MATTHIAS TROYER is a man at the top of his field. A specialist in quantum computing, he is helping to build the next generation of hardware that will replace the computers we use today. But over lunch with one of his colleagues earlier this year, a thought began to nag at him: could one of his machines one day replace him?
Computers aren’t putting scientists out of a job just yet. But within the past year, they have shown remarkable promise at resolving one of the thorniest problems facing physicists who study the way matter behaves at the quantum mechanical level.
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That problem stems from electrons, those carriers of negative charge that orbit the atomic nucleus. Their behaviour dictates many of the properties of matter, from how readily it conducts electricity and heat to its flexibility and ability to reflect light. They are also the basis for computing: last century’s breakthroughs paved the way for the computer revolution, allowing us to design circuits that could manipulate electrons carefully enough to perform calculations.
The behaviour of individual electrons is small beer, however. If we could understand how they behave in groups, we could identify and construct new forms of matter at will, from ultra-efficient batteries to wonder-drug medical treatments. And while the equations for a group of particles are exponentially more complicated than for any one on its own, well-known shortcuts can simplify the maths.
But electrons have a dirty secret. Once they gang up, negative signs start appearing in the calculations, making them impossible to solve. And “impossible” isn’t just a convenient shorthand. Even the world’s most powerful supercomputers are fundamentally incapable of producing a universal solution – leading to an impenetrable mathematical wall around whole areas of physics.
Now, advances in machine learning are allowing computers to see through this wall without actually doing any of the sums. If these techniques bear fruit, the age of the artificial physicist could be upon us.
Subatomic particles, like people, are easier to handle as individuals than in groups. The behaviour of any single particle, isolated from outside influence, can readily be worked out, even without a computer. But this tractability begins to disappear as soon as quantum particles start interacting with each other. Any complete mathematical description needs to encode every possible state that the system could occupy, causing the complexity of the description to rapidly mushroom.
Even for systems involving just tens of entangled particles, the calculations necessary to precisely predict their behaviour are too much for today’s computers. Doing the maths is like searching for a needle in a near-infinite haystack: there just isn’t enough time to grind out an exact solution, no matter how large your processor.
This intractability around entangled particles, known as the quantum many-body problem, is frustrating. It puts a computational roadblock in front of what physicists can know with certainty about matter at its most fundamental level. Of course, that doesn’t stop us stumbling across remarkable phenomena in the lab. Superconductivity was discovered in 1911 “essentially by accident”, says Andrew Millis, a physicist at Columbia University, New York.
“There is an impenetrable mathematical wall around whole areas of physics”
But we are still far from being able to predict whether a given arrangement of atoms will, say, produce a superconductor or a battery. “It would be nice to have a theoretical structure that lets you know where to look,” says Millis. And an accurate, efficiently computable solution to the quantum many-body problem would be a big help.
The trouble is that few, if any, physicists believe such a solution exists. Many, like Troyer at the Swiss Federal Institute of Technology in Zurich, are banking on the development of an entirely new breed of computers that can overcome the limitations of today’s technology. Unlike classical computers, which represent information in “bits” that can only have a value of 1 or 0, quantum computers use “qubits” capable of representing 1 and 0 at the same time. Theoretically, at least, this ability to encode two pieces of information simultaneously could expand their computing power by enough to bring the quantum many-body problem under control. For starters, a quantum computer would be able to use its qubits to store a complete description of a many-body system, precisely mirroring its complexities.
In the meantime, our best hope is to look for shortcuts. Imagine trying to calculate the odds of obtaining a full house in a game of poker. One way of tackling the problem is to work through the tedious probabilistic calculations. A less mathematical way is to simply deal a couple of thousand hands and keep track of how many full houses turn up. This kind of statistical technique, first used in the 1940s to work out the behaviour of neutrons, was given the name Monte Carlo after the celebrated Monaco casino.
By extension, Quantum Monte Carlo (or QMC) samples a selection of states that the system is most likely to occupy and uses them to approximate a solution. These approximations have allowed physicists to investigate quantum systems with billions of interacting particles – although these are still far too small to be seen with the naked eye. That is still far better than we could ever get with exact calculation. “Quantum Monte Carlo is the most successful numerical strategy we have,” says Roger Melko, a physicist at the University of Waterloo in Canada. “Nothing can touch it.”
By far the most interesting scenario to tackle with QMC is one that involves large quantities of electrons. But for all their potential, electrons are also a mathematical pain in the rear: they are notorious for introducing negative signs into the Monte Carlo equations. The result? Instead of the super-accurate estimates the technique is renowned for, it spits out mathematical nonsense such as negative probabilities.
This “sign problem” is more than merely inconvenient. It makes approximating the properties of many-electron systems just as intractable as solving the maths exactly. Which means the impenetrable wall of numbers rears up again, shutting down the science. “If you see a sign problem, most people just jump and say ‘Oh, this is too hard, let’s go’,” says Shailesh Chandrasekharan, who studies the mathematics of many-electron systems at Duke University in North Carolina.
Quantum computers would vault right over the sign problem – their theorised ability to “store” all of the inherent mathematical complexity of many-body systems in their qubits would make QMC’s approximations unnecessary. But some physicists, like Chandrasekharan, see the sign problem as a challenge to their mathematical ability. He has pioneered techniques that involve smothering negative terms in the Monte Carlo equations underneath enough positive ones to conceal their very existence, a technique capable of overcoming the sign problem for certain specific configurations of electrons.
Melko, meanwhile, is happy to ignore the sign problem’s undoable maths altogether. Like many physicists, he is tired of waiting around for quantum computation or pioneering mathematics to save the day. “My research studies things that I can solve, on today’s computers, with the algorithms that we can build for them,” he says. Faced with the sign problem, Melko wanted an algorithm that could skip the maths and get straight to the science.
In 2016, he had no idea what such an algorithm might look like. Then he heard news that Google’s machine-learning software had beaten one of the world’s best human players at Go. This ancient board game had long been considered too mathematically complex for computers to ever play competently, much less competitively. With a total number of possible games far greater than the number of atoms in the universe, Google’s AlphaGo software must have found a way of skipping past the brute-force calculations.
What gave AlphaGo this ability was a technique known as deep learning, which has revolutionised what computers can do. This involves training a computer to recognise patterns by passing them through multiple layers in a digital network, each of which is coded to search for specific features. These neural networks, inspired by the structure of neurons in our own brains, need to be fed large numbers of pre-labelled examples in order to pick out patterns, but can display remarkable speed at detecting similar patterns in brand new data. From translating speech into text on the fly to ensuring consistency on a production line, such networks are playing an increasingly important role in the world around us.
“The AI had X-ray vision: it could see through the maths to the physics”
It’s this ability to extract patterns that first got Melko’s pulse racing. “We basically started adopting these industry standard machine-learning algorithms, and just throwing our problems at them,” he says. As far as he was concerned, exactly how the neural network learned to distinguish between different electron configurations was irrelevant, so long as it did the job.
In the same way a computer might learn how to tell a cat from a dog by studying pixels in a photograph, the network learned how to tell a metal from an insulator by reading the raw mathematics of the quantum system.
With their neural network fattened up on a sign problem-free diet, Melko and his collaborators decided to move on to a real test. They fished out a quantum system whose sign problem had only recently been cracked, providing them with an answer they could compare with the machine’s. The two matched. The network seemed to have mathematical X-ray vision: it could “see through” the sign problem to glimpse the physical phenomena that Melko was actually interested in. “That’s where we were truly surprised,” he says.
But the sign problem is just the tip of the iceberg. If a computer can classify configurations of electrons simply by looking at them, then it should be able to do it for any large number of particles. That means physicists might be able to create many-body quantum models even more powerful than their current statistical approximations. Which means we might not have to wait for quantum computers to arrive in order to start simulating wonder materials from the atoms up. “I’m basically in a race now with the hardware people,” Melko says. “Maybe machine learning can keep me ahead of the race for another few years.”
Even as far as the hardware people are concerned, Melko’s results have been impressive. “I was sceptical,” says Troyer, “but it worked amazingly well.”
If we could get neural networks to somehow bypass the undoable maths of the quantum world, then would anything be left for human physicists? After all, isn’t the computer’s insight into the systems it studies somehow analogous to the machine “knowing” – or even “discovering” – something fundamental about physics without us?
Further reflection – at another lunchtime discussion with colleagues, of course – proved Troyer’s self-doubt to be fleeting. “The machine may be better at coming up with the way to store the quantum wave functions,” he says, “but the scientist still has to use that tool to solve some problem. If a machine becomes conscious, then it would be doing physics.” Scientists’ jobs are safe. It seems that someone, in the end, still has to feed the machine.
This article appeared in print under the headline “Artificial physicists”
