This extreme fragility might make quantum computing sound hopeless. But in 1995, the applied mathematician Peter Shor discovered a clever way to store quantum information. His encoding had two key properties. First, it could tolerate errors that only affected individual qubits. Second, it came with a procedure for correcting errors as they occurred, preventing them from piling up and derailing a computation. Shor’s discovery was the first example of a quantum error-correcting code, and its two key properties are the defining features of all such codes.
The first property stems from a simple principle: Secret information is less vulnerable when it’s divided up. Spy networks employ a similar strategy. Each spy knows very little about the network as a whole, so the organization remains safe even if any individual is captured. But quantum error-correcting codes take this logic to the extreme. In a quantum spy network, no single spy would know anything at all, yet together they’d know a lot.
Each quantum error-correcting code is a specific recipe for distributing quantum information across many qubits in a collective superposition state. This procedure effectively transforms a cluster of physical qubits into a single virtual qubit. Repeat the process many times with a large array of qubits, and you’ll get many virtual qubits that you can use to perform computations.
The physical qubits that make up each virtual qubit are like those oblivious quantum spies. Measure any one of them and you’ll learn nothing about the state of the virtual qubit it’s a part of—a property called local indistinguishability. Since each physical qubit encodes no information, errors in single qubits won’t ruin a computation. The information that matters is somehow everywhere, yet nowhere in particular.
“You can’t pin it down to any individual qubit,” Cubitt said.
All quantum error-correcting codes can absorb at least one error without any effect on the encoded information, but they will all eventually succumb as errors accumulate. That’s where the second property of quantum error-correcting codes kicks in—the actual error correction. This is closely related to local indistinguishability: Because errors in individual qubits don’t destroy any information, it’s always possible to reverse any error using established procedures specific to each code.
Taken for a Ride
Zhi Li, a postdoc at the Perimeter Institute for Theoretical Physics in Waterloo, Canada, was well versed in the theory of quantum error correction. But the subject was far from his mind when he struck up a conversation with his colleague Latham Boyle. It was the fall of 2022, and the two physicists were on an evening shuttle from Waterloo to Toronto. Boyle, an expert in aperiodic tilings who lived in Toronto at the time and is now at the University of Edinburgh, was a familiar face on those shuttle rides, which often got stuck in heavy traffic.
“Normally they could be very miserable,” Boyle said. “This was like the greatest one of all time.”
Before that fateful evening, Li and Boyle knew of each other’s work, but their research areas didn’t directly overlap, and they’d never had a one-on-one conversation. But like countless researchers in unrelated fields, Li was curious about aperiodic tilings. “It’s very hard to be not interested,” he said.
But there may be opportunities to indirectly spot the signatures of those gravitons.
One strategy Vafa and his collaborators are pursuing draws on large-scale cosmological surveys that chart the distribution of galaxies and matter. In those distributions, there might be “small differences in clustering behavior,” Obied said, that would signal the presence of dark gravitons.
When heavier dark gravitons decay, they produce a pair of lighter dark gravitons with a combined mass that is slightly less than that of their parent particle. The missing mass is converted to kinetic energy (in keeping with Einstein’s formula, E = mc2), which gives the newly created gravitons a bit of a boost—a “kick velocity” that’s estimated to be about one-ten-thousandth of the speed of light.
These kick velocities, in turn, could affect how galaxies form. According to the standard cosmological model, galaxies start with a clump of matter whose gravitational pull attracts more matter. But gravitons with a sufficient kick velocity can escape this gravitational grip. If they do, the resulting galaxy will be slightly less massive than the standard cosmological model predicts. Astronomers can look for this difference.
Recent observations of cosmic structure from the Kilo-Degree Survey are so far consistent with the dark dimension: An analysis of data from that survey placed an upper bound on the kick velocity that was very close to the value predicted by Obied and his coauthors. A more stringent test will come from the Euclid space telescope, which launched last July.
Meanwhile, physicists are also planning to test the dark dimension idea in the laboratory. If gravity is leaking into a dark dimension that measures 1 micron across, one could, in principle, look for any deviations from the expected gravitational force between two objects separated by that same distance. It’s not an easy experiment to carry out, said Armin Shayeghi, a physicist at the Austrian Academy of Sciences who is conducting the test. But “there’s a simple reason for why we have to do this experiment,” he added: We won’t know how gravity behaves at such close distances until we look.
The closest measurement to date—carried out in 2020 at the University of Washington—involved a 52-micron separation between two test bodies. The Austrian group is hoping to eventually attain the 1-micron range predicted for the dark dimension.
While physicists find the dark dimension proposal intriguing, some are skeptical that it will work out. “Searching for extra dimensions through more precise experiments is a very interesting thing to do,” said Juan Maldacena, a physicist at the Institute for Advanced Study, “though I think that the probability of finding them is low.”
Joseph Conlon, a physicist at Oxford, shares that skepticism: “There are many ideas that would be important if true, but are probably not. This is one of them. The conjectures it is based on are somewhat ambitious, and I think the current evidence for them is rather weak.”
Of course, the weight of evidence can change, which is why we do experiments in the first place. The dark dimension proposal, if supported by upcoming tests, has the potential to bring us closer to understanding what dark matter is, how it is linked to both dark energy and gravity, and why gravity appears feeble compared to the other known forces. “Theorists are always trying to do this ‘tying together.’ The dark dimension is one of the most promising ideas I have heard in this direction,” Gopakumar said.
But in an ironic twist, the one thing the dark dimension hypothesis cannot explain is why the cosmological constant is so staggeringly small—a puzzling fact that essentially initiated this whole line of inquiry. “It’s true that this program does not explain that fact,” Vafa admitted. “But what we can say, drawing from this scenario, is that if lambda is small—and you spell out the consequences of that—a whole set of amazing things could fall into place.”
Original storyreprinted with permission from Quanta Magazine, an editorially independent publication of theSimons Foundationwhose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
“We know from direct imaging searches of young stars that very few stars have giant planets in [wide] orbits,” Bate said. “It is difficult to accept that there were many large planetary systems in Orion to disrupt.”
Rogue Objects Abound
At this point, many researchers suspect there’s more than one way to make these strange in-between objects. For instance, with some fiddling, theorists might find that supernova shock waves can compress smaller gas clouds and help them to collapse into pairs of tiny stars more readily than expected. And Wang’s simulations have shown that booting giant planets in pairs is, at least in some cases, theoretically unavoidable.
While many questions remain, the multitude of free-floating worlds discovered in the past two years has taught researchers two things. First, they form quickly—over millions of years, rather than billions. In Orion, gas clouds have collapsed and planets have formed, and some, perhaps, have even been dragged into the abyss by passing stars, all during the time in which modern humans were evolving on Earth.
Sean Raymond developed simulations that show how large planets can punt their siblings into space, thus providing one potential explanation for the free-floating worlds.
Photograph: Laurence Honnorat
“Forming a planet in 1 million years is hard with current models,” van der Marel said. “This [discovery] would add another piece to that puzzle.”
Second, there are a ton of untethered worlds out there. And the heavy gas giants are the hardest to evict from their systems, much as a bowling ball would be the hardest object to knock off a billiard table. This observation suggests that for every Jupiter spotted, numerous free-floating Neptunes and Earths are going unnoticed.
We likely live in a galaxy teeming with banished worlds of all sizes.
Now, nearly half a millennium after Galileo marveled at the myriad pinpricks of light—moons, planets, and stars—in Earth’s skies, his successors are getting acquainted with the brightest tip of the iceberg of darker objects adrift between them. The tiny stars, the starless worlds, invisible asteroids, alien comets, and more.
“We know there’s a whole bunch of crap between stars,” Raymond said. This kind of research is “opening a window on all of that, not just free-floating planets but free-floating stuff in general.”
Original storyreprinted with permission from Quanta Magazine, an editorially independent publication of theSimons Foundationwhose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
The original version ofthis storyappeared in Quanta Magazine.
In 1917, the Japanese mathematician Sōichi Kakeya posed what at first seemed like nothing more than a fun exercise in geometry. Lay an infinitely thin, inch-long needle on a flat surface, then rotate it so that it points in every direction in turn. What’s the smallest area the needle can sweep out?
If you simply spin it around its center, you’ll get a circle. But it’s possible to move the needle in inventive ways, so that you carve out a much smaller amount of space. Mathematicians have since posed a related version of this question, called the Kakeya conjecture. In their attempts to solve it, they have uncovered surprising connections to harmonic analysis, number theory, and even physics.
“Somehow, this geometry of lines pointing in many different directions is ubiquitous in a large portion of mathematics,” said Jonathan Hickman of the University of Edinburgh.
But it’s also something that mathematicians still don’t fully understand. In the past few years, they’ve proved variations of the Kakeya conjecture in easier settings, but the question remains unsolved in normal, three-dimensional space. For some time, it seemed as if all progress had stalled on that version of the conjecture, even though it has numerous mathematical consequences.
Now, two mathematicians have moved the needle, so to speak. Their new proof strikes down a major obstacle that has stood for decades—rekindling hope that a solution might finally be in sight.
What’s the Small Deal?
Kakeya was interested in sets in the plane that contain a line segment of length 1 in every direction. There are many examples of such sets, the simplest being a disk with a diameter of 1. Kakeya wanted to know what the smallest such set would look like.
He proposed a triangle with slightly caved-in sides, called a deltoid, which has half the area of the disk. It turned out, however, that it’s possible to do much, much better.
The deltoid to the right is half the size of the circle, though both needles rotate through every direction.Video: Merrill Sherman/Quanta Magazine
In 1919, just a couple of years after Kakeya posed his problem, the Russian mathematician Abram Besicovitch showed that if you arrange your needles in a very particular way, you can construct a thorny-looking set that has an arbitrarily small area. (Due to World War I and the Russian Revolution, his result wouldn’t reach the rest of the mathematical world for a number of years.)
To see how this might work, take a triangle and split it along its base into thinner triangular pieces. Then slide those pieces around so that they overlap as much as possible but protrude in slightly different directions. By repeating the process over and over again—subdividing your triangle into thinner and thinner fragments and carefully rearranging them in space—you can make your set as small as you want. In the infinite limit, you can obtain a set that mathematically has no area but can still, paradoxically, accommodate a needle pointing in any direction.
“That’s kind of surprising and counterintuitive,” said Ruixiang Zhang of the University of California, Berkeley. “It’s a set that’s very pathological.”
Seeking his counsel, Arjuna asks Krishna to reveal his universal form. Krishna obliges, and in verse 12 of the Gita he manifests as a sublime, terrifying being of many mouths and eyes. It is this moment that entered Oppenheimer’s mind in July 1945. “If the radiance of a thousand suns were to burst at once into the sky, that would be like the splendor of the mighty one,” was Oppenheimer’s translation of that moment in the desert of New Mexico.
In Hinduism, which has a non-linear concept of time, the great god is involved in not only the creation, but also the dissolution. In verse 32, Krishna says the famous line. In it “death” literally translates as “world-destroying time,” says Thompson, adding that Oppenheimer’s Sanskrit teacher chose to translate “world-destroying time” as “death,” a common interpretation. Its meaning is simple: Irrespective of what Arjuna does, everything is in the hands of the divine.
“Arjuna is a soldier, he has a duty to fight. Krishna, not Arjuna, will determine who lives and who dies and Arjuna should neither mourn nor rejoice over what fate has in store, but should be sublimely unattached to such results,” says Thompson. “And ultimately the most important thing is he should be devoted to Krishna. His faith will save Arjuna’s soul.” But Oppenheimer, seemingly, was never able to achieve this peace. “In some sort of crude sense which no vulgarity, no humor, no overstatements can quite extinguish,” he said, two years after the Trinity explosion, “the physicists have known sin; and this is a knowledge which they cannot lose.”
“He doesn’t seem to believe that the soul is eternal, whereas Arjuna does,” says Thompson. “The fourth argument in the Gita is really that death is an illusion, that we’re not born and we don’t die. That’s the philosophy, really. That there’s only one consciousness and that the whole of creation is a wonderful play.” Oppenheimer, perhaps, never believed that the people killed in Hiroshima and Nagasaki would not suffer. While he carried out his work dutifully, he could never accept that this could liberate him from the cycle of life and death. In stark contrast, Arjuna realizes his error and decides to join the battle.
“Krishna is saying you have to simply do your duty as a warrior,” says Thompson. “If you were a priest you wouldn’t have to do this, but you are a warrior and you have to perform it. In the larger scheme of things, presumably, the bomb represented the path of the battle against the forces of evil, which were epitomized by the forces of fascism.”
For Arjuna, it may have been comparatively easy to be indifferent to war because he believed the souls of his opponents would live on regardless. But Oppenheimer felt the consequences of the atomic bomb acutely. “He hadn’t got that confidence that the destruction, ultimately, was an illusion,” says Thompson. Oppenheimer’s apparent inability to accept the idea of an immortal soul would always weigh heavy on his mind.
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