster than you could do it with a classical computer.

Twenty-seven years ago, Shor showed how to do all this for the problem of factoring integers, which breaks the widely used cryptographic codes underlying much of online commerce. We now know how to do it for some other problems, too, but only by exploiting the special mathematical structures in those problems. It’s not just a matter of trying all possible answers at once.

Compounding the difficulty is that, if you want to talk honestly about quantum computing, then you also need the conceptual vocabulary of theoretical computer science. I’m often asked how many times faster a quantum computer will be than today’s computers. A million times? A billion?

This question misses the point of quantum computers, which is to achieve better “scaling behavior,” or running time as a function of n, the number of bits of input data. This could mean taking a problem where the best classical algorithm needs a number of steps that grows exponentially with n, and solving it using a number of steps that grows only as n2. In such cases, for small n, solving the problem with a quantum computer will actually be slower and more expensive than solving it classically. It’s only as n grows that the quantum speedup first appears and then eventually comes to dominate.

But how can we know that there’s no classical shortcut — a conventional algorithm that would have similar scaling behavior to the quantum algorithm’s? Though typically ignored in popular accounts, this question is central to quantum algorithms research, where often the difficulty is not so much proving that a quantum computer can do something quickly, but convincingly arguing that a classical computer can’t. Alas, it turns out to be staggeringly hard to prove that problems are hard, as illustrated by the famous P versus NP problem (which asks, roughly, whether every problem with quickly checkable solutions can also be quickly solved). This is not just an academic issue, a matter of dotting i’s: Over the past few decades, conjectured quantum speedups have repeatedly gone away when classical algorithms were found with similar performance.

Note that, after explaining all this, I still haven’t said a word about the practical difficulty of building quantum computers. The problem, in a word, is decoherence, which means unwanted interaction between a quantum computer and its environment — nearby electric fields, warm objects, and other things that can record information about the qubits. This can result in premature “measurement” of the qubits, which collapses them down to classical bits that are either definitely 0 or definitely 1. The only known solution to this problem is quantum error correction: a scheme, proposed in the mid-1990s, that cleverly encodes each qubit of the quantum computation into the collective state of dozens or even thousands of physical qubits. But researchers are only now starting to make such error correction work in the real world, and actually putting it to use will take much longer. When you read about the latest experiment with 50 or 60 physical qubits, it’s important to understand that the qubits aren’t error-corrected. Until they are, we don’t expect to be able to scale beyond a few hundred qubits.

Once someone understands these concepts, I’d say they’re ready to start reading — or possibly even writing — an article on the latest claimed advance in quantum computing. They’ll know which questions to ask in the constant struggle to distinguish reality from hype. Understanding this stuff really is possible — after all, it isn’t rocket science; it’s just quantum computing!

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Secret Workings of Smell Receptors Revealed for First Time



Smell, rather than sight, reigns as the supreme sense for most animals. It allows them to find food, avoid danger and attract mates; it dominates their perceptions and guides their behavior; it dictates how they interpret and respond to the deluge of sensory information all around them.

“How we as biological creatures interface with chemistry in the world is profoundly important for understanding who we are and how we navigate the universe,” said Bob Datta, a neurobiologist at Harvard Medical School.

Yet olfaction might also be the least well understood of our senses, in part because of the complexity of the inputs it must reckon with. What we might label as a single odor — the smell of coffee in the morning, of wet grass after a summer storm, of shampoo or perfume — is often a mixture of hundreds of types of chemicals. For an animal to detect and discriminate between the many scents that are key to its survival, the limited repertoire of receptors on its olfactory sensory neurons must somehow recognize a vast number of compounds. So an individual receptor has to be able to respond to many diverse, seemingly unrelated odor molecules.

That versatility is at odds with the traditional lock-and-key model governing how selective chemical interactions tend to work. “In high school biology, that’s what I learned about ligand-receptor interactions,” said Annika Barber, a molecular biologist at Rutgers University. “Something has to fit precisely in a site, and then it changes the [protein’s atomic arrangement], and then it works.”

Now, new work has taken a crucial and much anticipated step forward in elucidating the beginning stages of the olfactory process. In a preprint posted online earlier this year, a team of researchers at Rockefeller University in New York provided the first molecular view of an olfactory receptor as it bound to an odor molecule. “That’s been a dream in the field” ever since olfactory receptors were discovered 30 years ago, said Richard Benton, a biologist at the University of Lausanne in Switzerland who was not involved in the new study.

“It’s unequivocally a landmark paper,” Datta said. “Although we’ve had access to receptors as molecules for a long time, no one’s ever actually seen with their eyes what it looks like when an odor binds to a receptor.”

The result goes a long way toward confirming how animals identify and discriminate among astronomical numbers of smells. It also sheds light on key principles of receptor activity that might have far-reaching implications — for the evolution of chemical perception, for our understanding of how other neurological systems and processes work, and for practical applications like the development of targeted drugs and insect repellents.

Several hypotheses have competed to explain how olfactory receptors achieve the necessary flexibility. Some scientists proposed that receptors respond to a single feature of odor molecules, such as shape or size; the brain might then identify an odor from some combination of those inputs. Other researchers posited that each receptor has multiple binding sites, enabling different kinds of compounds to dock. But to figure out which of these ideas was correct, they needed to see the receptor’s actual structure.

The Rockefeller team turned to receptor interactions in the jumping bristletail, an ancestral ground-dwelling insect that has a particularly simple olfactory receptor system.

In insects, olfactory receptors are ion channels that activate when an odor molecule binds to them. They may be the largest and most divergent family of ion channels in nature, with millions of variants across the world’s insect species. And so they must carefully balance generality against specificity, staying flexible enough to detect an enormous number of potential odors while being selective enough to reliably recognize the important ones, which could differ considerably from one species or environment to another.

What was the mechanism that allowed them to navigate that fine line, and to evolve that way? “It’s a crazy system to think about,” said Vanessa Ruta, a neuroscientist at Rockefeller University who led the research reported in the recent preprint. “So we realized that the best way to gain insight into this problem would probably be through structural methods.”

Traditional methods for pinning down the three-dimensional molecular structure of proteins don’t work well on olfactory receptors, which tend to misfold, behave abnormally or become difficult to distinguish under the conditions that those analyses require. But recent technological advances, most notably an imaging technique called cryo-electron microscopy, made it possible for Ruta and her colleagues to try.

They looked at the structure of a jumping bristletail olfactory receptor in three different configurations: by itself, and bound to either a common odor molecule called eugenol (which smells like clove to humans) or the insect repellant DEET. They then compared those structures, down to their individual atoms, to understand how odor binding opened the ion channel, and how a single receptor could detect chemicals of very different shapes and sizes.

“It actually is very beautiful,” Ruta said.

The researchers found that although DEET and eugenol don’t have much in common as molecules, they both docked at the same site within the receptor. That turned out to be a deep, geometrically simple pocket, lined with many amino acids that facilitate loose, weak interactions; eugenol and DEET took advantage of different interactions to lodge within it. Further computational modeling showed that each molecule was able to bind in many different orientations — and that many other kinds of odor compounds, though not all, could bind to the receptor in a similar way. This was no lock-and-key mechanism, but a one-size-fits-many approach.

The receptor “is doing a more holistic recognition of the molecule, as opposed to just detecting any specific structural feature,” Ruta said. “It’s just a very different chemical logic.”

When Ruta and her team introduced changes to the receptor’s pocket, they found that mutations of even a single amino acid were enough to alter its binding properties. And that, in turn, was enough to affect the receptor’s interactions with many compounds, entirely reconfiguring what the receptor responded to.

Widening the pocket, for instance, increased its affinity for DEET, a larger molecule, while decreasing its affinity for eugenol, which may not have been able to fit as snugly due to its smaller size. Such changes would have many downstream effects on the receptor’s broader odor-detecting palette, too, which the researchers were not set up to identify.

The team’s observations may explain how insect olfactory receptors can generally evolve so rapidly and diverge so much among species. Every insect species may have evolved “its unique repertoire of receptors that are really well suited to its particular chemical niche,” Ruta said.

“It tells us that more is going on than just the idea that receptors loosely interact with a bunch of ligands,” Datta said. A receptor built around a single binding pocket, with a response profile that can be retuned by the smallest of tweaks, could speed up evolution by freeing it to explore a broad spectrum of chemical repertoires.

The architecture of the receptor also supported this view. Ruta and her colleagues found that it consisted of four protein subunits loosely bound at the channel’s central pore, like the petals of a flower. Only the central region needed to be conserved as the receptor diversified and evolved; the genetic sequences governing the rest of the receptor units were less constrained. This structural organization meant the receptor could accommodate a wide degree of diversification.

Such light evolutionary constraints at the receptor level probably impose substantial selective pressure downstream on the neural circuits for olfaction: Nervous systems need good mechanisms for decoding the messy patterns of receptor activity. “Effectively, olfactory systems have evolved to take arbitrary patterns of receptor activation and endow them with meaning through learning and experience,” Ruta said.

Intriguingly, though, nervous systems don’t seem to be making the problem easier for themselves. Scientists had widely supposed that all the receptors on an individual olfactory neuron were of the same class, and that neurons for different classes went to segregated processing regions of the brain. In a pair of preprints posted last November, however, researchers reported that in both flies and mosquitoes, individual olfactory neurons express multiple classes of receptors. “Which is really surprising, and would increase the diversity of sensory perception even more,” Barber said.

The findings from Ruta’s team are far from the last word on how olfactory receptors work. Insects use many other classes of ion channel olfactory receptors, including ones that are much more complex and much more specific than those of the jumping bristletail. In mammals, the olfactory receptor is not even an ion channel; it belongs to an entirely different family of proteins.

“This is the first structure of odorant recognition in any receptor from any species. But it’s probably not the only mechanism of odorant recognition,” Ruta said. “This is just one solution to the problem. It would be very unlikely that it’s the only solution.”

Even so, she and other researchers think there are many more general lessons to learn from the jumping bristletail’s olfactory receptor. It’s tempting, for instance, to imagine how this mechanism might apply to other receptors in the brains of animals — from those that detect neuromodulators like dopamine to those that are affected by various kinds of anesthetics — “and how imprecise they are ‘allowed’ to be,” Barber said. “It offers a fascinating model for continuing to explore nonspecific binding interactions.”

Perhaps this flexible-binding approach should be considered in other contexts as well, she added. Research published in the Proceedings of the National Academy of Sciences in March, for example, suggested that even canonical lock-and-key ion channel receptors might not be as strictly selective as scientists thought.

If many different kinds of proteins bind to receptors through flexible, weak interactions within some type of pocket, that principle could guide rational drug design for various diseases, particularly neurological conditions. At the very least, Ruta’s work on the binding of DEET to an insect olfactory receptor could provide insights into how to develop targeted repellents. “The mosquito is still the deadliest animal on Earth” because of the diseases it carries, Ruta said.

Her findings actually clarify a debate more than a half-century old about how DEET works. DEET is one of the most effective insect repellents, but scientists haven’t understood why — whether it smells bad to insects, for instance, or whether it impairs their olfactory signaling. The work by Ruta and her colleagues elevates a different theory: that DEET confuses insects by activating lots of different receptors and flooding their olfactory system with meaningless signals.

“The mystery of chemical recognition is something that we now have a structural lens to think about,” Ruta said. “Structural biology, at its best, is beautiful and clarifying and has amazing explanatory power. My lab does a lot of work in more cellular and systems neuroscience, and very few experiments have as much explanatory power as a structure does.”

Datta agreed about the structural biology approach. “I think it’s really a harbinger of things to come,” he said. “It feels like the future.”

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Mathematicians Prove 2D Version of Quantum Gravity Really Works



Alexander Polyakov, a theoretical physicist now at Princeton University, caught a glimpse of the future of quantum theory in 1981. A range of mysteries, from the wiggling of strings to the binding of quarks into protons, demanded a new mathematical tool whose silhouette he could just make out.

“There are methods and formulae in science which serve as master keys to many apparently different problems,” he wrote in the introduction to a now famous four-page letter in Physics Letters B. “At the present time we have to develop an art of handling sums over random surfaces.”

Polyakov’s proposal proved powerful. In his paper he sketched out a formula that roughly described how to calculate averages of a wildly chaotic type of surface, the “Liouville field.” His work brought physicists into a new mathematical arena, one essential for unlocking the behavior of theoretical objects called strings and building a simplified model of quantum gravity.

Years of toil would lead Polyakov to breakthrough solutions for other theories in physics, but he never fully understood the mathematics behind the Liouville field.

Over the last seven years, however, a group of mathematicians has done what many researchers thought impossible. In a trilogy of landmark publications, they have recast Polyakov’s formula using fully rigorous mathematical language and proved that the Liouville field flawlessly models the phenomena Polyakov thought it would.

“It took us 40 years in math to make sense of four pages,” said Vincent Vargas, a mathematician at the French National Center for Scientific Research and co-author of the research with Rémi Rhodes of Aix-Marseille University, Antti Kupiainen of the University of Helsinki, François David of the French National Center for Scientific Research, and Colin Guillarmou of Paris-Saclay University.

The three papers forge a bridge between the pristine world of mathematics and the messy reality of physics — and they do so by breaking new ground in the mathematical field of probability theory. The work also touches on philosophical questions regarding the objects that take center stage in the leading theories of fundamental physics: quantum fields.

“This is a masterpiece in mathematical physics,” said Xin Sun, a mathematician at the University of Pennsylvania.

Infinite Fields

In physics today, the main actors in the most successful theories are fields — objects that fill space, taking on different values from place to place.

In classical physics, for example, a single field tells you everything about how a force pushes objects around. Take Earth’s magnetic field: The twitches of a compass needle reveal the field’s influence (its strength and direction) at every point on the planet.

Fields are central to quantum physics, too. However, the situation here is more complicated due to the deep randomness of quantum theory. From the quantum perspective, Earth doesn’t generate one magnetic field, but rather an infinite number of different ones. Some look almost like the field we observe in classical physics, but others are wildly different.

But physicists still want to make predictions — predictions that ideally match, in this case, what a mountaineer reads on a compass. Assimilating the infinite forms of a quantum field into a single prediction is the formidable task of a “quantum field theory,” or QFT. This is a model of how one or more quantum fields, each with their infinite variations, act and interact.

Driven by immense experimental support, QFTs have become the basic language of particle physics. The Standard Model is one such QFT, depicting fundamental particles like electrons as fuzzy bumps that emerge from an infinitude of electron fields. It has passed every experimental test to date (although various groups may be on the verge of finding the first holes).

Physicists play with many different QFTs. Some, like the Standard Model, aspire to model real particles moving through the four dimensions of our universe (three spatial dimensions plus one dimension of time). Others describe exotic particles in strange universes, from two-dimensional flatlands to six-dimensional uber-worlds. Their connection to reality is remote, but physicists study them in the hopes of gaining insights they can carry back into our own world.

Polyakov’s Liouville field theory is one such example.

Gravity’s Field

The Liouville field, which is based on an equation from complex analysis developed in the 1800s by the French mathematician Joseph Liouville, describes a completely random two-dimensional surface — that is, a surface, like Earth’s crust, but one in which the height of every point is chosen randomly. Such a planet would erupt with mountain ranges of infinitely tall peaks, each assigned by rolling a die with infinite faces.

Such an object might not seem like an informative model for physics, but randomness is not devoid of patterns. The bell curve, for example, tells you how likely you are to randomly pass a seven-foot basketball player on the street. Similarly, bulbous clouds and crinkly coastlines follow random patterns, but it’s nevertheless possible to discern consistent relationships between their large-scale and small-scale features.

Liouville theory can be used to identify patterns in the endless landscape of all possible random, jagged surfaces. Polyakov realized this chaotic topography was essential for modeling strings, which trace out surfaces as they move. The theory has also been applied to describe quantum gravity in a two-dimensional world. Einstein defined gravity as space-time’s curvature, but translating his description into the language of quantum field theory creates an infinite number of space-times — much as the Earth produces an infinite collection of magnetic fields. Liouville theory packages all those surfaces together into one object. It gives physicists the tools to measure the curvature —and hence, gravitation — at every location on a random 2D surface.

“Quantum gravity basically means random geometry, because quantum means random and gravity means geometry,” said Sun.

Polyakov’s first step in exploring the world of random surfaces was to write down an expression defining the odds of finding a particular spiky planet, much as the bell curve defines the odds of meeting someone of a particular height. But his formula did not lead to useful numerical predictions.

To solve a quantum field theory is to be able to use the field to predict observations. In practice, this means calculating a field’s “correlation functions,” which capture the field’s behavior by describing the extent to which a measurement of the field at one point relates, or correlates, to a measurement at another point. Calculating correlation functions in the photon field, for instance, can give you the textbook laws of quantum electromagnetism.

Polyakov was after something more abstract: the essence of random surfaces, similar to the statistical relationships that make a cloud a cloud or a coastline a coastline. He needed the correlations between the haphazard heights of the Liouville field. Over the decades he tried two different ways of calculating them. He started with a technique called the Feynman path integral and ended up developing a workaround known as the bootstrap. Both methods came up short in different ways, until the mathematicians behind the new work united them in a more precise formulation.

Add ’Em Up

You might imagine that accounting for the infinitely many forms a quantum field can take is next to impossible. And you would be right. In the 1940s Richard Feynman, a quantum physics pioneer, developed one prescription for dealing with this bewildering situation, but the method proved severely limited.

Take, again, Earth’s magnetic field. Your goal is to use quantum field theory to predict what you’ll observe when you take a compass reading at a particular location. To do this, Feynman proposed summing all the field’s forms together. He argued that your reading will represent some average of all the field’s possible forms. The procedure for adding up these infinite field configurations with the proper weighting is known as the Feynman path integral.

It’s an elegant idea that yields concrete answers only for select quantum fields. No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe. Mathematicians question its very existence as a valid operation and are bothered by the way physicists rely on it.

“I’m disturbed as a mathematician by something which is not defined,” said Eveliina Peltola, a mathematician at the University of Bonn in Germany.

Physicists can harness Feynman’s path integral to calculate exact correlation functions for only the most boring of fields — free fields, which do not interact with other fields or even with themselves. Otherwise, they have to fudge it, pretending the fields are free and adding in mild interactions, or “perturbations.” This procedure, known as perturbation theory, gets them correlation functions for most of the fields in the Standard Model, because nature’s forces happen to be quite feeble.

But it didn’t work for Polyakov. Although he initially speculated that the Liouville field might be amenable to the standard hack of adding mild perturbations, he found that it interacted with itself too strongly. Compared to a free field, the Liouville field seemed mathematically inscrutable, and its correlation functions appeared unattainable.

Up by the Bootstraps

Polyakov soon began looking for a workaround. In 1984, he teamed up with Alexander Belavin and Alexander Zamolodchikov to develop a technique called the bootstrap — a mathematical ladder that gradually leads to a field’s correlation functions.

To start climbing the ladder, you need a function which expresses the correlations between measurements at a mere three points in the field. This “three-point correlation function,” plus some additional information about the energies a particle of the field can take, forms the bottom rung of the bootstrap ladder.

From there you climb one point at a time: Use the three-point function to construct the four-point function, use the four-point function to construct the five-point function, and so on. But the procedure generates conflicting results if you start with the wrong three-point correlation function in the first rung.

Polyakov, Belavin and Zamolodchikov used the bootstrap to successfully solve a variety of simple QFT theories, but just as with the Feynman path integral, they couldn’t make it work for the Liouville field.

Then in the 1990s two pairs of physicists — Harald Dorn and Hans-Jörg Otto, and Zamolodchikov and his brother Alexei — managed to hit on the three-point correlation function that made it possible to scale the ladder, completely solving the Liouville field (and its simple description of quantum gravity). Their result, known by their initials as the DOZZ formula, let physicists make any prediction involving the Liouville field. But even the authors knew they had arrived at it partially by chance, not through sound mathematics.

“They were these kind of geniuses who guessed formulas,” said Vargas.

Educated guesses are useful in physics, but they don’t satisfy mathematicians, who afterward wanted to know where the DOZZ formula came from. The equation that solved the Liouville field should have come from some description of the field itself, even if no one had the faintest idea how to get it.

“It looked to me like science fiction,” said Kupiainen. “This is never going to be proven by anybody.”

Taming Wild Surfaces

In the early 2010s, Vargas and Kupiainen joined forces with the probability theorist Rémi Rhodes and the physicist François David. Their goal was to tie up the mathematical loose ends of the Liouville field — to formalize the Feynman path integral that Polyakov had abandoned and, just maybe, demystify the DOZZ formula.

As they began, they realized that a French mathematician named Jean-Pierre Kahane had discovered, decades earlier, what would turn out to be the key to Polyakov’s master theory.

“In some sense it’s completely crazy that Liouville was not defined before us,” Vargas said. “All the ingredients were there.”

The insight led to three milestone papers in mathematical physics completed between 2014 and 2020.

They first polished off the path integral, which had failed Polyakov because the Liouville field interacts strongly with itself, making it incompatible with Feynman’s perturbative tools. So instead, the mathematicians used Kahane’s ideas to recast the wild Liouville field as a somewhat milder random object known as the Gaussian free field. The peaks in the Gaussian free field don’t fluctuate to the same random extremes as the peaks in the Liouville field, making it possible for the mathematicians to calculate averages and other statistical measures in sensible ways.

“Somehow it’s all just using the Gaussian free field,” Peltola said. “From that they can construct everything in the theory.”

In 2014, they unveiled their result: a new and improved version of the path integral Polyakov had written down in 1981, but fully defined in terms of the trusted Gaussian free field. It’s a rare instance in which Feynman’s path integral philosophy has found a solid mathematical execution.

“Path integrals can exist, do exist,” said Jörg Teschner, a physicist at the German Electron Synchrotron.

With a rigorously defined path integral in hand, the researchers then tried to see if they could use it to get answers from the Liouville field and to derive its correlation functions. The target was the mythical DOZZ formula — but the gulf between it and the path integral seemed vast.

“We’d write in our papers, just for propaganda reasons, that we want to understand the DOZZ formula,” said Kupiainen.

The team spent years prodding their probabilistic path integral, confirming that it truly had all the features needed to make the bootstrap work. As they did so, they built on earlier work by Teschner. Eventually, Vargas, Kupiainen and Rhodes succeeded with a paper posted in 2017 and another in October 2020, with Colin Guillarmou. They derived DOZZ and other correlation functions from the path integral and showed that these formulas perfectly matched the equations physicists had reached using the bootstrap.

“Now we’re done,” Vargas said. “Both objects are the same.”

The work explains the origins of the DOZZ formula and connects the bootstrap procedure —which mathematicians had considered sketchy — with verified mathematical objects. Altogether, it resolves the final mysteries of the Liouville field.

“It’s somehow the end of an era,” said Peltola. “But I hope it’s also the beginning of some new, interesting things.”

New Hope for QFTs

Vargas and his collaborators now have a unicorn on their hands, a strongly interacting QFT perfectly described in a nonperturbative way by a brief mathematical formula that also makes numerical predictions.

Now the literal million-dollar question is: How far can these probabilistic methods go? Can they generate tidy formulas for all QFTs? Vargas is quick to dash such hopes, insisting that their tools are specific to the two-dimensional environment of Liouville theory. In higher dimensions, even free fields are too irregular, so he doubts the group’s methods will ever be able to handle the quantum behavior of gravitational fields in our universe.

But the fresh minting of Polyakov’s “master key” will open other doors. Its effects are already being felt in probability theory, where mathematicians can now wield previously dodgy physics formulas with impunity. Emboldened by the Liouville work, Sun and his collaborators have already imported equations from physics to solve two problems regarding random curves.

Physicists await tangible benefits too, further down the road. The rigorous construction of the Liouville field could inspire mathematicians to try their hand at proving features of other seemingly intractable QFTs — not just toy theories of gravity but descriptions of real particles and forces that bear directly on the deepest physical secrets of reality.

“[Mathematicians] will do things that we can’t even imagine,” said Davide Gaiotto, a theoretical physicist at the Perimeter Institute.

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Warm-starting quantum optimization



Daniel J. Egger1, Jakub Mareček2, and Stefan Woerner1

1IBM Quantum, IBM Research – Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
2Czech Technical University, Karlovo nam. 13, Prague 2, the Czech Republic

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There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

Many optimization problems in binary decision variables are hard to solve. In this work, we demonstrate how to leverage decades of research in classical optimization algorithms to warm-start quantum optimization algorithms. This allows the quantum algorithm to inherit the performance guarantees from the classical algorithm used in the warm-start.

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Experimental localisation of quantum entanglement through monitored classical mediator



Soham Pal1, Priya Batra1, Tanjung Krisnanda2, Tomasz Paterek2,3,4, and T. S. Mahesh1

1Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India
2School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
3MajuLab, International Joint Research Unit UMI 3654, CNRS, Université Côte d’Azur, Sorbonne Université, National University of Singapore, Nanyang Technological University, Singapore
4Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland

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Quantum entanglement is a form of correlation between quantum particles that cannot be increased via local operations and classical communication. It has therefore been proposed that an increment of quantum entanglement between probes that are interacting solely via a mediator implies non-classicality of the mediator. Indeed, under certain assumptions regarding the initial state, entanglement gain between the probes indicates quantum coherence in the mediator. Going beyond such assumptions, there exist other initial states which produce entanglement between the probes via only local interactions with a classical mediator. In this process the initial entanglement between any probe and the rest of the system “flows through” the classical mediator and gets localised between the probes. Here we theoretically characterise maximal entanglement gain via classical mediator and experimentally demonstrate, using liquid-state NMR spectroscopy, the optimal growth of quantum correlations between two nuclear spin qubits interacting through a mediator qubit in a classical state. We additionally monitor, i.e., dephase, the mediator in order to emphasise its classical character. Our results indicate the necessity of verifying features of the initial state if entanglement gain between the probes is used as a figure of merit for witnessing non-classical mediator. Such methods were proposed to have exemplary applications in quantum optomechanics, quantum biology and quantum gravity.

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The above citations are from SAO/NASA ADS (last updated successfully 2021-06-17 13:33:33). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2021-06-17 13:33:31: Could not fetch cited-by data for 10.22331/q-2021-06-17-478 from Crossref. This is normal if the DOI was registered recently.

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