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Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

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Benjamin Zanger1, Christian B. Mendl1,2, Martin Schulz1,3, and Martin Schreiber1

1Technical University of Munich, Department of Informatics, Boltzmannstraße 3, 85748 Garching, Germany
2TUM Institute for Advanced Study, Lichtenbergstraße 2a, 85748 Garching, Germany
3Leibniz Supercomputing Centre, Boltzmannstraße 1, 85748 Garching, Germany

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Abstract

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6$^{mathrm{th}}$ order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an “oracle” within quantum search algorithms for inverse problems.

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Cited by

[1] Martin Knudsen and Christian B. Mendl, “Solving Differential Equations via Continuous-Variable Quantum Computers”, arXiv:2012.12220.

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Quantum

Quantum walk-based portfolio optimisation

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N. Slate, E. Matwiejew, S. Marsh, and J. B. Wang

Department of Physics, The University of Western Australia, Perth WA 6009, Australia

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Abstract

This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at near-term noisy intermediate-scale quantum computers. Recent work by Hodson et al. (2019) explored potential application of hybrid quantum-classical algorithms to the problem of financial portfolio rebalancing. In particular, they deal with the portfolio optimisation problem using the Quantum Approximate Optimisation Algorithm and the Quantum Alternating Operator Ansatz. In this paper, we demonstrate substantially better performance using a newly developed Quantum Walk Optimisation Algorithm in finding high-quality solutions to the portfolio optimisation problem.

► BibTeX data

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Cited by

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Quantum

An efficient quantum algorithm for the time evolution of parameterized circuits

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Stefano Barison, Filippo Vicentini, and Giuseppe Carleo

Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

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Abstract

We introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. The method, named “projected – Variational Quantum Dynamics” (p-VQD) realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this is equivalent to the McLachlan’s variational principle. Our approach is efficient in the sense that it exhibits an optimal linear scaling with the total number of variational parameters. Furthermore, it is global in the sense that it uses the variational principle to optimize all parameters at once. The global nature of our approach then significantly extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle that, due to a demanding quadratic scaling with parameter numbers, are unsuitable for large parameterized quantum circuits.

► BibTeX data

► References

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[3] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, “Experimental Quantum Learning of a Spectral Decomposition”, arXiv:2104.03295.

[4] Kian Hwee Lim, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, “Fast-Forwarding with NISQ Processors without Feedback Loop”, arXiv:2104.01931.

[5] Tobias Haug and M. S. Kim, “Optimal training of variational quantum algorithms without barren plateaus”, arXiv:2104.14543.

[6] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, “Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information”, arXiv:2103.09232.

[7] Yongdan Yang, Bing-Nan Lu, and Ying Li, “Accelerated quantum Monte Carlo with mitigated error on noisy quantum computer”, arXiv:2106.09880.

[8] Paolo P. Mazza, Dominik Zietlow, Federico Carollo, Sabine Andergassen, Georg Martius, and Igor Lesanovsky, “Machine learning time-local generators of open quantum dynamics”, Physical Review Research 3 2, 023084 (2021).

[9] Refik Mansuroglu, Samuel Wilkinson, Ludwig Nützel, and Michael J. Hartmann, “Classical Variational Optimization of Gate Sequences for Time Evolution of Large Quantum Systems”, arXiv:2106.03680.

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[12] Rouven Koch and Jose L. Lado, “Neural network enhanced hybrid quantum many-body dynamical distributions”, arXiv:2105.03129.

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Quantum

Causal and compositional structure of unitary transformations

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Robin Lorenz1,2 and Jonathan Barrett1

1Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
2Cambridge Quantum Computing Ltd, 17 Beaumont Street, Oxford OX1 2NA, UK

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Abstract

The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system $A$ to output system $B$, system $A$ cannot influence system $B$. Conversely, given a unitary $U$ with a no-influence relation from input $A$ to output $B$, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of $U$ with no path from $A$ to $B$. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident $textit{simultaneously}$. To address this, we introduce a new formalism of `extended circuit diagrams’, which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A $textit{causally faithful}$ extended circuit decomposition, representing a unitary $U$, is then one for which there is a path from an input $A$ to an output $B$ if and only if there actually is influence from $A$ to $B$ in $U$. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary’s respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

Whenever one is able to draw an intuitive picture that succinctly captures the essential features of a complicated whole, then not only does it usually help communication, arguably, this also is a sign of having achieved good conceptual understanding. Also for quantum theory there is a rich history of developing and employing diagrammatic reasoning, where quantum circuit diagrams capture aspects of how quantum systems interact with each other and evolve over time. Circuit diagrams are the basic ingredient to one of the main paradigms of quantum computation, and have also proven useful in research ranging from the foundations of quantum theory to the design of algorithms and efficient quantum compilers. One particular area of research of foundational, as well as applied importance is causality. Unitary transformations that play a fundamental role in the quantum formalism, describe the evolution of a set of systems and have a clear notion of causal structure: which of the systems can causally influence which other systems.

Combining all these lines of thought, this paper asks: can one understand the causal structure of a unitary transformation in compositional terms, i.e. through an intuitive diagram, where its components are connected up in such a way that lays bare where and how causal influence goes? We first show that this is generally not possible with ordinary circuit diagrams. We then derive new kinds of decompositions for many causal structures, as well as introduce a new graphical language of ‘extended circuit diagrams’ to visualise them. This novel perspective to study causal structure and its ramifications is of conceptual significance, as well as likely to facilitate progress with other open problems, which it in fact already has in the study of indefinite causal order, a much debated area of quantum physics.

► BibTeX data

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Cited by

[1] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, “Quantum Causal Models”, arXiv:1906.10726.

[2] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, “Cyclic quantum causal models”, Nature Communications 12, 885 (2021).

[3] Wataru Yokojima, Marco Túlio Quintino, Akihito Soeda, and Mio Murao, “Consequences of preserving reversibility in quantum superchannels”, arXiv:2003.05682.

[4] Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett, “Routed quantum circuits”, arXiv:2011.08120.

[5] Paolo Perinotti, “Causal influence in operational probabilistic theories”, arXiv:2012.15213.

The above citations are from SAO/NASA ADS (last updated successfully 2021-07-28 09:46:13). 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 2021-07-28 09:46:11: Could not fetch cited-by data for 10.22331/q-2021-07-28-511 from Crossref. This is normal if the DOI was registered recently.

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Quantum

A Soil-Science Revolution Upends Plans to Fight Climate Change

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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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Source: https://www.quantamagazine.org/a-soil-science-revolution-upends-plans-to-fight-climate-change-20210727/

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