a The striking new evidence in quantum computational complexity can be best understood with a playful thought experiment. Take a bath, then put a bunch of floating magnets in the water. Each magnet will flip back and forth trying to align its orientation with that of its neighbors. This will push and pull other magnets and in turn will be pushed and pulled. Now try to answer this: What will be the final arrangement of the system?
This problem and others like it, it turns out, are impossibly complex. With more than a few hundred magnets, computer simulations would take a long time to respond.
Now make those magnets quantum – individual atoms are subject to the Byzantine rules of the quantum world. As you can guess, the problem becomes even more difficult. “The conversation gets more complicated,” said Henry Yuen of Columbia University. “A more complex constraint occurs when two neighboring ‘quantum magnets’ are happy.”
These simple-seeming systems have provided extraordinary insight into the limits of computation in both the classical and quantum versions. In the case of classical or non-quantum systems, a milestone theorem from computer science takes us further. Called the PCP theorem (for “probably verifiable proof”), it states that not only is it incredibly difficult to calculate the final state of magnets (or aspects related to it), but there are many steps leading to it. . The complexity of the situation is even more stark, in other words, with the final situation surrounded by a realm of mystery.
Another version of the PCP theorem, which has not yet been proven, deals specifically with the quantum matter. Computer scientists suspect that the quantum PCP conjecture is correct, and proving it will change our understanding of the complexity of quantum problems. It is considered to be arguably the most important open problem in quantum computational complexity theory. But so far it is out of reach.
Nine years ago, two researchers identified an intermediate goal to help us get there. they came together a simple idea, which is known as the “no low-energy trivial state” (NLTS) conjecture, which would have to be true if the quantum PCP conjecture was true. Proving it won’t make it any easier to prove the quantum PCP conjecture, but it will solve some of its most interesting questions.
Then in June of 2022, in a paper posted on the scientific preprint site arxiv.org, three computer scientists NLTS conjecture proved, The result has important implications for computer science and quantum physics.
“It’s very exciting,” said dorit aharonov of the Hebrew University of Jerusalem. “It would encourage people to look at the difficult problem of quantum PCP estimation.”
To understand the new result, start by drawing a quantum system such as a set of atoms. Each atom has a property called spin, which is somewhat similar to the alignment of a magnet in that it points along an axis. But unlike the alignment of a magnet, an atom’s spin can be in a position that mixes different directions together, a phenomenon known as superposition. Furthermore, it may be impossible to describe the spin of an atom without taking into account the orbits of other atoms from distant regions. When this happens, those interconnected atoms are said to be in a state of quantum entanglement. The entanglement is remarkable, but also fragile and easily disrupted by thermal interactions. The more heat in a system, the more difficult it is to entangle it.
Now imagine that a group of atoms is cooled until they reach absolute zero. As the system cools and the entanglement patterns become more stable, its energy decreases. The minimum possible energy, or “ground energy”, provides a brief description of the complex final state of the entire system. Or at least it would be, if it could be calculated.
In the late 1990s, researchers found that for some systems, this ground energy could not be calculated in any reasonable time frame.
However, physicists thought that an energy level close to the ground energy (but not quite) should be easy to calculate, as the system would be hotter and less entangled, and therefore simpler.
Computer scientists disagreed. According to the classical PCP theorem, computing the energy close to the final state is as difficult as the final energy. And so the quantum version of the PCP theorem, if true, would say that computing the precursor energy to the ground energy would be as difficult as computing the ground energy. Since the classical PCP theorem is true, many researchers think that the quantum version must be true as well. “Certainly, a quantum version must be true,” Yuen said.
The physical implications of such a theorem would be deep. This would mean that there are quantum systems that maintain their entanglement at high temperatures – completely contrary to what physicists expected. But no one could prove that such a system existed.
In 2013, Michael Friedman and Matthew Hastings, both working at Microsoft Research’s Station Q in Santa Barbara, California, downplayed the problem. They decided to look for systems whose lowest and almost lowest energies are difficult to calculate according to just one metric: how much circuitry the computer would need to simulate them. These quantum systems, if they could find them, would have to maintain rich patterns of entanglement in all their lowest energies. The existence of such systems would not prove the quantum PCP conjecture – there may be other stiffness metrics to consider – but it would be counted as progress.
Computer scientists didn’t know of any such systems, but they knew where to go: in a field of study called quantum error correction, where researchers create entanglement recipes that protect atoms from perturbations. are designed. Each recipe is known as a cod, and there are several codes of both major and minor statures.
At the end of 2021, computer scientists achieved a great success In creating quantum error-correcting codes of an essentially ideal nature. In the coming months, several other groups of researchers built on those results to produce different versions.
The paper’s three authors, who had been collaborating on related projects over the past two years, came together to prove that one of the new codes had all the properties needed to build a quantum system similar to the hypothesis of Friedman and Hastings. . In doing so, he proved the NLTS conjecture.
Their result shows that entanglement is not necessarily as sensitive and sensitive to temperature as the physicists had thought. And it supports the quantum PCP conjecture, suggesting that even far from the ground energy, it can be nearly impossible to calculate the energy of a quantum system.
“It tells us that what was not likely to be true is true,” said Isaac Kim University of California, Davis K. “Although in some very strange system.”
The researchers believe that a variety of technical tools will be needed to prove the full quantum PCP conjecture. However, they see being optimistic that the current results will bring them closer.
They are perhaps most concerned with whether the newly discovered NLTS quantum systems—though possible in principle—can actually be created in nature, and what they would look like. According to the current result, they would require complex patterns of long-range entanglement that have never been generated in the laboratory, and which can be created using only astronomical numbers of atoms.
“These are highly engineered objects,” said Chinamay Nirkhewith a computer scientist at the University of California, Berkeley, and a co-author of the paper Anshu of Harvard University and Nicholas Brookman of University College London.
“If you have the ability to pair really far and wide, I believe you can feel the system,” Anshu said. “But there’s actually another journey to go into the low-energy spectrum.” Added Brookman, “There’s probably some part of the universe that is NLTS. I don’t know.”
Lead Image: Christina Armitage for Quanta Magazine.