The Era of Quantum Computing Is Here.


 Quantum computers should soon be able to beat classical computers at certain basic tasks. But before they’re truly powerful, researchers have to overcome a number of fundamental roadblocks.

Quantum computers have to deal with the problem of noise, which can quickly derail any calculation.

Quantum computers have to deal with the problem of noise, which can quickly derail any calculation.

After decades of heavy slog with no promise of success, quantum computing is suddenly buzzing with almost feverish excitement and activity. Nearly two years ago, IBM made a quantum computer available to the world: the 5-quantum-bit (qubit) resource they now call (a little awkwardly) the IBM Q experience. That seemed more like a toy for researchers than a way of getting any serious number crunching done. But 70,000 users worldwide have registered for it, and the qubit count in this resource has now quadrupled. In the past few months, IBM and Intel have announced that they have made quantum computers with 50 and 49 qubits, respectively, and Google is thought to have one waiting in the wings. “There is a lot of energy in the community, and the recent progress is immense,” said physicist Jens Eisert of the Free University of Berlin.

It would be tempting to conclude from all this that the basic problems are solved in principle and the path to a future of ubiquitous quantum computing is now just a matter of engineering. But that would be a mistake. The fundamental physics of quantum computing is far from solved and can’t be readily disentangled from its implementation.

Even if we soon pass the quantum supremacy milestone, the next year or two might be the real crunch time for whether quantum computers will revolutionize computing. There’s still everything to play for and no guarantee of reaching the big goal.

IBM’s quantum computing center at the Thomas J. Watson Research Center in Yorktown Heights, New York, holds quantum computers in large cryogenic tanks (far right) that are cooled to a fraction of a degree above absolute zero.

IBM’s quantum computing center at the Thomas J. Watson Research Center in Yorktown Heights, New York, holds quantum computers in large cryogenic tanks (far right) that are cooled to a fraction of a degree above absolute zero.

Connie Zhou for IBM

Shut Up and Compute

Both the benefits and the challenges of quantum computing are inherent in the physics that permits it. The basic story has been told many times, though not always with the nuance that quantum mechanics demands. Classical computers encode and manipulate information as strings of binary digits — 1 or 0. Quantum bits do the same, except that they may be placed in a so-called superposition of the states 1 and 0, which means that a measurement of the qubit’s state could elicit the answer 1 or 0 with some well-defined probability.

To perform a computation with many such qubits, they must all be sustained in interdependent superpositions of states — a “quantum-coherent” state, in which the qubits are said to be entangled. That way, a tweak to one qubit may influence all the others. This means that somehow computational operations on qubits count for more than they do for classical bits. The computational resources increase in simple proportion to the number of bits for a classical device, but adding an extra qubit potentially doubles the resources of a quantum computer. This is why the difference between a 5-qubit and a 50-qubit machine is so significant.

Note that I’ve not said — as it often is said — that a quantum computer has an advantage because the availability of superpositions hugely increases the number of states it can encode, relative to classical bits. Nor have I said that entanglement permits many calculations to be carried out in parallel. (Indeed, a strong degree of qubit entanglement isn’t essential.) There’s an element of truth in those descriptions — some of the time — but none captures the essence of quantum computing.

Inside one of IBM’s cryostats wired for a 50-qubit quantum system.

Inside one of IBM’s cryostats wired for a 50-qubit quantum system.

 

It’s hard to say qualitatively why quantum computing is so powerful precisely because it is hard to specify what quantum mechanics means at all. The equations of quantum theory certainly show that it will work: that, at least for some classes of computation such as factorization or database searches, there is tremendous speedup of the calculation. But how exactly?

Perhaps the safest way to describe quantum computing is to say that quantum mechanics somehow creates a “resource” for computation that is unavailable to classical devices. As quantum theorist Daniel Gottesman of the Perimeter Institute in Waterloo, Canada, put it, “If you have enough quantum mechanics available, in some sense, then you have speedup, and if not, you don’t.”

Some things are clear, though. To carry out a quantum computation, you need to keep all your qubits coherent. And this is very hard. Interactions of a system of quantum-coherent entities with their surrounding environment create channels through which the coherence rapidly “leaks out” in a process called decoherence. Researchers seeking to build quantum computers must stave off decoherence, which they can currently do only for a fraction of a second. That challenge gets ever greater as the number of qubits — and hence the potential to interact with the environment — increases. This is largely why, even though quantum computing was first proposed by Richard Feynman in 1982 and the theory was worked out in the early 1990s, it has taken until now to make devices that can actually perform a meaningful computation.

Quantum Errors

There’s a second fundamental reason why quantum computing is so difficult. Like just about every other process in nature, it is noisy. Random fluctuations, from heat in the qubits, say, or from fundamentally quantum-mechanical processes, will occasionally flip or randomize the state of a qubit, potentially derailing a calculation. This is a hazard in classical computing too, but it’s not hard to deal with — you just keep two or more backup copies of each bit so that a randomly flipped bit stands out as the odd one out.

Researchers working on quantum computers have created strategies for how to deal with the noise. But these strategies impose a huge debt of computational overhead — all your computing power goes to correcting errors and not to running your algorithms. “Current error rates significantly limit the lengths of computations that can be performed,” said Andrew Childs, the codirector of the Joint Center for Quantum Information and Computer Science at the University of Maryland. “We’ll have to do a lot better if we want to do something interesting.”

Andrew Childs, a quantum theorist at the University of Maryland, cautions that error rates are a fundamental concern for quantum computers.

Andrew Childs, a quantum theorist at the University of Maryland, cautions that error rates are a fundamental concern for quantum computers.

A lot of research on the fundamentals of quantum computing has been devoted to error correction. Part of the difficulty stems from another of the key properties of quantum systems: Superpositions can only be sustained as long as you don’t measure the qubit’s value. If you make a measurement, the superposition collapses to a definite value: 1 or 0. So how can you find out if a qubit has an error if you don’t know what state it is in?

One ingenious scheme involves looking indirectly, by coupling the qubit to another “ancilla” qubit that doesn’t take part in the calculation but that can be probed without collapsing the state of the main qubit itself. It’s complicated to implement, though. Such solutions mean that, to construct a genuine “logical qubit” on which computation with error correction can be performed, you need many physical qubits.

How many? Quantum theorist Alán Aspuru-Guzik of Harvard University estimates that around 10,000 of today’s physical qubits would be needed to make a single logical qubit — a totally impractical number. If the qubits get much better, he said, this number could come down to a few thousand or even hundreds. Eisert is less pessimistic, saying that on the order of 800 physical qubits might already be enough, but even so he agrees that “the overhead is heavy,” and for the moment we need to find ways of coping with error-prone qubits.

An alternative to correcting errors is avoiding them or canceling out their influence: so-called error mitigation. Researchers at IBM, for example, are developing schemes for figuring out mathematically how much error is likely to have been incurred in a computation and then extrapolating the output of a computation to the “zero noise” limit.

Some researchers think that the problem of error correction will prove intractable and will prevent quantum computers from achieving the grand goals predicted for them. “The task of creating quantum error-correcting codes is harder than the task of demonstrating quantum supremacy,” said mathematician Gil Kalai of the Hebrew University of Jerusalem in Israel. And he adds that “devices without error correction are computationally very primitive, and primitive-based supremacy is not possible.” In other words, you’ll never do better than classical computers while you’ve still got errors.

Others believe the problem will be cracked eventually. According to Jay Gambetta, a quantum information scientist at IBM’s Thomas J. Watson Research Center, “Our recent experiments at IBM have demonstrated the basic elements of quantum error correction on small devices, paving the way towards larger-scale devices where qubits can reliably store quantum information for a long period of time in the presence of noise.” Even so, he admits that “a universal fault-tolerant quantum computer, which has to use logical qubits, is still a long way off.” Such developments make Childs cautiously optimistic. “I’m sure we’ll see improved experimental demonstrations of [error correction], but I think it will be quite a while before we see it used for a real computation,” he said.

Living With Errors

For the time being, quantum computers are going to be error-prone, and the question is how to live with that. At IBM, researchers are talking about “approximate quantum computing” as the way the field will look in the near term: finding ways of accommodating the noise.

This calls for algorithms that tolerate errors, getting the correct result despite them. It’s a bit like working out the outcome of an election regardless of a few wrongly counted ballot papers. “A sufficiently large and high-fidelity quantum computation should have some advantage [over a classical computation] even if it is not fully fault-tolerant,” said Gambetta.

One of the most immediate error-tolerant applications seems likely to be of more value to scientists than to the world at large: to simulate stuff at the atomic level. (This, in fact, was the motivation that led Feynman to propose quantum computing in the first place.) The equations of quantum mechanics prescribe a way to calculate the properties — such as stability and chemical reactivity — of a molecule such as a drug. But they can’t be solved classically without making lots of simplifications.

In contrast, the quantum behavior of electrons and atoms, said Childs, “is relatively close to the native behavior of a quantum computer.” So one could then construct an exact computer model of such a molecule. “Many in the community, including me, believe that quantum chemistry and materials science will be one of the first useful applications of such devices,” said Aspuru-Guzik, who has been at the forefront of efforts to push quantum computing in this direction.

Quantum simulations are proving their worth even on the very small quantum computers available so far. A team of researchers including Aspuru-Guzik has developed an algorithm that they call the variational quantum eigensolver (VQE), which can efficiently find the lowest-energy states of molecules even with noisy qubits. So far it can only handle very small molecules with few electrons, which classical computers can already simulate accurately. But the capabilities are getting better, as Gambetta and coworkers showed last September when they used a 6-qubit device at IBM to calculate the electronic structures of molecules, including lithium hydride and beryllium hydride. The work was “a significant leap forward for the quantum regime,” according to physical chemist Markus Reiher of the Swiss Federal Institute of Technology in Zurich, Switzerland. “The use of the VQE for the simulation of small molecules is a great example of the possibility of near-term heuristic algorithms,” said Gambetta.

But even for this application, Aspuru-Guzik confesses that logical qubits with error correction will probably be needed before quantum computers truly begin to surpass classical devices. “I would be really excited when error-corrected quantum computing begins to become a reality,” he said.

“If we had more than 200 logical qubits, we could do things in quantum chemistry beyond standard approaches,” Reiher adds. “And if we had about 5,000 such qubits, then the quantum computer would be transformative in this field.”

What’s Your Volume?

Despite the challenges of reaching those goals, the fast growth of quantum computers from 5 to 50 qubits in barely more than a year has raised hopes. But we shouldn’t get too fixated on these numbers, because they tell only part of the story. What matters is not just — or even mainly — how many qubits you have, but how good they are, and how efficient your algorithms are.

Any quantum computation has to be completed before decoherence kicks in and scrambles the qubits. Typically, the groups of qubits assembled so far have decoherence times of a few microseconds. The number of logic operations you can carry out during that fleeting moment depends on how quickly the quantum gates can be switched — if this time is too slow, it really doesn’t matter how many qubits you have at your disposal. The number of gate operations needed for a calculation is called its depth: Low-depth (shallow) algorithms are more feasible than high-depth ones, but the question is whether they can be used to perform useful calculations.

What’s more, not all qubits are equally noisy. In theory it should be possible to make very low-noise qubits from so-called topological electronic states of certain materials, in which the “shape” of the electron states used for encoding binary information confers a kind of protection against random noise. Researchers at Microsoft, most prominently, are seeking such topological states in exotic quantum materials, but there’s no guarantee that they’ll be found or will be controllable.

Researchers at IBM have suggested that the power of a quantum computation on a given device be expressed as a number called the “quantum volume,” which bundles up all the relevant factors: number and connectivity of qubits, depth of algorithm, and other measures of the gate quality, such as noisiness. It’s really this quantum volume that characterizes the power of a quantum computation, and Gambetta said that the best way forward right now is to develop quantum-computational hardware that increases the available quantum volume.

This is one reason why the much vaunted notion of quantum supremacy is more slippery than it seems. The image of a 50-qubit (or so) quantum computer outperforming a state-of-the-art supercomputer sounds alluring, but it leaves a lot of questions hanging. Outperforming for which problem? How do you know the quantum computer has got the right answer if you can’t check it with a tried-and-tested classical device? And how can you be sure that the classical machine wouldn’t do better if you could find the right algorithm?

So quantum supremacy is a concept to handle with care. Some researchers prefer now to talk about “quantum advantage,” which refers to the speedup that quantum devices offer without making definitive claims about what is best. An aversion to the word “supremacy” has also arisen because of the racial and political implications.

Whatever you choose to call it, a demonstration that quantum computers can do things beyond current classical means would be psychologically significant for the field. “Demonstrating an unambiguous quantum advantage will be an important milestone,” said Eisert — it would prove that quantum computers really can extend what is technologically possible.

That might still be more of a symbolic gesture than a transformation in useful computing resources. But such things may matter, because if quantum computing is going to succeed, it won’t be simply by the likes of IBM and Google suddenly offering their classy new machines for sale. Rather, it’ll happen through an interactive and perhaps messy collaboration between developers and users, and the skill set will evolve in the latter only if they have sufficient faith that the effort is worth it. This is why both IBM and Google are keen to make their devices available as soon as they’re ready. As well as a 16-qubit IBM Q experience offered to anyone who registers online, IBM now has a 20-qubit version for corporate clients, including JP Morgan Chase, Daimler, Honda, Samsung and the University of Oxford. Not only will that help clients discover what’s in it for them; it should create a quantum-literate community of programmers who will devise resources and solve problems beyond what any individual company could muster.

“For quantum computing to take traction and blossom, we must enable the world to use and to learn it,” said Gambetta. “This period is for the world of scientists and industry to focus on getting quantum-ready.”

The Argument Against Quantum Computers


  The mathematician Gil Kalai believes that quantum computers can’t possibly work, even in principle.

Sixteen years ago, on a cold February day at Yale University, a poster caught Gil Kalai’s eye. It advertised a series of lectures by Michel Devoret, a well-known expert on experimental efforts in quantum computing. The talks promised to explore the question “Quantum Computer: Miracle or Mirage?” Kalai expected a vigorous discussion of the pros and cons of quantum computing. Instead, he recalled, “the skeptical direction was a little bit neglected.” He set out to explore that skeptical view himself.

Today, Kalai, a mathematician at Hebrew University in Jerusalem, is one of the most prominent of a loose group of mathematicians, physicists and computer scientists arguing that quantum computing, for all its theoretical promise, is something of a mirage. Some argue that there exist good theoretical reasons why the innards of a quantum computer — the “qubits” — will never be able to consistently perform the complex choreography asked of them. Others say that the machines will never work in practice, or that if they are built, their advantages won’t be great enough to make up for the expense.

Kalai has approached the issue from the perspective of a mathematician and computer scientist. He has analyzed the issue by looking at computational complexity and, critically, the issue of noise. All physical systems are noisy, he argues, and qubits kept in highly sensitive “superpositions” will inevitably be corrupted by any interaction with the outside world. Getting the noise down isn’t just a matter of engineering, he says. Doing so would violate certain fundamental theorems of computation.

Kalai knows that his is a minority view. Companies like IBM, Intel and Microsoft have invested heavily in quantum computing; venture capitalists are funding quantum computing startups (such as Quantum Circuits, a firm set up by Devoret and two of his Yale colleagues). Other nations — most notably China — are pouring billions of dollars into the sector.

Quanta Magazine recently spoke with Kalai about quantum computing, noise and the possibility that a decade of work will be proven wrong within a matter of weeks. A condensed and edited version of that conversation follows.

When did you first have doubts about quantum computers?

At first, I was quite enthusiastic, like everybody else. But at a lecture in 2002 by Michel Devoret called “Quantum Computer: Miracle or Mirage,” I had a feeling that the skeptical direction was a little bit neglected. Unlike the title, the talk was very much the usual rhetoric about how wonderful quantum computing is. The side of the mirage was not well-presented.

And so you began to research the mirage.

Only in 2005 did I decide to work on it myself. I saw a scientific opportunity and some possible connection with my earlier work from 1999 with Itai Benjamini and Oded Schramm on concepts called noise sensitivity and noise stability.

What do you mean by “noise”?

By noise I mean the errors in a process, and sensitivity to noise is a measure of how likely the noise — the errors — will affect the outcome of this process. Quantum computing is like any similar process in nature — noisy, with random fluctuations and errors. When a quantum computer executes an action, in every computer cycle there is some probability that a qubit will get corrupted.

Kalai argues that limiting the noise in a quantum computer will also limit the computational power of the system.

Video: Kalai argues that limiting the noise in a quantum computer will also limit the computational power of the system.

And so this corruption is the key problem?

We need what’s known as quantum error correction. But this will require 100 or even 500 “physical” qubits to represent a single “logical” qubit of very high quality. And then to build and use such quantum error-correcting codes, the amount of noise has to go below a certain level, or threshold.

To determine the required threshold mathematically, we must effectively model the noise. I thought it would be an interesting challenge.

What exactly did you do?

I tried to understand what happens if the errors due to noise are correlated — or connected. There is a Hebrew proverb that says that trouble comes in clusters. In English you would say: When it rains, it pours. In other words, interacting systems will have a tendency for errors to be correlated. There will be a probability that errors will affect many qubits all at once.

So over the past decade or so, I’ve been studying what kind of correlations emerge from complicated quantum computations and what kind of correlations will cause a quantum computer to fail.

In my earlier work on noise we used a mathematical approach called Fourier analysis, which says that it’s possible to break down complex waveforms into simpler components. We found that if the frequencies of these broken-up waves are low, the process is stable, and if they are high, the process is prone to error.

That previous work brought me to my more recent paper that I wrote in 2014 with a Hebrew University computer scientist, Guy Kindler. Our calculations suggest that the noise in a quantum computer will kill all the high-frequency waves in the Fourier decomposition. If you think about the computational process as a Beethoven symphony, the noise will allow us to hear only the basses, but not the cellos, violas and violins.

These results also give good reasons to think that noise levels cannot be sufficiently reduced; they will still be much higher than what is needed to demonstrate quantum supremacy and quantum error correction.

Why can’t we push the noise level below this threshold?

Many researchers believe that we can go beyond the threshold, and that constructing a quantum computer is merely an engineering challenge of lowering it. However, our first result shows that the noise level cannot be reduced, because doing so will contradict an insight from the theory of computing about the power of primitive computational devices. Noisy quantum computers in the small and intermediate scale deliver primitive computational power. They are too primitive to reach “quantum supremacy” — and if quantum supremacy is not possible, then creating quantum error-correcting codes, which is harder, is also impossible.

What do your critics say to that?

Critics point out that my work with Kindler deals with a restricted form of quantum computing and argue that our model for noise is not physical, but a mathematical simplification of an actual physical situation. I’m quite certain that what we have demonstrated for our simplified model is a real and general phenomenon.

My critics also point to two things that they find strange in my analysis: The first is my attempt to draw conclusions about engineering of physical devices from considerations about computation. The second is drawing conclusions about small-scale quantum systems from insights of the theory of computation that are usually applied to large systems. I agree that these are unusual and perhaps even strange lines of analysis.

And finally, they argue that these engineering difficulties are not fundamental barriers, and that with sufficient hard work and resources, the noise can be driven down to as close to zero as needed. But I think that the effort required to obtain a low enough error level for any implementation of universal quantum circuits increases exponentially with the number of qubits, and thus, quantum computers are not possible.

How can you be certain?

I am pretty certain, while a little nervous to be proven wrong. Our results state that noise will corrupt the computation, and that the noisy outcomes will be very easy to simulate on a classical computer. This prediction can already be tested; you don’t even need 50 qubits for that, I believe that 10 to 20 qubits will suffice. For quantum computers of the kind Google and IBM are building, when you run, as they plan to do, certain computational processes, they expect robust outcomes that are increasingly hard to simulate on a classical computer. Well, I expect very different outcomes. So I don’t need to be certain, I can simply wait and see.

Job One for Quantum Computers: Boost Artificial Intelligence


The fusion of quantum computing and machine learning has become a booming research area. Can it possibly live up to its high expectations?

In the early ’90s, Elizabeth Behrman, a physics professor at Wichita State University, began working to combine quantum physics with artificial intelligence — in particular, the then-maverick technology of neural networks. Most people thought she was mixing oil and water. “I had a heck of a time getting published,” she recalled. “The neural-network journals would say, ‘What is this quantum mechanics?’ and the physics journals would say, ‘What is this neural-network garbage?’”

Today the mashup of the two seems the most natural thing in the world. Neural networks and other machine-learning systems have become the most disruptive technology of the 21st century. They out-human humans, beating us not just at tasks most of us were never really good at, such as chess and data-mining, but also at the very types of things our brains evolved for, such as recognizing faces, translating languages and negotiating four-way stops. These systems have been made possible by vast computing power, so it was inevitable that tech companies would seek out computers that were not just bigger, but a new class of machine altogether.

Quantum computers, after decades of research, have nearly enough oomph to perform calculations beyond any other computer on Earth. Their killer app is usually said to be factoring large numbers, which are the key to modern encryption. That’s still another decade off, at least. But even today’s rudimentary quantum processors are uncannily matched to the needs of machine learning. They manipulate vast arrays of data in a single step, pick out subtle patterns that classical computers are blind to, and don’t choke on incomplete or uncertain data. “There is a natural combination between the intrinsic statistical nature of quantum computing … and machine learning,” said Johannes Otterbach, a physicist at Rigetti Computing, a quantum-computer company in Berkeley, California.

If anything, the pendulum has now swung to the other extreme. Google, Microsoft, IBM and other tech giants are pouring money into quantum machine learning, and a startup incubator at the University of Toronto is devoted to it. “‘Machine learning’ is becoming a buzzword,” said Jacob Biamonte, a quantum physicist at the Skolkovo Institute of Science and Technology in Moscow. “When you mix that with ‘quantum,’ it becomes a mega-buzzword.”

Yet nothing with the word “quantum” in it is ever quite what it seems. Although you might think a quantum machine-learning system should be powerful, it suffers from a kind of locked-in syndrome. It operates on quantum states, not on human-readable data, and translating between the two can negate its apparent advantages. It’s like an iPhone X that, for all its impressive specs, ends up being just as slow as your old phone, because your network is as awful as ever. For a few special cases, physicists can overcome this input-output bottleneck, but whether those cases arise in practical machine-learning tasks is still unknown. “We don’t have clear answers yet,” said Scott Aaronson, a computer scientist at the University of Texas, Austin, who is always the voice of sobriety when it comes to quantum computing. “People have often been very cavalier about whether these algorithms give a speedup.”

Quantum Neurons

The main job of a neural network, be it classical or quantum, is to recognize patterns. Inspired by the human brain, it is a grid of basic computing units — the “neurons.” Each can be as simple as an on-off device. A neuron monitors the output of multiple other neurons, as if taking a vote, and switches on if enough of them are on. Typically, the neurons are arranged in layers. An initial layer accepts input (such as image pixels), intermediate layers create various combinations of the input (representing structures such as edges and geometric shapes) and a final layer produces output (a high-level description of the image content).

 

Crucially, the wiring is not fixed in advance, but adapts in a process of trial and error. The network might be fed images labeled “kitten” or “puppy.” For each image, it assigns a label, checks whether it was right, and tweaks the neuronal connections if not. Its guesses are random at first, but get better; after perhaps 10,000 examples, it knows its pets. A serious neural network can have a billion interconnections, all of which need to be tuned.

On a classical computer, all these interconnections are represented by a ginormous matrix of numbers, and running the network means doing matrix algebra. Conventionally, these matrix operations are outsourced to a specialized chip such as a graphics processing unit. But nothing does matrices like a quantum computer. “Manipulation of large matrices and large vectors are exponentially faster on a quantum computer,” said Seth Lloyd, a physicist at the Massachusetts Institute of Technology and a quantum-computing pioneer.

For this task, quantum computers are able to take advantage of the exponential nature of a quantum system. The vast bulk of a quantum system’s information storage capacity resides not in its individual data units — its qubits, the quantum counterpart of classical computer bits — but in the collective properties of those qubits. Two qubits have four joint states: both on, both off, on/off, and off/on. Each has a certain weighting, or “amplitude,” that can represent a neuron. If you add a third qubit, you can represent eight neurons; a fourth, 16. The capacity of the machine grows exponentially. In effect, the neurons are smeared out over the entire system. When you act on a state of four qubits, you are processing 16 numbers at a stroke, whereas a classical computer would have to go through those numbers one by one.

Lloyd estimates that 60 qubits would be enough to encode an amount of data equivalent to that produced by humanity in a year, and 300 could carry the classical information content of the observable universe. (The biggest quantum computers at the moment, built by IBM, Intel and Google, have 50-ish qubits.) And that’s assuming each amplitude is just a single classical bit. In fact, amplitudes are continuous quantities (and, indeed, complex numbers) and, for a plausible experimental precision, one might store as many as 15 bits, Aaronson said.

But a quantum computer’s ability to store information compactly doesn’t make it faster. You need to be able to use those qubits. In 2008, Lloyd, the physicist Aram Harrow of MIT and Avinatan Hassidim, a computer scientist at Bar-Ilan University in Israel, showed how to do the crucial algebraic operation of inverting a matrix. They broke it down into a sequence of logic operations that can be executed on a quantum computer. Their algorithm works for a huge variety of machine-learning techniques. And it doesn’t require nearly as many algorithmic steps as, say, factoring a large number does. A computer could zip through a classification task before noise — the big limiting factor with today’s technology — has a chance to foul it up. “You might have a quantum advantage before you have a fully universal, fault-tolerant quantum computer,” said Kristan Temme of IBM’s Thomas J. Watson Research Center.

Let Nature Solve the Problem

So far, though, machine learning based on quantum matrix algebra has been demonstrated only on machines with just four qubits. Most of the experimental successes of quantum machine learning to date have taken a different approach, in which the quantum system does not merely simulate the network; it is the network. Each qubit stands for one neuron. Though lacking the power of exponentiation, a device like this can avail itself of other features of quantum physics.

The largest such device, with some 2,000 qubits, is the quantum processor manufactured by D-Wave Systems, based near Vancouver, British Columbia. It is not what most people think of as a computer. Instead of starting with some input data, executing a series of operations and displaying the output, it works by finding internal consistency. Each of its qubits is a superconducting electric loop that acts as a tiny electromagnet oriented up, down, or up and down — a superposition. Qubits are “wired” together by allowing them to interact magnetically.

Processors made by D-Wave Systems are being used for machine learning applications.

Processors made by D-Wave Systems are being used for machine learning applications.

To run the system, you first impose a horizontal magnetic field, which initializes the qubits to an equal superposition of up and down — the equivalent of a blank slate. There are a couple of ways to enter data. In some cases, you fix a layer of qubits to the desired input values; more often, you incorporate the input into the strength of the interactions. Then you let the qubits interact. Some seek to align in the same direction, some in the opposite direction, and under the influence of the horizontal field, they flip to their preferred orientation. In so doing, they might trigger other qubits to flip. Initially that happens a lot, since so many of them are misaligned. Over time, though, they settle down, and you can turn off the horizontal field to lock them in place. At that point, the qubits are in a pattern of up and down that ensures the output follows from the input.

It’s not at all obvious what the final arrangement of qubits will be, and that’s the point. The system, just by doing what comes naturally, is solving a problem that an ordinary computer would struggle with. “We don’t need an algorithm,” explained Hidetoshi Nishimori, a physicist at the Tokyo Institute of Technology who developed the principles on which D-Wave machines operate. “It’s completely different from conventional programming. Nature solves the problem.”

The qubit-flipping is driven by quantum tunneling, a natural tendency that quantum systems have to seek out their optimal configuration, rather than settle for second best. You could build a classical network that worked on analogous principles, using random jiggling rather than tunneling to get bits to flip, and in some cases it would actually work better. But, interestingly, for the types of problems that arise in machine learning, the quantum network seems to reach the optimum faster.

The D-Wave machine has had its detractors. It is extremely noisy and, in its current incarnation, can perform only a limited menu of operations. Machine-learning algorithms, though, are noise-tolerant by their very nature. They’re useful precisely because they can make sense of a messy reality, sorting kittens from puppies against a backdrop of red herrings. “Neural networks are famously robust to noise,” Behrman said.

In 2009 a team led by Hartmut Neven, a computer scientist at Google who pioneered augmented reality — he co-founded the Google Glass project — and then took up quantum information processing, showed how an early D-Wave machine could do a respectable machine-learning task. They used it as, essentially, a single-layer neural network that sorted images into two classes: “car” or “no car” in a library of 20,000 street scenes. The machine had only 52 working qubits, far too few to take in a whole image. (Remember: the D-Wave machine is of a very different type than in the state-of-the-art 50-qubit systems coming online in 2018.) So Neven’s team combined the machine with a classical computer, which analyzed various statistical quantities of the images and calculated how sensitive these quantities were to the presence of a car — usually not very, but at least better than a coin flip. Some combination of these quantities could, together, spot a car reliably, but it wasn’t obvious which. It was the network’s job to find out.

The team assigned a qubit to each quantity. If that qubit settled into a value of 1, it flagged the corresponding quantity as useful; 0 meant don’t bother. The qubits’ magnetic interactions encoded the demands of the problem, such as including only the most discriminating quantities, so as to keep the final selection as compact as possible. The result was able to spot a car.

Last year a group led by Maria Spiropulu, a particle physicist at the California Institute of Technology, and Daniel Lidar, a physicist at USC, applied the algorithm to a practical physics problem: classifying proton collisions as “Higgs boson” or “no Higgs boson.” Limiting their attention to collisions that spat out photons, they used basic particle theory to predict which photon properties might betray the fleeting existence of the Higgs, such as momentum in excess of some threshold. They considered eight such properties and 28 combinations thereof, for a total of 36 candidate signals, and let a late-model D-Wave at the University of Southern California find the optimal selection. It identified 16 of the variables as useful and three as the absolute best. The quantum machine needed less data than standard procedures to perform an accurate identification. “Provided that the training set was small, then the quantum approach did provide an accuracy advantage over traditional methods used in the high-energy physics community,” Lidar said.

Maria Spiropulu, a physicist at the California Institute of Technology, used quantum machine learning to find Higgs bosons.

Maria Spiropulu, a physicist at the California Institute of Technology, used quantum machine learning to find Higgs bosons.

 

In December, Rigetti demonstrated a way to automatically group objects using a general-purpose quantum computer with 19 qubits. The researchers did the equivalent of feeding the machine a list of cities and the distances between them, and asked it to sort the cities into two geographic regions. What makes this problem hard is that the designation of one city depends on the designation of all the others, so you have to solve the whole system at once.

The Rigetti team effectively assigned each city a qubit, indicating which group it was assigned to. Through the interactions of the qubits (which, in Rigetti’s system, are electrical rather than magnetic), each pair of qubits sought to take on opposite values — their energy was minimized when they did so. Clearly, for any system with more than two qubits, some pairs of qubits had to consent to be assigned to the same group. Nearby cities assented more readily since the energetic cost for them to be in the same group was lower than for more-distant cities.

To drive the system to its lowest energy, the Rigetti team took an approach similar in some ways to the D-Wave annealer. They initialized the qubits to a superposition of all possible cluster assignments. They allowed qubits to interact briefly, which biased them toward assuming the same or opposite values. Then they applied the analogue of a horizontal magnetic field, allowing the qubits to flip if they were so inclined, pushing the system a little way toward its lowest-energy state. They repeated this two-step process — interact then flip — until the system minimized its energy, thus sorting the cities into two distinct regions.

These classification tasks are useful but straightforward. The real frontier of machine learning is in generative models, which do not simply recognize puppies and kittens, but can generate novel archetypes — animals that never existed, but are every bit as cute as those that did. They might even figure out the categories of “kitten” and “puppy” on their own, or reconstruct images missing a tail or paw. “These techniques are very powerful and very useful in machine learning, but they are very hard,” said Mohammad Amin, the chief scientist at D-Wave. A quantum assist would be most welcome.

D-Wave and other research teams have taken on this challenge. Training such a model means tuning the magnetic or electrical interactions among qubits so the network can reproduce some sample data. To do this, you combine the network with an ordinary computer. The network does the heavy lifting — figuring out what a given choice of interactions means for the final network configuration — and its partner computer uses this information to adjust the interactions. In one demonstration last year, Alejandro Perdomo-Ortiz, a researcher at NASA’s Quantum Artificial Intelligence Lab, and his team exposed a D-Wave system to images of handwritten digits. It discerned that there were 10 categories, matching the digits 0 through 9, and generated its own scrawled numbers.

Bottlenecks Into the Tunnels

Well, that’s the good news. The bad is that it doesn’t much matter how awesome your processor is if you can’t get your data into it. In matrix-algebra algorithms, a single operation may manipulate a matrix of 16 numbers, but it still takes 16 operations to load the matrix. “State preparation — putting classical data into a quantum state — is completely shunned, and I think this is one of the most important parts,” said Maria Schuld, a researcher at the quantum-computing startup Xanadu and one of the first people to receive a doctorate in quantum machine learning. Machine-learning systems that are laid out in physical form face parallel difficulties of how to embed a problem in a network of qubits and get the qubits to interact as they should.

Once you do manage to enter your data, you need to store it in such a way that a quantum system can interact with it without collapsing the ongoing calculation. Lloyd and his colleagues have proposed a quantum RAM that uses photons, but no one has an analogous contraption for superconducting qubits or trapped ions, the technologies found in the leading quantum computers. “That’s an additional huge technological problem beyond the problem of building a quantum computer itself,” Aaronson said. “The impression I get from the experimentalists I talk to is that they are frightened. They have no idea how to begin to build this.”

And finally, how do you get your data out? That means measuring the quantum state of the machine, and not only does a measurement return only a single number at a time, drawn at random, it collapses the whole state, wiping out the rest of the data before you even have a chance to retrieve it. You’d have to run the algorithm over and over again to extract all the information.

Yet all is not lost. For some types of problems, you can exploit quantum interference. That is, you can choreograph the operations so that wrong answers cancel themselves out and right ones reinforce themselves; that way, when you go to measure the quantum state, it won’t give you just any random value, but the desired answer. But only a few algorithms, such as brute-force search, can make good use of interference, and the speedup is usually modest.

In some cases, researchers have found shortcuts to getting data in and out. In 2015 Lloyd, Silvano Garnerone of the University of Waterloo in Canada, and Paolo Zanardi at USC showed that, for some kinds of statistical analysis, you don’t need to enter or store the entire data set. Likewise, you don’t need to read out all the data when a few key values would suffice. For instance, tech companies use machine learning to suggest shows to watch or things to buy based on a humongous matrix of consumer habits. “If you’re Netflix or Amazon or whatever, you don’t actually need the matrix written down anywhere,” Aaronson said. “What you really need is just to generate recommendations for a user.”

All this invites the question: If a quantum machine is powerful only in special cases, might a classical machine also be powerful in those cases? This is the major unresolved question of the field. Ordinary computers are, after all, extremely capable. The usual method of choice for handling large data sets — random sampling — is actually very similar in spirit to a quantum computer, which, whatever may go on inside it, ends up returning a random result. Schuld remarked: “I’ve done a lot of algorithms where I felt, ‘This is amazing. We’ve got this speedup,’ and then I actually, just for fun, write a sampling technique for a classical computer, and I realize you can do the same thing with sampling.”

If you look back at the successes that quantum machine learning has had so far, they all come with asterisks. Take the D-Wave machine. When classifying car images and Higgs bosons, it was no faster than a classical machine. “One of the things we do not talk about in this paper is quantum speedup,” said Alex Mott, a computer scientist at Google DeepMind who was a member of the Higgs research team. Matrix-algebra approaches such as the Harrow-Hassidim-Lloyd algorithm show a speedup only if the matrices are sparse — mostly filled with zeroes. “No one ever asks, are sparse data sets actually interesting in machine learning?” Schuld noted.

Quantum Intelligence

On the other hand, even the occasional incremental improvement over existing techniques would make tech companies happy. “These advantages that you end up seeing, they’re modest; they’re not exponential, but they are quadratic,” said Nathan Wiebe, a quantum-computing researcher at Microsoft Research. “Given a big enough and fast enough quantum computer, we could revolutionize many areas of machine learning.” And in the course of using the systems, computer scientists might solve the theoretical puzzle of whether they are inherently faster, and for what.

Schuld also sees scope for innovation on the software side. Machine learning is more than a bunch of calculations. It is a complex of problems that have their own particular structure. “The algorithms that people construct are removed from the things that make machine learning interesting and beautiful,” she said. “This is why I started to work the other way around and think: If have this quantum computer already — these small-scale ones — what machine-learning model actually can it generally implement? Maybe it is a model that has not been invented yet.” If physicists want to impress machine-learning experts, they’ll need to do more than just make quantum versions of existing models

Just as many neuroscientists now think that the structure of human thought reflects the requirements of having a body, so, too, are machine-learning systems embodied. The images, language and most other data that flow through them come from the physical world and reflect its qualities. Quantum machine learning is similarly embodied — but in a richer world than ours. The one area where it will undoubtedly shine is in processing data that is already quantum. When the data is not an image, but the product of a physics or chemistry experiment, the quantum machine will be in its element. The input problem goes away, and classical computers are left in the dust.

In a neatly self-referential loop, the first quantum machine-learning systems may help to design their successors. “One way we might actually want to use these systems is to build quantum computers themselves,” Wiebe said. “For some debugging tasks, it’s the only approach that we have.” Maybe they could even debug us. Leaving aside whether the human brain is a quantum computer — a highly contentious question — it sometimes acts as if it were one. Human behavior is notoriously contextual; our preferences are formed by the choices we are given, in ways that defy logic. In this, we are like quantum particles. “The way you ask questions and the ordering matters, and that is something that is very typical in quantum data sets,” Perdomo-Ortiz said. So a quantum machine-learning system might be a natural way to study human cognitive biases

Neural networks and quantum processors have one thing in common: It is amazing they work at all. It was never obvious that you could train a network, and for decades most people doubted it would ever be possible. Likewise, it is not obvious that quantum physics could ever be harnessed for computation, since the distinctive effects of quantum physics are so well hidden from us. And yet both work — not always, but more often than we had any right to expect. On this precedent, it seems likely that their union will also find its place.

Quantum Computing Without Qubits


A quantum computing pioneer explains why analog simulators may beat out general-purpose digital quantum machines — for now.

Ivan Deutsch is building quantum computers out of base-16 “qudits,” quantum information units that can assume any number of “d” states.

Ivan Deutsch is building quantum computers out of base-16 “qudits,” quantum information units that can assume any number of “d” states.

For more than 20 years, Ivan H. Deutsch has struggled to design the guts of a working quantum computer. He has not been alone. The quest to harness the computational might of quantum weirdness continues to occupy hundreds of researchers around the world. Why hasn’t there been more to show for their work? As physicists have known since quantum computing’s beginnings, the same characteristics that make quantum computing exponentially powerful also make it devilishly difficult to control. The quantum computing “nightmare” has always been that a quantum computer’s advantages in speed would be wiped out by the machine’s complexity.

Yet progress is arriving on two main fronts. First, researchers are developing unique quantum error-correction techniques that will help keep quantum processors up and running for the time needed to complete a calculation. Second, physicists are working with so-called analog quantum simulators — machines that can’t act like a general-purpose computer, but rather are designed to explore specific problems in quantum physics. A classical computer would have to run for thousands of years to compute the quantum equations of motion for just 100 atoms. A quantum simulator could do it in less than a second.

Quanta Magazine spoke with Deutsch about recent progress in the field, his hopes for the near future, and his own work at the University of New Mexico’s Center for Quantum Information and Control on scaling up binary quantum bits into base-16 digits.

QUANTA MAGAZINE: Why would a universal quantum machine be so uniquely powerful?

IVAN DEUTSCH: In a classical computer, information is stored in retrievable bits binary coded as 0 or 1. But in a quantum computer, elementary particles inhabit a probabilistic limbo called superposition where a “qubit” can be coded as 0 and 1.

Here is the magic: Each qubit can be entangled with the other qubits in the machine. The intertwining of quantum “states” exponentially increases the number of 0s and 1s that can be simultaneously processed by an array of qubits. Machines that can harness the power of quantum logic can deal with exponentially greater levels of complexity than the most powerful classical computer. Problems that would take a state-of-the-art classical computer the age of our universe to solve, can, in theory, be solved by a universal quantum computer in hours.

What is the quantum computing “nightmare”?

The same quantum effects that make a quantum computer so blazingly fast also make it incredibly difficult to operate. From the beginning, it has not been clear whether the exponential speed up provided by a quantum computer would be cancelled out by the exponential complexity needed to protect the system from crashing.

Is the situation hopeless?

Not at all. We now know that a universal quantum computer will not require exponential complexity in design. But it is still very hard.

So what’s the problem, and how do we get around it?

The hardware problem is that the superposition is so fragile that the random interaction of a single qubit with the molecules composing its immediate surroundings can cause the entire network of entangled qubits to delink or collapse. The ongoing calculation is destroyed as each qubit transforms into a digitized classical bit holding a single value: 0 or 1.

In classical computers, we reduce the inevitable loss of information by designing a lot of redundancy into the system. Error-correcting algorithms compare multiple copies of the output. They select the most frequent answer and discard the rest of the data as noise. We cannot do that with a quantum computer, because trying to directly compare qubits will crash the program. But we are gradually learning how to keep systems of entangled qubits from collapsing.

The major obstacle, to my mind, is creating error-correcting software that can keep data from being corrupted as the calculation proceeds toward the final readout. The great trick is to design and implement an algorithm that only measures the errors and not the data, thus preserving the superposition that contains the correct answer.

Will that end the nightmare?

It turns out that the error correction technique itself introduces errors. One of the most wonderful advances in quantum computing was recognizing that, in theory, we can correct the new errors without requiring 100 percent precision, allowing minor background noise to pollute the calculation as it rolls along. We cannot actually do this — yet. The main reason that we do not have a working universal quantum computer is that we are still experimenting with how to implant such a “fault-tolerant” algorithm into a quantum circuit. Right now we can control 10 qubits reasonably well. But there is no error-correcting technique, to my knowledge, capable of controlling the thousands of qubits needed to construct a universal machine.

Is that what you’re working on?

I study the information processing capabilities of trapped atoms. My colleague Poul Jessen at the University of Arizona and I are pushing the logical power beyond binary-based qubits. For example, what if we can control the superposition of an atom with, say, 16 different energy levels? Using base 16, we can then store what we call a “qudit” in a single atom. That would move us beyond the information processing speed obtainable by a base 2 system, the qubit.

What other options do we have?

There may be significant applications available for making non-universal machines: Special purpose, analog quantum simulators designed to solve specific problems, such as how room-temperature superconductors work or how a particular protein folds.

Are these actually computers?

They are not universal machines capable of solving any type of question. But say that I want to model global climate change. One way to do this is to write down a mathematical model and then solve the equations on a digital computer. That is typically what climate scientists do. Another way is to try to simulate some aspect of the earth’s climate in a controllable experiment. I can create a simple physical system that obeys the same laws of motion as the system I’m trying to model — mixing nitrogen, oxygen, and hydrogen in a tank, for example. What goes on inside the tank is a real-world computation that tells me something about atmospheric turbulence under certain conditions.

It is the same with an analog quantum simulator — I use one controllable physical system to simulate another. For example, successfully simulating a superconductor with such a device would reveal the quantum mechanics of high-temperature superconductivity. That could lead to the manufacture of non-brittle superconducting materials for many uses, including building less-fragile quantum circuits. Hopefully, we can learn how to build a robust universal digital computer by experimenting with analog simulators.

Has anyone built a working analog quantum simulator?

In 2002, a group at the Max Planck Institute in Germany built an optical lattice — a super-chilled egg carton made of light — and controlled it by pulsing different strengths of laser beams at it. This was a fundamentally analog device designed to obey quantum mechanical equations of motion. The short story is that it successfully simulated how atoms transition between acting as superfluids or insulators. That experiment has sparked a lot of research in analog quantum computing with optical lattices and cold atom traps.

What are the main challenges for these quantum simulators?

Because the evolution of the analog simulation is not digitized, the software cannot correct the tiny errors that accumulate during the calculation as we could error-correct noise on a universal machine. The analog device must keep a quantum superposition intact long enough for the simulation to run its course without resorting to digital error correction. This is a particular challenge for the analog approach to quantum simulation.

Is the D-Wave machine a quantum simulator?

The D-Wave prototype is not a universal quantum computer. It is not digital, nor error-correcting, nor fault tolerant. It is a purely analog machine designed to solve a particular optimization problem. It is unclear if it qualifies as a quantum device.

Will a scalable quantum computer be deployed during your lifetime?

We are pushing past the nightmare. Around the world, many university-based labs are working hard to remove or bypass the road block of fault tolerance. Academic researchers are leading the way, intellectually. For example, the groups of Rob Schoelkopf and Michel H. Devoret at Yale are taking superconducting technologies close to fault-tolerance.

But constructing a working universal digital quantum computer will likely require mobilizing industrial-scale resources. To that end, IBM is exploring quantum computing with superconducting circuits with personnel largely from the Yale groups. Google is working with John Martinis’s lab at the University of California, Santa Barbara. HRL Laboratories is working on silicon-based quantum computing. Lockheed Martin is exploring ion traps. And who knows what the National Security Agency is up to.

But generally in academic labs, without these industrial-scale resources, scientists are focusing more and more on learning how to control analog quantum simulators. There is short-term fruit to be picked in that arena — both intellectually and in the currency of academics: publishable papers.

Are you willing to settle for analog?

I favor pursuing the digital approach full force. Before I die, I would love to see just one universal logical qubit that can be indefinitely error corrected. It would instantly be classified by the government, of course. But I dream on, regardless.

Why Quantum Computers Might Not Break Cryptography


A new paper claims that a common digital security system could be tweaked to withstand attacks even from a powerful quantum computer.

Math is hard. Indeed, much of the modern infrastructure for secure communication depends heavily on the difficulty of elementary mathematics — of factoring, to be exact. It’s easy to reduce a small number like 15 to its prime factors (3 x 5), but factoring numbers with a few hundred digits is still exceedingly difficult. For this reason, the RSA cryptosystem, an encryption scheme that derives its security from the difficulty of integer factorization, remains a popular tool for secure communication.

Research suggests, however, that a quantum computer would be able to factor a large number far more quickly than the best available methods today. If researchers could build a quantum computer that could outperform classical supercomputers, the thinking goes, cryptographers could use a particular algorithm called Shor’s algorithm to render the RSA cryptosystem unsalvageable. The deadline to avert this may arrive sooner than we think: Google recently claimed that its quantum computers will be able to perform a calculation that’s beyond the reach of any classical computer by the end of the year. In light of this, cryptographers are scrambling to find a new quantum-proof security standard.

Yet perhaps RSA isn’t in as much trouble as researchers have assumed. A few weeks ago, a paper surfaced on the Cryptology ePrint Archive that asked: “Is it actually true that quantum computers will kill RSA?” The authors note that even though a quantum computer running Shor’s algorithm would be faster than a classical computer, the RSA algorithm is faster than both. And the larger the RSA “key” — the number that must be factored — the greater the speed difference.

The authors of the paper estimate that attacking a terabyte-size key using Shor’s algorithm would require around 2100 operations on a quantum computer, an enormous number comparable to the total number of bacterial cells on Earth. The authors don’t convert this to a concrete time estimate, but current research suggests that a real quantum computer wouldn’t be able to accomplish this in any reasonable amount of time. “RSA is not entirely dead even if quantum computers are practical,” said Nadia Heninger, an assistant professor of computer and information science at the University of Pennsylvania and a co-author of the paper. The paper also shows how to implement such a massive RSA key, which had not been done before.

Still, a terabyte-size key isn’t exactly easy to work with. (The largest RSA keys right now are a few thousand bits; a terabyte is many trillions of bits.) The authors report that generating a terabyte-size RSA key and carrying out the encryption-decryption process takes about five days. “The encryption and decryption cost is terrible for most applications,” said Scott Aaronson, the director of the Quantum Information Center at the University of Texas, Austin. What’s more, the security we gain from using enormous RSA keys is “extremely precarious, vulnerable to even a modest improvement in algorithms or hardware, or a determined and well-funded-enough adversary.”

“Scott is thinking in a theoretical sense,” said Heninger, who maintains that the gap is enough “from a concrete security point of view.” “More importantly,” the paper states, “it is interesting to see that the conventional wisdom is wrong.”

The Era of Quantum Computing Is Here. Outlook: Cloudy


Quantum computers should soon be able to beat classical computers at certain basic tasks. But before they’re truly powerful, researchers have to overcome a number of fundamental roadblocks.

Quantum computers have to deal with the problem of noise, which can quickly derail any calculation.

Quantum computers have to deal with the problem of noise, which can quickly derail any calculation.

 

After decades of heavy slog with no promise of success, quantum computing is suddenly buzzing with almost feverish excitement and activity. Nearly two years ago, IBM made a quantum computer available to the world: the 5-quantum-bit (qubit) resource they now call (a little awkwardly) the IBM Q experience. That seemed more like a toy for researchers than a way of getting any serious number crunching done. But 70,000 users worldwide have registered for it, and the qubit count in this resource has now quadrupled. In the past few months, IBM and Intel have announced that they have made quantum computers with 50 and 49 qubits, respectively, and Google is thought to have one waiting in the wings. “There is a lot of energy in the community, and the recent progress is immense,” said physicist Jens Eisert of the Free University of Berlin.

There is now talk of impending “quantum supremacy”: the moment when a quantum computer can carry out a task beyond the means of today’s best classical supercomputers. That might sound absurd when you compare the bare numbers: 50 qubits versus the billions of classical bits in your laptop. But the whole point of quantum computing is that a quantum bit counts for much, much more than a classical bit. Fifty qubits has long been considered the approximate number at which quantum computing becomes capable of calculations that would take an unfeasibly long time classically. Midway through 2017, researchers at Google announced that they hoped to have demonstrated quantum supremacy by the end of the year. (When pressed for an update, a spokesperson recently said that “we hope to announce results as soon as we can, but we’re going through all the detailed work to ensure we have a solid result before we announce.”)

It would be tempting to conclude from all this that the basic problems are solved in principle and the path to a future of ubiquitous quantum computing is now just a matter of engineering. But that would be a mistake. The fundamental physics of quantum computing is far from solved and can’t be readily disentangled from its implementation.

Even if we soon pass the quantum supremacy milestone, the next year or two might be the real crunch time for whether quantum computers will revolutionize computing. There’s still everything to play for and no guarantee of reaching the big goal.

IBM’s quantum computing center at the Thomas J. Watson Research Center in Yorktown Heights, New York, holds quantum computers in large cryogenic tanks (far right) that are cooled to a fraction of a degree above absolute zero.

IBM’s quantum computing center at the Thomas J. Watson Research Center in Yorktown Heights, New York, holds quantum computers in large cryogenic tanks (far right) that are cooled to a fraction of a degree above absolute zero.

Connie Zhou for IBM

Shut Up and Compute

Both the benefits and the challenges of quantum computing are inherent in the physics that permits it. The basic story has been told many times, though not always with the nuance that quantum mechanics demands. Classical computers encode and manipulate information as strings of binary digits — 1 or 0. Quantum bits do the same, except that they may be placed in a so-called superposition of the states 1 and 0, which means that a measurement of the qubit’s state could elicit the answer 1 or 0 with some well-defined probability.

To perform a computation with many such qubits, they must all be sustained in interdependent superpositions of states — a “quantum-coherent” state, in which the qubits are said to be entangled. That way, a tweak to one qubit may influence all the others. This means that somehow computational operations on qubits count for more than they do for classical bits. The computational resources increase in simple proportion to the number of bits for a classical device, but adding an extra qubit potentially doubles the resources of a quantum computer. This is why the difference between a 5-qubit and a 50-qubit machine is so significant.

Note that I’ve not said — as it often is said — that a quantum computer has an advantage because the availability of superpositions hugely increases the number of states it can encode, relative to classical bits. Nor have I said that entanglement permits many calculations to be carried out in parallel. (Indeed, a strong degree of qubit entanglement isn’t essential.) There’s an element of truth in those descriptions — some of the time — but none captures the essence of quantum computing.

Inside one of IBM’s cryostats wired for a 50-qubit quantum system.

Inside one of IBM’s cryostats wired for a 50-qubit quantum system.

Connie Zhou for IBM

It’s hard to say qualitatively why quantum computing is so powerful precisely because it is hard to specify what quantum mechanics means at all. The equations of quantum theory certainly show that it will work: that, at least for some classes of computation such as factorization or database searches, there is tremendous speedup of the calculation. But how exactly?

Perhaps the safest way to describe quantum computing is to say that quantum mechanics somehow creates a “resource” for computation that is unavailable to classical devices. As quantum theorist Daniel Gottesman of the Perimeter Institute in Waterloo, Canada, put it, “If you have enough quantum mechanics available, in some sense, then you have speedup, and if not, you don’t.”

Some things are clear, though. To carry out a quantum computation, you need to keep all your qubits coherent. And this is very hard. Interactions of a system of quantum-coherent entities with their surrounding environment create channels through which the coherence rapidly “leaks out” in a process called decoherence. Researchers seeking to build quantum computers must stave off decoherence, which they can currently do only for a fraction of a second. That challenge gets ever greater as the number of qubits — and hence the potential to interact with the environment — increases. This is largely why, even though quantum computing was first proposed by Richard Feynman in 1982 and the theory was worked out in the early 1990s, it has taken until now to make devices that can actually perform a meaningful computation.

Quantum Errors

There’s a second fundamental reason why quantum computing is so difficult. Like just about every other process in nature, it is noisy. Random fluctuations, from heat in the qubits, say, or from fundamentally quantum-mechanical processes, will occasionally flip or randomize the state of a qubit, potentially derailing a calculation. This is a hazard in classical computing too, but it’s not hard to deal with — you just keep two or more backup copies of each bit so that a randomly flipped bit stands out as the odd one out.

Researchers working on quantum computers have created strategies for how to deal with the noise. But these strategies impose a huge debt of computational overhead — all your computing power goes to correcting errors and not to running your algorithms. “Current error rates significantly limit the lengths of computations that can be performed,” said Andrew Childs, the codirector of the Joint Center for Quantum Information and Computer Science at the University of Maryland. “We’ll have to do a lot better if we want to do something interesting.”

Andrew Childs, a quantum theorist at the University of Maryland, cautions that error rates are a fundamental concern for quantum computers.

Andrew Childs, a quantum theorist at the University of Maryland, cautions that error rates are a fundamental concern for quantum computers.

Photo by John T. Consoli/University of Maryland

A lot of research on the fundamentals of quantum computing has been devoted to error correction. Part of the difficulty stems from another of the key properties of quantum systems: Superpositions can only be sustained as long as you don’t measure the qubit’s value. If you make a measurement, the superposition collapses to a definite value: 1 or 0. So how can you find out if a qubit has an error if you don’t know what state it is in?

One ingenious scheme involves looking indirectly, by coupling the qubit to another “ancilla” qubit that doesn’t take part in the calculation but that can be probed without collapsing the state of the main qubit itself. It’s complicated to implement, though. Such solutions mean that, to construct a genuine “logical qubit” on which computation with error correction can be performed, you need many physical qubits.

How many? Quantum theorist Alán Aspuru-Guzik of Harvard University estimates that around 10,000 of today’s physical qubits would be needed to make a single logical qubit — a totally impractical number. If the qubits get much better, he said, this number could come down to a few thousand or even hundreds. Eisert is less pessimistic, saying that on the order of 800 physical qubits might already be enough, but even so he agrees that “the overhead is heavy,” and for the moment we need to find ways of coping with error-prone qubits.

An alternative to correcting errors is avoiding them or canceling out their influence: so-called error mitigation. Researchers at IBM, for example, are developing schemes for figuring out mathematically how much error is likely to have been incurred in a computation and then extrapolating the output of a computation to the “zero noise” limit.

Some researchers think that the problem of error correction will prove intractable and will prevent quantum computers from achieving the grand goals predicted for them. “The task of creating quantum error-correcting codes is harder than the task of demonstrating quantum supremacy,” said mathematician Gil Kalai of the Hebrew University of Jerusalem in Israel. And he adds that “devices without error correction are computationally very primitive, and primitive-based supremacy is not possible.” In other words, you’ll never do better than classical computers while you’ve still got errors.

Others believe the problem will be cracked eventually. According to Jay Gambetta, a quantum information scientist at IBM’s Thomas J. Watson Research Center, “Our recent experiments at IBM have demonstrated the basic elements of quantum error correction on small devices, paving the way towards larger-scale devices where qubits can reliably store quantum information for a long period of time in the presence of noise.” Even so, he admits that “a universal fault-tolerant quantum computer, which has to use logical qubits, is still a long way off.” Such developments make Childs cautiously optimistic. “I’m sure we’ll see improved experimental demonstrations of [error correction], but I think it will be quite a while before we see it used for a real computation,” he said.

Living With Errors

For the time being, quantum computers are going to be error-prone, and the question is how to live with that. At IBM, researchers are talking about “approximate quantum computing” as the way the field will look in the near term: finding ways of accommodating the noise.

This calls for algorithms that tolerate errors, getting the correct result despite them. It’s a bit like working out the outcome of an election regardless of a few wrongly counted ballot papers. “A sufficiently large and high-fidelity quantum computation should have some advantage [over a classical computation] even if it is not fully fault-tolerant,” said Gambetta.

Lucy Reading-Ikkanda/Quanta Magazine

One of the most immediate error-tolerant applications seems likely to be of more value to scientists than to the world at large: to simulate stuff at the atomic level. (This, in fact, was the motivation that led Feynman to propose quantum computing in the first place.) The equations of quantum mechanics prescribe a way to calculate the properties — such as stability and chemical reactivity — of a molecule such as a drug. But they can’t be solved classically without making lots of simplifications.

In contrast, the quantum behavior of electrons and atoms, said Childs, “is relatively close to the native behavior of a quantum computer.” So one could then construct an exact computer model of such a molecule. “Many in the community, including me, believe that quantum chemistry and materials science will be one of the first useful applications of such devices,” said Aspuru-Guzik, who has been at the forefront of efforts to push quantum computing in this direction.

Quantum simulations are proving their worth even on the very small quantum computers available so far. A team of researchers including Aspuru-Guzik has developed an algorithm that they call the variational quantum eigensolver (VQE), which can efficiently find the lowest-energy states of molecules even with noisy qubits. So far it can only handle very small molecules with few electrons, which classical computers can already simulate accurately. But the capabilities are getting better, as Gambetta and coworkers showed last Septemberwhen they used a 6-qubit device at IBM to calculate the electronic structures of molecules, including lithium hydride and beryllium hydride. The work was “a significant leap forward for the quantum regime,” according to physical chemist Markus Reiher of the Swiss Federal Institute of Technology in Zurich, Switzerland. “The use of the VQE for the simulation of small molecules is a great example of the possibility of near-term heuristic algorithms,” said Gambetta.

But even for this application, Aspuru-Guzik confesses that logical qubits with error correction will probably be needed before quantum computers truly begin to surpass classical devices. “I would be really excited when error-corrected quantum computing begins to become a reality,” he said.

“If we had more than 200 logical qubits, we could do things in quantum chemistry beyond standard approaches,” Reiher adds. “And if we had about 5,000 such qubits, then the quantum computer would be transformative in this field.”

What’s Your Volume?

Despite the challenges of reaching those goals, the fast growth of quantum computers from 5 to 50 qubits in barely more than a year has raised hopes. But we shouldn’t get too fixated on these numbers, because they tell only part of the story. What matters is not just — or even mainly — how many qubits you have, but how good they are, and how efficient your algorithms are.

Any quantum computation has to be completed before decoherence kicks in and scrambles the qubits. Typically, the groups of qubits assembled so far have decoherence times of a few microseconds. The number of logic operations you can carry out during that fleeting moment depends on how quickly the quantum gates can be switched — if this time is too slow, it really doesn’t matter how many qubits you have at your disposal. The number of gate operations needed for a calculation is called its depth: Low-depth (shallow) algorithms are more feasible than high-depth ones, but the question is whether they can be used to perform useful calculations.

What’s more, not all qubits are equally noisy. In theory it should be possible to make very low-noise qubits from so-called topological electronic states of certain materials, in which the “shape” of the electron states used for encoding binary information confers a kind of protection against random noise. Researchers at Microsoft, most prominently, are seeking such topological states in exotic quantum materials, but there’s no guarantee that they’ll be found or will be controllable.

Researchers at IBM have suggested that the power of a quantum computation on a given device be expressed as a number called the “quantum volume,” which bundles up all the relevant factors: number and connectivity of qubits, depth of algorithm, and other measures of the gate quality, such as noisiness. It’s really this quantum volume that characterizes the power of a quantum computation, and Gambetta said that the best way forward right now is to develop quantum-computational hardware that increases the available quantum volume.

This is one reason why the much vaunted notion of quantum supremacy is more slippery than it seems. The image of a 50-qubit (or so) quantum computer outperforming a state-of-the-art supercomputer sounds alluring, but it leaves a lot of questions hanging. Outperforming for which problem? How do you know the quantum computer has got the right answer if you can’t check it with a tried-and-tested classical device? And how can you be sure that the classical machine wouldn’t do better if you could find the right algorithm?

So quantum supremacy is a concept to handle with care. Some researchers prefer now to talk about “quantum advantage,” which refers to the speedup that quantum devices offer without making definitive claims about what is best. An aversion to the word “supremacy” has also arisen because of the racial and political implications.

Whatever you choose to call it, a demonstration that quantum computers can do things beyond current classical means would be psychologically significant for the field. “Demonstrating an unambiguous quantum advantage will be an important milestone,” said Eisert — it would prove that quantum computers really can extend what is technologically possible.

That might still be more of a symbolic gesture than a transformation in useful computing resources. But such things may matter, because if quantum computing is going to succeed, it won’t be simply by the likes of IBM and Google suddenly offering their classy new machines for sale. Rather, it’ll happen through an interactive and perhaps messy collaboration between developers and users, and the skill set will evolve in the latter only if they have sufficient faith that the effort is worth it. This is why both IBM and Google are keen to make their devices available as soon as they’re ready. As well as a 16-qubit IBM Q experience offered to anyone who registers online, IBM now has a 20-qubit version for corporate clients, including JP Morgan Chase, Daimler, Honda, Samsung and the University of Oxford. Not only will that help clients discover what’s in it for them; it should create a quantum-literate community of programmers who will devise resources and solve problems beyond what any individual company could muster.

“For quantum computing to take traction and blossom, we must enable the world to use and to learn it,” said Gambetta. “This period is for the world of scientists and industry to focus on getting quantum-ready.”

This New Device Enables Quantum Computers to Calculate Much More Complex Questions that Today’s Computers Can’t Handle


Princeton University scientist have built a device for Quantum computers in which a single electron can pass its quantum information to a particle of light. This will allow silicon-based quantum computers to be built and let them solve much more complicated problems.

The future of computers lies in quantum computing. No longer will we be working with 1’s and 0’s, but qubits that are capable of being both 1 and 0 at the same time. In terms of capability, this means that a quantum computer is able to calculate much more complex problems simultaneously, unlike your standard computer.

But, as great as that all sounds, it’s not easy programming a quantum computer to be able to solve these difficult equations. However, one recent study may have just made the process a little easier. The involved study scientists at Princeton University and basically consisted of the group creating a device that was able to pass a quantum state from particle to particle. One of the researchers involved in the study and graduate student in Princeton’s Department of Physics, Xiao Mi, said, “We now have the ability actually to transmit the quantum state to a photon. This has never been done before in a semiconductor device because the quantum state was lost before it could transfer its information.”

The qubit consists of a single electron that is trapped below the surface of a silicon chip (gray). The green, pink and purple wires on top of the silicon structure deliver precise voltages to the qubit. The purple plate reduces electronic interference that can destroy the qubit’s quantum information. By adjusting the voltages in the wires, the researchers can trap a single electron in a double quantum dot and adjust its energy so that it can communicate its quantum information to a nearby photon. 
A fully packaged device for trapping and manipulating single electrons and photons. A series of on-chip electrodes (lower left and upper right) lead to the formation of a double quantum dot that confines a single electron below the surface of the chip. The photon, which is free to move within the full 7-millimeter span of the cavity, exchanges quantum information with the electron inside the double quantum dot.

The whole research project went on for five years with details of the passing of information from an electron to a photon disclosed just recently. The device used to do this was constructed by HRL Laboratories and is owned by General Motors and Boeing. Its simplistic design uses tow commonly used materials: silicon and silicon-germanium. Made from layers of silicon and nanowires placed at the top, these chips are great for transferring quantum information across. Photons were used as the medium for exchanging quantum information between electrons as they’re less sensitive to disruption and also have the ability to travel well through optic cables and circuits that could be used in a quantum supercomputer.

Jason Petta is a professor of physics at Princeton, and he says, “Just like in human interactions, to have good communication a number of things need to work out – it helps to speak the same language and so forth. We can bring the energy of the electronic state into resonance with the light particle so that the two can talk to each other.” So, hopefully, it won’t be too many more years we have to wait to see a quantum computer, but for now, it’s not quite there yet.

Now Quantum Computers Can Send Information Using a Single Particle of Light


Physicists at Princeton University have revealed a device they’ve created that will allow a single electron to transfer its quantum information to a photon. This is a revolutionary breakthrough for the team as it gets them one step closer to producing the ultimate quantum computer. The device is the result of five years worth of research and could accelerate the world of quantum computing no end.

Xiao Mi, a graduate student in Princeton’s Department of Physics, says, “We now have the ability to transmit the quantum state to a photon. This has never been done in a semiconductor device because the quantum state was lost before it could transfer its information.” The way in which the device works is by trapping an electron and photon within a device built by HRL Laboratories and owned by Boeing and General Motors. The semiconducting chip is made up of several layers of silicon and silicon-germanium, and across the top are a number of nanowires that are used to deliver energy to the chip.

Quantum computing brings with it a whole range of advantages; the main one being its capability to calculate many problems simultaneously resulting in much faster computing than anything we’re used to. Jason Petta, professor of physics at Princeton, said, “In our device, the state of the qubit is encoded in the position of the electron. The electron is trapped in a double well potential where the electron can occupy the left well, the right well, or be in a superposition state: both left and right at the same time. The information is therefore stored in the position of a single electron.”

The researchers discovered the best way for them to pass information between electrons without destroying them in the process was to use photons as they’re less sensitive to environmental factors and would be perfect for helping to form the circuits for a quantum computer. Petta goes on to say,” Just like in human interactions, to have good communication a number of things need to work out – it helps to speak the same language and so forth. We are able to bring the energy of the electronic state into resonance with the light particle so that the two can talk to each other.” The researchers will continue to do further work with the device to fine tune it to allow greater control over the transferring of information between qubits.

Quantum Computers Could Crush Today’s Top Encryption in 15 Years


Quantum computers could bring about a quantum leap in processing power, with countless benefits for fields like data science and AI. But there’s also a dark side: this extra power will make it simple to crack the encryption keeping everything from our emails to our online banking secure.

A recent report from the Global Risk Institute predicted that there is a one in seven chance vital cryptography tools will be rendered useless by 2026, rising to a 50% chance by 2031. In the meantime, hackers and spies can hoover up data encrypted using current approaches and simply wait until quantum computers powerful enough to crack the code have been developed.

quantum-computers-encryption-7The threat to encryption from quantum computers stems from the fact that some of the most prevalent approaches rely on solving fiendishly complicated mathematical problems. Unfortunately, this is something quantum computers are expected to be incredibly good at.

While traditional computers use binary systems with bits that can either be represented as 0 or 1, a quantum bit—or “qubit”—can be simultaneously 0 and 1 thanks to a phenomenon known as superposition. As you add qubits to the systems this means the power of the computer grows exponentially, making quantum computers far more efficient.

In 1994 Peter Shor of Bell Laboratories created a quantum algorithm that can solve a problem called integer factorization. As a report from the National Institute of Standards and Technology (NIST) released in April notes, this algorithm can be used to efficiently solve the mathematical problems at the heart of three of the most widely-used encryption approaches: Diffie-Hellman key exchange, RSA, and elliptic curve cryptography.

The threat is not imminent, though; building quantum computers is difficult. Most designs rely on complex and expensive technology like superconductors, lasers and cryogenics and have yet to make it out of the lab. Google, IBM and Microsoft are all working on commercializing the technology. Canadian company D-Wave is already selling quantum computers, but capabilities are still limited.

The very laws of quantum mechanics that makes these computers so powerful also provide a way to circumvent the danger. Quantum cryptography uses qubits in the form of photons to transmit information securely by encoding it into the particles’ quantum states. Attempting to measure any property of a quantum state will alter another property, which means attempts to intercept and read the message can be easily detected by the recipient.

quantum-computers-encryption-2The most promising application of this approach is called quantum key distribution, which uses quantum communication to securely share keys that can be used to decrypt messages sent over conventional networks. City-wide networks have already been demonstrated in the US, Europe and Japan, and China’s newest satellite is quantum communication-enabled.

But the systems are held back by low bandwidth and the fact they only work over short distances. China is trying to build a 2,000km-long quantum network between Shanghai and Beijing, but this will require 32 “trusted nodes” to decode the key and retransmit it, introducing complexity and potential weaknesses to the system.

There’s also no guarantee quantum communication will be widely adopted by the time encryption-cracking quantum computers become viable. And importantly, building a single powerful encryption-busting quantum computer would require considerably less resources than restructuring entire communication networks to accommodate quantum cryptography.

Fortunately, there are other approaches to the problem that do not rely on quantum physics. So-called symmetric-key algorithms are likely to be resistant to quantum attacks if the key lengths are doubled, and new approaches like lattice-based, code-based and multi-variate cryptography all look likely to be uncrackable by quantum computers.

Symmetric-keys only work in a limited number of applications, though, and the other methods are still at the research stage. On the back of its report the NIST announced that it would launch a public competition to help drive development of these new approaches. It also recommends organizations focus on “crypto agility” so they can easily swap out their encryption systems as quantum-hardened ones become available.

But the document also highlighted the fact that it has taken roughly 20 years to deploy our current cryptography infrastructure. Just a month before the release of the report, researchers from MIT and the University of Innsbruck in Austria demonstrated a five-atom quantum computer capable of running Shor’s algorithm to factor the number 15.

Crucially, their approach is readily scalable, which the team says means building a more powerful quantum computer is now an engineering challenge rather than a conceptual one. Needless to say, the race is on.

New chip architecture may provide foundation for quantum computer


New chip architecture may provide foundation for quantum computer
A photograph of the completed BGA trap assembly. The trap chip is at the center, sitting atop the larger interposer chip that fans out the wiring. The trap chip surface area is 1mm x 3mm, while the interposer is roughly 1 cm square. Credit: D. Youngner, Honeywell

Quantum computers are in theory capable of simulating the interactions of molecules at a level of detail far beyond the capabilities of even the largest supercomputers today. Such simulations could revolutionize chemistry, biology and material science, but the development of quantum computers has been limited by the ability to increase the number of quantum bits, or qubits, that encode, store and access large amounts of data.

In a paper appearing this week in the Journal of Applied Physics, from AIP Publishing, a team of researchers at Georgia Tech Research Institute and Honeywell International have demonstrated a new device that allows more electrodes to be placed on a chip—an important step that could help increase qubit densities and bring us one step closer to a quantum computer that can simulate molecules or perform other algorithms of interest.

“To write down the quantum state of a system of just 300 qubits, you would need 2^300 numbers, roughly the number of protons in the known universe, so no amount of Moore’s Law scaling will ever make it possible for a classical computer to process that many numbers,” said Nicholas Guise, who led the research. “This is why it’s impossible to fully simulate even a modest sized quantum system, let alone something like chemistry of complex molecules, unless we can build a quantum computer to do it.”

While existing computers use classical bits of information, quantum computers use “” or qubits to store information. Classical bits use either a 0 or 1, but a qubit, exploiting a weird quantum property called superposition, can actually be in both 0 and 1 simultaneously, allowing much more information to be encoded. Since qubits can be correlated with each other in a way that classical bits cannot, they allow a new sort of massively parallel computation, but only if many qubits at a time can be produced and controlled. The challenge that the field has faced is scaling this technology up, much like moving from the first transistors to the first computers.

New chip architecture may provide foundation for quantum computer
Fluorescence images of calcium ions confined in the BGA trap. 

Creating the Building Blocks for Quantum Computing

One leading qubit candidate is individual ions trapped inside a vacuum chamber and manipulated with lasers. The scalability of current trap architectures is limited since the connections for the electrodes needed to generate the trapping fields come at the edge of the chip, and their number are therefore limited by the chip perimeter.

The GTRI/Honeywell approach uses new microfabrication techniques that allow more electrodes to fit onto the chip while preserving the laser access needed.

The team’s design borrows ideas from a type of packaging called a ball grid array (BGA) that is used to mount integrated circuits. The ball grid array’s key feature is that it can bring electrical signals directly from the backside of the mount to the surface, thus increasing the potential density of electrical connections.

The researchers also freed up more chip space by replacing area-intensive surface or edge capacitors with trench capacitors and strategically moving wire connections.

New chip architecture may provide foundation for quantum computer
SEMs of the trench capacitor and TSV (thru-substrate via) structures fabricated into the trap chip. These make electrical connections to the trap electrodes while filtering out RF pickup.
The space-saving moves allowed tight focusing of an addressing laser beam for fast operations on single qubits. Despite early difficulties bonding the chips, a solution was developed in collaboration with Honeywell, and the device was trapping ions from the very first day.

The team was excited with the results. “Ions are very sensitive to stray electric fields and other noise sources, and a few microns of the wrong material in the wrong place can ruin a trap. But when we ran the BGA trap through a series of benchmarking tests we were pleasantly surprised that it performed at least as well as all our previous traps,” Guise said.

Working with trapped ion currently requires a room full of bulky equipment and several graduate students to make it all run properly, so the researchers say much work remains to be done to shrink the technology. The BGA project demonstrated that it’s possible to fit more and more electrodes on a surface trap chip while wiring them from the back of the chip in a compact and extensible way. However, there are a host of engineering challenges that still need to be addressed to turn this into a miniaturized, robust and nicely packaged system that would enable , the researchers say.

In the meantime, these advances have applications beyond quantum computing. “We all hope that someday quantum computers will fulfill their vast promise, and this research gets us one step closer to that,” Guise said. “But another reason that we work on such difficult problems is that it forces us to come up with solutions that may be useful elsewhere. For example, like those demonstrated here for ion traps are also very relevant for making miniature atomic devices like sensors, magnetometers and chip-scale atomic clocks.”