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Mathematicians Prove Symmetry of Phase Transitions



For more than 50 years, mathematicians have been searching for a rigorous way to prove that an unusually strong symmetry is universal across physical systems at the mysterious juncture where they’re changing from one state into another. The powerful symmetry, known as conformal invariance, is actually a package of three separate symmetries that are all wrapped up within it.

Now, in a proof posted in December, a team of five mathematicians has come closer than ever before to proving that conformal invariance is a necessary feature of these physical systems as they transition between phases. The work establishes that rotational invariance — one of the three symmetries contained within conformal invariance — is present at the boundary between states in a wide range of physical systems.

“It’s a major contribution. This was open for a long time,” said Gady Kozma of the Weizmann Institute of Science in Israel.

Rotational invariance is a symmetry exhibited by the circle: Rotate it any number of degrees and it looks the same. In the context of physical systems on the brink of phase changes, it means many properties of the system behave the same regardless of how a model of the system is rotated.

Earlier results had established that rotational invariance holds for two specific models, but their methods were not flexible enough to be used for other models. The new proof breaks from this history and marks the first time that rotational invariance has been proved to be a universal phenomenon across a broad class of models.

“This universality result is even more intriguing” because it means that the same patterns emerge regardless of the differences between models of physical systems, said Hugo Duminil-Copin of the Institute of Advanced Scientific Studies (IHES) and the University of Geneva.

Duminil-Copin is a co-author of the work along with Karol Kajetan Kozlowski of the École Normale Supérieure in Lyon, Dmitry Krachun of the University of Geneva, Ioan Manolescu of the University of Fribourg and Mendes Oulamara of IHES and Paris-Saclay University.

The new work also raises hopes that mathematicians might be closing in on an even more ambitious result: proving that these physical models are conformally invariant. Over the last several decades mathematicians have proved that conformal invariance holds for a few particular models, but they’ve been unable to prove that it holds for all of them, as they suspect it does. This new proof lays the foundation for sweeping results along those lines.

“It’s already a very big breakthrough,” said Stanislav Smirnov of the University of Geneva. “[Conformal invariance] now looks within reach.”

Magic Moments

Transitions between one state and another are some of the most mesmerizing events in the natural world. Some are abrupt, like the transformation of water when it heats into vapor or cools into ice. Others, like the phase transitions studied in the new work, evolve gradually, with a murky boundary between two states. It’s here, at these critical points, that the system hangs in the balance and is neither quite what it was nor what it’s about to become.

Mathematicians try to bottle this magic in simplified models.

Take, for example, what happens as you heat iron. Above a certain temperature it loses its magnetic attraction. The change occurs as millions of sizzling atoms acting as miniature magnets flip and no longer align with the magnetic positions of their neighbors. Around 1,000 degrees Fahrenheit, heat wins out and a magnet reduces to a mere piece of metal.

Mathematicians study this process with the Ising model. It imagines a block of iron as a two-dimensional square lattice, much like the grid on a piece of graph paper. The model situates the iron atoms at the intersections of the lattice lines and represents them as arrows pointing up or down.

The Ising model came into widespread use in the 1950s as a tool to represent physical systems near critical points. These included metals losing magnetism and also the gas-liquid transition in air and the switch between order and disorder in alloys. These are all very different types of systems that behave in very different ways at the microscopic level.

Then, in 1970, the young physicist Alexander Polyakov predicted that despite their apparent differences, these systems all exhibit conformal invariance at their critical points. Decades of subsequent analysis convinced physicists that Polyakov was right. But mathematicians have been left with the difficult job of rigorously proving that it’s true.

The Symmetry of Symmetries

Conformal invariance consists of three types of symmetries rolled into one more extensive symmetry. You can shift objects that exhibit it (translational symmetry), rotate them by any number of degrees (rotational symmetry or invariance), or change their size (scale symmetry), all without changing any of their angles.

“[Conformal invariance] is what sometimes I call ‘the symmetry to rule them all’ because it’s an overall symmetry, which is stronger than the three others,” said Duminil-Copin.

Conformal invariance shows up in physical models in a more subtle way. In the Ising model, when magnetism is still intact and a phase transition hasn’t occurred yet, most arrows point up in one massive cluster. There are also some small clusters in which all arrows point down. But at the critical temperature, atoms can influence each other from greater distances than before. Suddenly, the alignment of atoms everywhere is unstable: Clusters of different sizes with arrows pointing either up or down appear all at once, and magnetism is about to be lost.

At this critical point, mathematicians look at the model from very far away and study correlations between arrows, which characterize the likelihood that any given pair points in the same direction. In this setting, conformal invariance means that you can translate, rotate and rescale the grid without distorting those correlations. That is, if two arrows have a 50% chance of pointing in the same direction, and then you apply those symmetries, the arrows that come to occupy the same positions in the lattice will also have a 50% chance of aligning.

The result is that if you compare your original lattice model with the new, transformed lattice, you won’t be able to tell which is which. Importantly, the same is not true of the Ising model before the phase transition. There, if you take the top corner of the lattice and blow it up to be the same size as the original (a scale transformation), you’ll also increase the typical size of the small islands of down arrows, making it obvious which lattice is the original.

The presence of conformal invariance has a direct physical meaning: It indicates that the global behavior of the system won’t change even if you tweak the microscopic details of the substance. It also hints at a certain mathematical elegance that sets in, for a brief interlude, just as the entire system is breaking its overarching form and becoming something else.

The First Proofs

In 2001 Smirnov produced the first rigorous mathematical proof of conformal invariance in a physical model. It applied to a model of percolation, which is the process of liquid passing through a maze in a porous medium, like a stone.

Smirnov looked at percolation on a triangular lattice, where water is allowed to flow only through vertices that are “open.” Initially, every vertex has the same probability of being open to the flow of water. When the probability is low, the chances of water having a path all the way through the stone is low.

But as you slowly increase the probability, there comes a point where enough vertices are open to create the first path spanning the stone. Smirnov proved that at the critical threshold, the triangular lattice is conformally invariant, meaning percolation occurs regardless of how you transform it with conformal symmetries.

Five years later, at the 2006 International Congress of Mathematicians, Smirnov announced that he had proved conformal invariance again, this time in the Ising model. Combined with his 2001 proof, this groundbreaking work earned him the Fields Medal, math’s highest honor.

In the years since, other proofs have trickled in on a case-by-case basis, establishing conformal invariance for specific models. None have come close to proving the universality that Polyakov envisioned.

“The previous proofs that worked were tailored to specific models,” said Federico Camia, a mathematical physicist at New York University Abu Dhabi. “You have a very specific tool to prove it for a very specific model.”

Smirnov himself acknowledged that both of his proofs relied on some sort of “magic” that was present in the two models he worked with but isn’t usually available.

“Since it used magic, it only works in situations where there is magic, and we weren’t able to find magic in other situations,” he said.

The new work is the first to disrupt this pattern — proving that rotational invariance, a core feature of conformal invariance, exists widely.

One at a Time

Duminil-Copin first began to think about proving universal conformal invariance in the late 2000s, when he was Smirnov’s graduate student at the University of Geneva. He had a unique understanding of the brilliance of his mentor’s techniques — and also of their limitations. Smirnov bypassed the need to prove all three symmetries separately and instead found a direct route to establishing conformal invariance — like a shortcut to a summit.

“He’s an amazing problem solver. He proved conformal invariance of two models of statistical physics by finding the entrance in this huge mountain, like this kind of crux that he went through,” said Duminil-Copin.

For years after graduate school, Duminil-Copin worked on building up a set of proofs that might eventually allow him to go beyond Smirnov’s work. By the time he and his co-authors set to work in earnest on conformal invariance, they were ready to take a different approach than Smirnov had. Rather than take their chances with magic, they returned to the original hypotheses about conformal invariance made by Polyakov and later physicists.

The physicists had required a proof in three steps, one for each symmetry present in conformal invariance: translational, rotational and scale invariance. Prove each of them separately, and you get conformal invariance as a consequence.

With this in mind, the authors set out to prove scale invariance first, believing that rotational invariance would be the most difficult symmetry and knowing that translational invariance was simple enough and wouldn’t require its own proof. In attempting this, they realized instead that they could prove the existence of rotational invariance at the critical point in a large variety of percolation models on square and rectangular grids.

They used a technique from probability theory, called coupling, that made it possible to directly compare the large-scale behavior of square lattices with rotated rectangular lattices. By combining this approach with ideas from another field of mathematics called integrability, which studies hidden structures in evolving systems, they were able to prove that the behavior at critical points was the same across the models — thus establishing rotational invariance. Then they proved that their results extended to other physical models where it’s possible to apply the same coupling.

The end result is a powerful proof that rotational invariance is a universal property of a large subset of known two-dimensional models. They believe the success of their work indicates that a similarly eclectic set of techniques, melded from various fields of math, will be necessary to make additional progress on conformal invariance.

“I think it’s going to be more and more true, in arguments of conformal invariance and the study of phase transitions, that you need a little bit of everything. You cannot just attack it with one angle of attack,” said Duminil-Copin.

Last Steps

For the first time since Smirnov’s 2001 result, mathematicians have new purchase on the long-standing challenge of proving the universality of conformal invariance. And unlike that earlier work, this new result opens new paths to follow. By following a bottom-up approach in which they aimed to prove one constituent symmetry at a time, the researchers hope they laid a foundation that will eventually support a universal set of results.

Now, with rotational invariance down, Duminil-Copin and his colleagues have their sights set on scale invariance, their original target. A proof of scale invariance, given the recent work on rotational symmetry and the fact that translational symmetry doesn’t need its own proof, would put mathematicians on the cusp of proving full conformal invariance. And the flexibility of their methods makes the researchers optimistic it can be done.

“I definitely think that step three is going to fall fairly soon,” said Duminil-Copin. “If it’s not us, it would be somebody smarter, but definitely, it’s going to happen very soon.”

The proof of rotational invariance took five years, though, so the next results may yet take some time. Still, Smirnov is hopeful that two-dimensional conformal invariance may finally be within reach.

“That might mean a week, or it might mean five years, but I’m much more optimistic than I was in November,” said Smirnov.

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A Soil-Science Revolution Upends Plans to Fight Climate Change



The hope was that the soil might save us. With civilization continuing to pump ever-increasing amounts of carbon dioxide into the atmosphere, perhaps plants — nature’s carbon scrubbers — might be able to package up some of that excess carbon and bury it underground for centuries or longer.

That hope has fueled increasingly ambitious climate change–mitigation plans. Researchers at the Salk Institute, for example, hope to bioengineer plants whose roots will churn out huge amounts of a carbon-rich, cork-like substance called suberin. Even after the plant dies, the thinking goes, the carbon in the suberin should stay buried for centuries. This Harnessing Plants Initiative is perhaps the brightest star in a crowded firmament of climate change solutions based on the brown stuff beneath our feet.

Such plans depend critically on the existence of large, stable, carbon-rich molecules that can last hundreds or thousands of years underground. Such molecules, collectively called humus, have long been a keystone of soil science; major agricultural practices and sophisticated climate models are built on them.

But over the past 10 years or so, soil science has undergone a quiet revolution, akin to what would happen if, in physics, relativity or quantum mechanics were overthrown. Except in this case, almost nobody has heard about it — including many who hope soils can rescue the climate. “There are a lot of people who are interested in sequestration who haven’t caught up yet,” said Margaret Torn, a soil scientist at Lawrence Berkeley National Laboratory.

A new generation of soil studies powered by modern microscopes and imaging technologies has revealed that whatever humus is, it is not the long-lasting substance scientists believed it to be. Soil researchers have concluded that even the largest, most complex molecules can be quickly devoured by soil’s abundant and voracious microbes. The magic molecule you can just stick in the soil and expect to stay there may not exist.

“I have The Nature and Properties of Soils in front of me — the standard textbook,” said Gregg Sanford, a soil researcher at the University of Wisconsin, Madison. “The theory of soil organic carbon accumulation that’s in that textbook has been proven mostly false … and we’re still teaching it.”

The consequences go far beyond carbon sequestration strategies. Major climate models such as those produced by the Intergovernmental Panel on Climate Change are based on this outdated understanding of soil. Several recent studies indicate that those models are underestimating the total amount of carbon that will be released from soil in a warming climate. In addition, computer models that predict the greenhouse gas impacts of farming practices — predictions that are being used in carbon markets — are probably overly optimistic about soil’s ability to trap and hold on to carbon.

It may still be possible to store carbon underground long term.  Indeed, radioactive dating measurements suggest that some amount of carbon can stay in the soil for centuries. But until soil scientists build a new paradigm to replace the old — a process now underway — no one will fully understand why.

The Death of Humus

Soil doesn’t give up its secrets easily. Its constituents are tiny, varied and outrageously numerous. At a bare minimum, it consists of minerals, decaying organic matter, air, water, and enormously complex ecosystems of microorganisms. One teaspoon of healthy soil contains more bacteria, fungi and other microbes than there are humans on Earth.

The German biologist Franz Karl Achard was an early pioneer in making sense of the chaos. In a seminal 1786 study, he used alkalis to extract molecules made of long carbon chains from peat soils. Over the centuries, scientists came to believe that such long chains, collectively called humus, constituted a large pool of soil carbon that resists decomposition and pretty much just sits there. A smaller fraction consisting of shorter molecules was thought to feed microbes, which respired carbon dioxide to the atmosphere.

This view was occasionally challenged, but by the mid-20th century, the humus paradigm was “the only game in town,” said Johannes Lehmann, a soil scientist at Cornell University. Farmers were instructed to adopt practices that were supposed to build humus. Indeed, the existence of humus is probably one of the few soil science facts that many non-scientists could recite.

What helped break humus’s hold on soil science was physics. In the second half of the 20th century, powerful new microscopes and techniques such as nuclear magnetic resonance and X-ray spectroscopy allowed soil scientists for the first time to peer directly into soil and see what was there, rather than pull things out and then look at them.

What they found — or, more specifically, what they didn’t find — was shocking: there were few or no long “recalcitrant” carbon molecules — the kind that don’t break down. Almost everything seemed to be small and, in principle, digestible.

“We don’t see any molecules in soil that are so recalcitrant that they can’t be broken down,” said Jennifer Pett-Ridge, a soil scientist at Lawrence Livermore National Laboratory. “Microbes will learn to break anything down — even really nasty chemicals.”

Lehmann, whose studies using advanced microscopy and spectroscopy were among the first to reveal the absence of humus, has become the concept’s debunker-in-chief. A 2015 Nature paper he co-authored states that “the available evidence does not support the formation of large-molecular-size and persistent ‘humic substances’ in soils.” In 2019, he gave a talk with a slide containing a mock death announcement for “our friend, the concept of Humus.”

Over the past decade or so, most soil scientists have come to accept this view. Yes, soil is enormously varied. And it contains a lot of carbon. But there’s no carbon in soil that can’t, in principle, be broken down by microorganisms and released into the atmosphere. The latest edition of The Nature and Properties of Soils, published in 2016, cites Lehmann’s 2015 paper and acknowledges that “our understanding of the nature and genesis of soil humus has advanced greatly since the turn of the century, requiring that some long-accepted concepts be revised or abandoned.”

Old ideas, however, can be very recalcitrant. Few outside the field of soil science have heard of humus’s demise.

Buried Promises

At the same time that soil scientists were rediscovering what exactly soil is, climate researchers were revealing that increasing amounts of carbon dioxide in the atmosphere were rapidly warming the climate, with potentially catastrophic consequences.

Thoughts soon turned to using soil as a giant carbon sink. Soils contain enormous amounts of carbon — more carbon than in Earth’s atmosphere and all its vegetation combined. And while certain practices such as plowing can stir up that carbon — farming, over human history, has released an estimated 133 billion metric tons of carbon into the atmosphere — soils can also take up carbon, as plants die and their roots decompose.

Scientists began to suggest that we might be able to coax large volumes of atmospheric carbon back into the soil to dampen or even reverse the damage of climate change.

In practice, this has proved difficult. An early idea to increase carbon stores — planting crops without tilling the soil — has mostly fallen flat. When farmers skipped the tilling and instead drilled seeds into the ground, carbon stores grew in upper soil layers, but they disappeared from lower layers. Most experts now believe that the practice redistributes carbon within the soil rather than increases it, though it can improve other factors such as water quality and soil health.

Efforts like the Harnessing Plants Initiative represent something like soil carbon sequestration 2.0: a more direct intervention to essentially jam a bunch of carbon into the ground.

The initiative emerged when two plant geneticists at the Salk Institute, Joanne Chory and Wolfgang Busch, came up with an idea: Create plants whose roots produce an excess of carbon-rich molecules. By their calculations, if grown widely, such plants might sequester up to 20% of the excess carbon dioxide that humans add to the atmosphere every year.

The researchers zeroed in on a complex, cork-like molecule called suberin, which is produced by many plant roots. Studies from the 1990s and 2000s had hinted that suberin and similar molecules could resist decomposition in soil.

With flashy marketing, the Harnessing Plants Initiative gained attention. An initial round of fundraising in 2019 brought in over $35 million. Last year, the multibillionaire Jeff Bezos contributed $30 million from his “Earth Fund.”

But as the project gained momentum, it attracted doubters. One group of researchers noted in 2016 that no one had actually observed the suberin decomposition process. When those authors did the relevant experiment, they found that much of the suberin decayed quickly.

In 2019, Chory described the project at a TED conference. Asmeret Asefaw Berhe, a soil scientist at the University of California, Merced, who spoke at the same conference, pointed out to Chory that according to modern soil science, suberin, like any carbon-containing compound, should break down in soil. (Berhe, who has been nominated to lead the U.S. Department of Energy’s Office of Science, declined an interview request.)

Around the same time, Hanna Poffenbarger, a soil researcher at the University of Kentucky, made a similar comment after hearing Busch speak at a workshop. “You should really get some soil scientists on board, because the assumption that we can breed for more recalcitrant roots — that may not be valid,” Poffenbarger recalls telling Busch.

Questions about the project surfaced publicly earlier this year, when Jonathan Sanderman, a soil scientist at the Woodwell Climate Research Center in Woods Hole, Massachusetts, tweeted, “I thought the soil biogeochem community had moved on from the idea that there is a magical recalcitrant plant compound. Am I missing some important new literature on suberin?” Another soil scientist responded, “Nope, the literature suggests that suberin will be broken down just like every other organic plant component. I’ve never understood why the @salkinstitute has based their Harnessing Plant Initiative on this premise.”

Busch, in an interview, acknowledged that “there is no unbreakable biomolecule.” But, citing published papers on suberin’s resistance to decomposition, he said, “We are still very optimistic when it comes to suberin.”

He also noted a second initiative Salk researchers are pursuing in parallel to enhancing suberin. They are trying to design plants with longer roots that could deposit carbon deeper in soil. Independent experts such as Sanderman agree that carbon tends to stick around longer in deeper soil layers, putting that solution on potentially firmer conceptual ground.

Chory and Busch have also launched collaborations with Berhe and Poffenbarger, respectively. Poffenbarger, for example, will analyze how soil samples containing suberin-rich plant roots change under different environmental conditions. But even those studies won’t answer questions about how long suberin sticks around, Poffenbarger said — important if the goal is to keep carbon out of the atmosphere long enough to make a dent in global warming.

Beyond the Salk project, momentum and money are flowing toward other climate projects that would rely on long-term carbon sequestration and storage in soils. In an April speech to Congress, for example, President Biden suggested paying farmers to plant cover crops, which are grown not for harvest but to nurture the soil in between plantings of cash crops. Evidence suggests that when cover crop roots break down, some of their carbon stays in the soil — although as with suberin, how long it lasts is an open question.

Not Enough Bugs in the Code

Recalcitrant carbon may also be warping climate prediction.

In the 1960s, scientists began writing large, complex computer programs to predict the global climate’s future. Because soil both takes up and releases carbon dioxide, climate models attempted to take into account soil’s interactions with the atmosphere. But the global climate is fantastically complex, and to enable the programs to run on the machines of the time, simplifications were necessary. For soil, scientists made a big one: They ignored microbes in the soil entirely. Instead, they basically divided soil carbon into short-term and long-term pools, in accordance with the humus paradigm.

More recent generations of models, including ones that the Intergovernmental Panel on Climate Change uses for its widely read reports, are essentially palimpsests built on earlier ones, said Torn. They still assume soil carbon exists in long-term and short-term pools. As a consequence, these models may be overestimating how much carbon will stick around in soils and underestimating how much carbon dioxide they will emit.

Last summer, a study published in Nature examined how much carbon dioxide was released when researchers artificially warmed the soil in a Panamanian rainforest to mimic the long-term effects of climate change. They found that the warmed soil released 55% more carbon than nearby unwarmed areas — a much larger release than predicted by most climate models. The researchers think that microbes in the soil grow more active at the warmer temperatures, leading to the increase.

The study was especially disheartening because most of the world’s soil carbon is in the tropics and the northern boreal zone. Despite this, leading soil models are calibrated to results of soil studies in temperate countries such as the U.S. and Europe, where most studies have historically been done. “We’re doing pretty bad in high latitudes and the tropics,” said Lehmann.

Even temperate climate models need improvement. Torn and colleagues reported earlier this year that, contrary to predictions, deep soil layers in a California forest released roughly a third of their carbon when warmed for five years.

Ultimately, Torn said, models need to represent soil as something closer to what it actually is: a complex, three-dimensional environment governed by a hyper-diverse community of carbon-gobbling bacteria, fungi and other microscopic beings. But even smaller steps would be welcome. Just adding microbes as a single class would be major progress for most models, she said.

Fertile Ground

If the humus paradigm is coming to an end, the question becomes: What will replace it?

One important and long-overlooked factor appears to be the three-dimensional structure of the soil environment. Scientists describe soil as a world unto itself, with the equivalent of continents, oceans and mountain ranges. This complex microgeography determines where microbes such as bacteria and fungi can go and where they can’t; what food they can gain access to and what is off limits.

A soil bacterium “may be only 10 microns away from a big chunk of organic matter that I’m sure they would love to degrade, but it’s on the other side of a cluster of minerals,” said Pett-Ridge. “It’s literally as if it’s on the other side of the planet.”

Another related, and poorly understood, ingredient in a new soil paradigm is the fate of carbon within the soil. Researchers now believe that almost all organic material that enters soil will get digested by microbes. “Now it’s really clear that soil organic matter is just this loose assemblage of plant matter in varying degrees of degradation,” said Sanderman. Some will then be respired into the atmosphere as carbon dioxide. What remains could be eaten by another microbe — and a third, and so on. Or it could bind to a bit of clay or get trapped inside a soil aggregate: a porous clump of particles that, from a microbe’s point of view, could be as large as a city and as impenetrable as a fortress. Studies of carbon isotopes have shown that a lot of carbon can stick around in soil for centuries or even longer. If humus isn’t doing the stabilizing, perhaps minerals and aggregates are.

Before soil science settles on a new theory, there will doubtless be more surprises. One may have been delivered recently by a group of researchers at Princeton University who constructed a simplified artificial soil using microfluidic devices — essentially, tiny plastic channels for moving around bits of fluid and cells. The researchers found that carbon they put inside an aggregate made of bits of clay was protected from bacteria. But when they added a digestive enzyme, the carbon was freed from the aggregate and quickly gobbled up. “To our surprise, no one had drawn this connection between enzymes, bacteria and trapped carbon,” said Howard Stone, an engineer who led the study.

Lehmann is pushing to replace the old dichotomy of stable and unstable carbon with a “soil continuum model” of carbon in progressive stages of decomposition. But this model and others like it are far from complete, and at this point, more conceptual than mathematically predictive.

Researchers agree that soil science is in the midst of a classic paradigm shift. What nobody knows is exactly where the field will land — what will be written in the next edition of the textbook. “We’re going through a conceptual revolution,” said Mark Bradford, a soil scientist at Yale University. “We haven’t really got a new cathedral yet. We have a whole bunch of churches that have popped up.”

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QED driven QAOA for network-flow optimization



Yuxuan Zhang1, Ruizhe Zhang2, and Andrew C. Potter1

1Center for Complex Quantum Systems, University of Texas at Austin, Austin, TX 78712, USA
2Department of Computer Science, University of Texas at Austin, Austin, TX 78712, USA

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We present a general framework for modifying quantum approximate optimization algorithms (QAOA) to solve constrained network flow problems. By exploiting an analogy between flow-constraints and Gauss’ law for electromagnetism, we design lattice quantum electrodynamics (QED)- inspired mixing Hamiltonians that preserve flow constraints throughout the QAOA process. This results in an exponential reduction in the size of the configuration space that needs to be explored, which we show through numerical simulations, yields higher quality approximate solutions compared to the original QAOA routine. We outline a specific implementation for edge-disjoint path (EDP) problems related to traffic congestion minimization, numerically analyze the effect of initial state choice, and explore trade-offs between circuit complexity and qubit resources via a particle-vortex duality mapping. Comparing the effect of initial states reveals that starting with an ergodic (unbiased) superposition of solutions yields better performance than beginning with the mixer ground-state, suggesting a departure from the “short-cut to adiabaticity” mechanism often used to motivate QAOA.

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[23] Zhihui Wang, Nicholas C Rubin, Jason M Dominy, and Eleanor G Rieffel. $xy$ mixers: Analytical and numerical results for the quantum alternating operator ansatz. Physical Review A, 101 (1): 012320, 2020. 10.1103/​PhysRevA.101.012320.

Cited by

[1] Bobak Toussi Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, and Seth Lloyd, “Quantum Earth Mover’s Distance: A New Approach to Learning Quantum Data”, arXiv:2101.03037.

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The ‘Weirdest’ Matter, Made of Partial Particles, Defies Description



Your desk is made up of individual, distinct atoms, but from far away its surface appears smooth. This simple idea is at the core of all our models of the physical world. We can describe what’s happening overall without getting bogged down in the complicated interactions between every atom and electron.

So when a new theoretical state of matter was discovered whose microscopic features stubbornly persist at all scales, many physicists refused to believe in its existence.

“When I first heard about fractons, I said there’s no way this could be true, because it completely defies my prejudice of how systems behave,” said Nathan Seiberg, a theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey. “But I was wrong. I realized I had been living in denial.”

The theoretical possibility of fractons surprised physicists in 2011. Recently, these strange states of matter have been leading physicists toward new theoretical frameworks that could help them tackle some of the grittiest problems in fundamental physics.

Fractons are quasiparticles — particle-like entities that emerge out of complicated interactions between many elementary particles inside a material. But fractons are bizarre even compared to other exotic quasiparticles, because they are totally immobile or able to move only in a limited way. There’s nothing in their environment that stops fractons from moving; rather it’s an inherent property of theirs. It means fractons’ microscopic structure influences their behavior over long distances.

“That’s totally shocking. For me it is the weirdest phase of matter,” said Xie Chen, a condensed matter theorist at the California Institute of Technology.

Partial Particles

In 2011, Jeongwan Haah, then a graduate student at Caltech, was searching for unusual phases of matter that were so stable they could be used to secure quantum memory, even at room temperature. Using a computer algorithm, he turned up a new theoretical phase that came to be called the Haah code. The phase quickly caught the attention of other physicists because of the strangely immovable quasiparticles that make it up.

They seemed, individually, like mere fractions of particles, only able to move in combination. Soon, more theoretical phases were found with similar characteristics, and so in 2015 Haah — along with Sagar Vijay and Liang Fucoined the term “fractons” for the strange partial quasiparticles. (An earlier but overlooked paper by Claudio Chamon is now credited with the original discovery of fracton behavior.)

To see what’s so exceptional about fracton phases, consider a more typical particle, such as an electron, moving freely through a material. The odd but customary way certain physicists understand this movement is that the electron moves because space is filled with electron-positron pairs momentarily popping into and out of existence. One such pair appears so that the positron (the electron’s oppositely charged antiparticle) is on top of the original electron, and they annihilate. This leaves behind the electron from the pair, displaced from the original electron. As there’s no way of distinguishing between the two electrons, all we perceive is a single electron moving.

Now instead imagine that pairs of particles and antiparticles can’t arise out of the vacuum but only squares of them. In this case, a square might arise so that one antiparticle lies on top of the original particle, annihilating that corner. A second square then pops out of the vacuum so that one of its sides annihilates with a side from the first square. This leaves behind the second square’s opposite side, also consisting of a particle and an antiparticle. The resultant movement is that of a particle-antiparticle pair moving sideways in a straight line. In this world — an example of a fracton phase — a single particle’s movement is restricted, but a pair can move easily.

The Haah code takes the phenomenon to the extreme: Particles can only move when new particles are summoned in never-ending repeating patterns called fractals. Say you have four particles arranged in a square, but when you zoom in to each corner you find another square of four particles that are close together. Zoom in on a corner again and you find another square, and so on. For such a structure to materialize in the vacuum requires so much energy that it’s impossible to move this type of fracton. This allows very stable qubits — the bits of quantum computing — to be stored in the system, as the environment can’t disrupt the qubits’ delicate state.

The immovability of fractons makes it very challenging to describe them as a smooth continuum from far away. Because particles can usually move freely, if you wait long enough they’ll jostle into a state of equilibrium, defined by bulk properties such as temperature or pressure. Particles’ initial locations cease to matter. But fractons are stuck at specific points or can only move in combination along certain lines or planes. Describing this motion requires keeping track of fractons’ distinct locations, and so the phases cannot shake off their microscopic character or submit to the usual continuum description.

Their resolute microscopic behavior makes it “a challenge to imagine examples of fractons and to think deeply about what is possible,” said Vijay, a theorist at the University of California, Santa Barbara. “Without a continuous description, how do we define these states of matter?”

“We’re missing a big chunk of things,” said Chen. “We have no idea how to describe them and what they mean.”

A New Fracton Framework

Fractons have yet to be made in the lab, but that will probably change. Certain crystals with immovable defects have been shown to be mathematically similar to fractons. And the theoretical fracton landscape has unfurled beyond what anyone anticipated, with new models popping up every month.

“Probably in the near future someone will take one of these proposals and say, ‘OK, let’s do some heroic experiment with cold atoms and exactly realize one of these fracton models,’” said Brian Skinner, a condensed matter physicist at Ohio State University who has devised fracton models.

Even without their experimental realization, the mere theoretical possibility of fractons rang alarm bells for Seiberg, a leading expert in quantum field theory, the theoretical framework in which almost all physical phenomena are currently described.

Quantum field theory depicts discrete particles as excitations in continuous fields that stretch across space and time. It’s the most successful physical theory ever discovered, and it encompasses the Standard Model of particle physics — the impressively accurate equation governing all known elementary particles.

“Fractons do not fit into this framework. So my take is that the framework is incomplete,” said Seiberg.

There are other good reasons for thinking that quantum field theory is incomplete — for one thing, it so far fails to account for the force of gravity. If they can figure out how to describe fractons in the quantum field theory framework, Seiberg and other theorists foresee new clues toward a viable quantum gravity theory.

“Fractons’ discreteness is potentially dangerous, as it can ruin the whole structure that we already have,” said Seiberg. “But either you say it’s a problem, or you say it’s an opportunity.”

He and his colleagues are developing novel quantum field theories that try to encompass the weirdness of fractons by allowing some discrete behavior on top of a bedrock of continuous space-time.

“Quantum field theory is a very delicate structure, so we would like to change the rules as little as possible,” he said. “We are walking on very thin ice, hoping to get to the other side.”

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Resource-Optimized Fermionic Local-Hamiltonian Simulation on a Quantum Computer for Quantum Chemistry



Qingfeng Wang1, Ming Li2, Christopher Monroe2,3, and Yunseong Nam2,4

1Chemical Physics Program and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
2IonQ, College Park, MD 20740, USA
3Joint Quantum Institute, Department of Physics, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA
4Department of Physics, University of Maryland, College Park, MD 20742, USA

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The ability to simulate a fermionic system on a quantum computer is expected to revolutionize chemical engineering, materials design, nuclear physics, to name a few. Thus, optimizing the simulation circuits is of significance in harnessing the power of quantum computers. Here, we address this problem in two aspects. In the fault-tolerant regime, we optimize the $R_z$ and $textit{T gate}$ counts along with the ancilla qubit counts required, assuming the use of a product-formula algorithm for implementation. We obtain a savings ratio of two in the gate counts and a savings ratio of eleven in the number of ancilla qubits required over the state of the art. In the pre-fault tolerant regime, we optimize the two-qubit gate counts, assuming the use of the variational quantum eigensolver (VQE) approach. Specific to the latter, we present a framework that enables bootstrapping the VQE progression towards the convergence of the ground-state energy of the fermionic system. This framework, based on perturbation theory, is capable of improving the energy estimate at each cycle of the VQE progression, by about a factor of three closer to the known ground-state energy compared to the standard VQE approach in the test-bed, classically-accessible system of the water molecule. The improved energy estimate in turn results in a commensurate level of savings of quantum resources, such as the number of qubits and quantum gates, required to be within a pre-specified tolerance from the known ground-state energy. We also explore a suite of generalized transformations of fermion to qubit operators and show that resource-requirement savings of up to more than $20%$, in small instances, is possible.

► BibTeX data

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Cited by

[1] Christopher David White, ChunJun Cao, and Brian Swingle, “Conformal field theories are magical”, Physical Review B 103 7, 075145 (2021).

[2] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake, “Fermionic partial tomography via classical shadows”, arXiv:2010.16094.

[3] Yordan S. Yordanov, David R. M. Arvidsson-Shukur, and Crispin H. W. Barnes, “Efficient quantum circuits for quantum computational chemistry”, Physical Review A 102 6, 062612 (2020).

[4] Saad Yalouz, Bruno Senjean, Jakob Günther, Francesco Buda, Thomas E. O’Brien, and Lucas Visscher, “A state-averaged orbital-optimized hybrid quantum-classical algorithm for a democratic description of ground and excited states”, Quantum Science and Technology 6 2, 024004 (2021).

[5] Brian C. Sawyer and Kenton R. Brown, “Wavelength-insensitive, multispecies entangling gate for group-2 atomic ions”, Physical Review A 103 2, 022427 (2021).

[6] Ning Bao, ChunJun Cao, and Vincent Paul Su, “Magic State Distillation from Entangled States”, arXiv:2106.12591.

[7] Nikodem Grzesiak, Andrii Maksymov, Pradeep Niroula, and Yunseong Nam, “Efficient quantum programming using EASE gates on a trapped-ion quantum computer”, arXiv:2107.07591.

The above citations are from SAO/NASA ADS (last updated successfully 2021-07-26 13:07:04). 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 2021-07-26 13:07:02: Could not fetch cited-by data for 10.22331/q-2021-07-26-509 from Crossref. This is normal if the DOI was registered recently.

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