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Topological quantum computation – from Wilczek’s dream to reality
Frank Wilczek, a professor at MIT, is both a genius and a lucky physicist. He won the Nobel Prize in Physics at the age of 52 for a paper he wrote at the age of 22 while a graduate student at Princeton. In 1982, at a time when it was considered a God-given truth among physicists that there are only two kinds of particles in the universe: bosons and fermions, he had a brilliant idea that ran counter to this widely-held belief. He argued that, if there were particles that could move only in a two-dimensional plane rather than the three-dimensional world we are living in, the those particles wouldn’t necessarily be bosons or fermions. Wilczek called this hypothetical new particle an anyon. Just as Samsung’s AnyCall was named to emphasize its ability to make calls anywhere, the “any” in the name of the new particle suggests that it can be anything, not just bosons or fermions 1.
Such a particle could easily have been nothing more than a figment of Wilczek’s imagination, however brilliant it may have been, in a way that the wonderful creature of a dragon can only be found in fairy tales. Much to everybody’s surprise including (I am sure) Wilczek’s own, the anyon was confirmed as a real particle as soon as Wilczek’s idea was published, when fractional quantum Hall states were discovered by condensed matter physicists in the laboratory. When a very strong magnetic field is applied perpendicularly to a very thin layer of two-dimensional electrons trapped at the interface between two solids, like jam in a sandwich, the motion of the electrons changes dramatically from rectilinear to circular, creating what is called a quantum Hall state. Basically, the playground where children used to run around randomly turns into a dance floor where people execute polished circular motion around the other dancers. More precisely, there are integer quantum Hall states where the electrons dance like amateurs, and fractional quantum Hall states, where the electronic dance parallels that of seasoned dancers expertly swinging other equally competent dancers. Among the two, it is the fractional quantum Hall states that interest us.
A well-coordinated group dance can give the illusion of watching a polished choreography performed by a single professional. A collection of particles has the similar ability to create ‘quasiparticles’ that behave like a single particle, while in fact it is a consequence of a large collection of underlying particles moving together in a coordinated fashion, perhaps like an army of highly coordinated ants. In fractional quantum Hall states, there are quasiparticles that behave as if an electron had been split into three pieces, each with the charge only one-third of the electron’s charge. This is why such a state is called a fractional quantum Hall state, because an electron appears to have been split into several fractional charges. Quasiparticles with such fractionalized charges turn out to have exactly the properties of anyons predicted by Wilczek. The theoretical paper in which Wilczek imagined the anyon was published in April 1982, and the experimental paper in which Dan Tsui and Horst Stormer discovered the existence of the fractional quantum Hall state was published in May of that year. A stroke of Wilczek’s imagination proved to be real in just a month, in a highly two-dimensional system synthesized by material scientists. Wilczek’s anyon is an abelian anyon, so named after a mathematical theory created by a famed Norwegian mathematician, Niels Henrik Abel. Like Agent Smith in the movie The Matrix that can be replicated without a fault, anyons are totally indistinguishable from each other that no one would notice if the positions of two anyons were swapped.
Once Wilczek’s fantasy of a new particle became a reality, theoretical research into anyons entered an explosive stage, and by 1991, a novel proposal was made that particles even more bizarre than Wilczek’s abelian anyon could exist in two dimensions. Greg Moore and Nicholas Read of Yale University and Xiao-Gang Wen of the Massachusetts Institute of Technology, among others, proposed a particle called the non-abelian anyon. They showed that if you swap the positions of two non-abelian anyons, the state after the swap can be different from the state before, even if all the anyons appear to be identical. This resulted in a simple but important insight: knowing the position of a particle does not tell you everything about the properties of the whole system. In a world of non-abelian anyons, if someone switches the positions of two non-abelian anyons while you’re not looking, you’d still be able to detect the swap had occurred. This is because swapping the positions of non-abelian anyons creates a new state, not the original state. Deep down, this is due to the fact that non-abelian anyons have information other than their “positions”. When two particles swap places, their positional information remains the same, but other information changes, so the fact that they swapped places can no longer be hidden. If we number the N non-abelian anyons from 1 to N, and start exchanging anyons 1 and 2, followed by the swap of 2 and 3, then anyons 4 and 6, and so on, we will keep creating new states unlike any of the previous states. The number of different states that can be created using all possible exchange processes grows exponentially with the number N of anyons. It was predicted in Moore-Read’s paper that certain fractional quantum Hall states will indeed possess non-abelian anyons as quasiparticles 2.
The non-abelian anyons predicted in Moore-Reed theory has yet to find conclusive experimental evidence of their existence. Nevertheless, the wildly fascinating prophecy of non-abelian anyons has been steadily gaining traction in a totally different field – that of quantum computing. This is thanks to the brilliant idea of topological quantum computation, based on using non-abelian anyons as tools of computation, quantum computation nonetheless. Topological quantum computation is a field championed by Alexei Kitaev in the late 1990s 3,4 and whose mathematical foundations were laid by Fields medalist Michael Freedman and others 5. As recognition spread that non-abelian particles could be the fundamental building blocks of quantum computers, Microsoft began investing in the possibility of creating a quantum computer using fractional quantum Hall states, which were a prime candidate for realizing non-abelian particles at the time, and created Station Q to spearhead its effort inside a building located at the University in California at Santa Barbara.
The effort to implement quantum computers through quantum Hall materials has what one might call a `control issue’. Quantum Hall matter, after all, consists of a lot of electrons. A small number of well-behaved non-abelian anyons are then created as quasiparticles out of these electrons, and then one applies carefully chosen controls over these quasiparticles to perform desired quantum operatitons. When all is said and done to perfection, one can hand over the quantum computer to Microsoft for commercialization. Before anything remotely analogous to a practical quantum computer can be built, one must face the fact that the underlying electrons are constantly executing a dancing motion and are in what physicists call the strongly interacting regime, where electrons jostle with other electrons with all the might of the Coulomb force. Exercising total control over these `mighty’ electrons is not easy. After all, electrons are the tiniest and the lightest of all the particles that we see in a laboratory. Maybe because of that, Microsoft has yet to make tangible breakthroughs in making a quantum computer.
To understand the relationship between non-abelian particles and quantum computers, we must first understand the basic elements that make a regular computer work: bits. A computer is literally a machine that calculates. They are machines that can perform complex mathematical calculations through simple programs much faster than the human brain can. Humans use neurons in their brains to make calculations, but computers use tiny pieces of materials (called devices) to make calculations. First, it converts all mathematical operations into operations based on two binary numbers, 0 and 1. Then we make sure that there are only two physical states in very small devices typically made with silicon. These tiny devices are called bits. The process of changing the state of a bit from 0 to 1, or 1 to 0, by passing electricity through it, is a mathematical operation that can be performed on a large scale and at high speed.
For example, let’s say there are five bits and they are in a state |00110>. This mean that the first bit is in state 0, the second bit in state 0, and so on. Replacing this state with another state, say |01010>, is the kind of operation that computers do all the time. In the example here, the states of the second and third bits are reversed. This manipulation is called a gate operation. A computer “computes” by constantly applying a few simple kinds of gate operations to keep changing the state of bits. One might compare it to getting a crowd of people together and making all sorts of patterns out of card sections. Just as humans make mistakes, sometimes a bit that should be a zero will accidentally become a one. When this happens, the error must be detected and corrected. Error correction is a process that’s happening constantly in our computers, and thanks to it, we can safely rely on computers to perform their tasks.
Quantum computation uses qubits instead of bits. The information represented by a qubit is not a binary number of 0 or 1, but rather a complex number that combines both real and imaginary numbers. There are only two binary numbers, 0 and 1, but there are an infinite variety of real and imaginary numbers, including 0, 0.1, 0.01……, which cannot even be counted. Furthermore, the states of a qubit can be superposed like a Schrödinger’s cat, which is neither alive nor dead, A qubit, as a non-biological analogue of Schrödinger’s cat, can be in a superposition of two possible states (dead-state and alive-state). With two qubits, the possibilities are dead-dead, dead-alive, alive-dead, alive-alive, a total of four states. With N qubits, the possibilities would have grown exponentially to 2x2x…x2=2^n different states. Being quantum means that the N-qubit state can be in a superposition among these exponentially many possibilities!
There are 32 possible states for five bits, ranging from |00000> to |11111>, and a typical computer can only be in one particular state out of the thirty-two. A computer’s computation consists of converting from one possible bit state to another. On the other hand, a five-qubit quantum computer can easily exist in a superposition of these thirty-two bit states. The superposition, reminiscent of Schrödinger’s cat, means that the quantum computer can be in this state or that state. A 5-qubit state is a superposition of all possible 5-bit states. To specify a 5-qubit state, we need to specify the probability of being in each 5-bit state (32 in total). The problem is that this probability is not a binary number like 0 or 1, but a real number that can take on any value between 0 and 1. This flexibility is the source of the computational power of quantum computers, but it’s also what makes them so difficult to build and operate. If you give a probability value of 0.2 when it should be 0.1, the quantum computer already contains an error. This is a much harder problem than correcting for a bit value of 1 when it should be 0. A quantum computer made of ten qubits must always be able to control all 2^10=1024 numbers representing the probabilities of that particular state occurring in a superposition. As the number of qubits increases, the number of probabilities to be placed under control grows exponentially. The ultimate problem, unique to the quantum computer, is that these numbers tend to change spontaneously, in a phenomenon called decoherence. Without a well-crafted plan to prevent decoherence, a quantum computer is but a useless machine that works for a short time and then breaks down.
There is a way to avoid this difficulty once and for all. It’s called topological quantum computation. The idea is that the probability of error is reduced by replacing an individual mistake with that of a group. For example, imagine ten people holding hands and playing a human merry-go-round. Let’s call it 0 when they turn clockwise and 1 when they turn counterclockwise. With ten people implementing the bit state instead of one person, there’s less room for error. Furthermore, the direction of rotation is not continuous. There are only two directions: clockwise or counterclockwise. In this setting bits can be implemented robustly and are less prone to error. Topological quantum computation is a way to mobilize many qubits to create a single collective qubit state. While it is cumbersome to mobilize a large number of qubits, it provides stability against various errors once it is created.
Bit-based operations take place in the space of binary numbers. It’s a mathematical space with such a simple structure that it’s hard to imagine anything simpler. In order for a computer to perform computations properly, it must be able to physically realize every possible state in this space of binary numbers. On the other hand, the mathematical space in which quantum operations are performed is called Hilbert space. For quantum computation to work well, it must be possible to represent every quantum state in a Hilbert space by qubits. If there are N non-abelian anyons, the size of the Hilbert space they create grows exponentially with N. For quantum computation to work well in this Hilbert space, we need to have access to an exponential number of all the states. The starting point for topological quantum computation is the mathematical proof that when there are N non-abelian anyons, all states of the Hilbert space can be realized by simply swapping the positions of these anyons. Microsoft wants to create quantum computers by creating non-abelian particles in quantum Hall matter and then manipulating them to swap their positions.
A non-abelian particle is a particle that remembers its past. Computations in an ordinary computer is accomplished through the physical process of changing the state of bits from 0 to 1 and 1 to 0. In topological quantum computer, we swap the positions of non-abelian anyons. The exact orbit involved in the swapping doesn’t matter. The only information that matters is whether the winding has taken place or not, and how many times. Even if there is some marginal error in the computation, it’s hard to undo the fact that winding has taken place. This kind of robustness is the reason why topological quantum computation is viewed with great hope for quantum computing.
The brilliance of Kitaev’s and other quantum theorists’ insight is that we don’t have to rely on nature’s gifts such as quantum Hall materials to create structures suitable for quantum computation. Rather, we can create quantum platforms, also called quantum circuits, artificially. For example, qubits are made using something called a Josephson device, which is based on superconducting materials. There are well-developed processes for fabricating Josephson devices in large quantities. It is possible to make Josephson devices that are about 0.1-0.2 microns across on a substrate, and then use electromagnetic waves to manipulate the quantum state of each Josephson device individually. The quantum computers being promoted by Google and IBM are based on Josephson devices and are currently recognized as one of the most high-performance quantum computers. Another way to build a quantum computer is with atoms. When one electron is removed from an atom, it becomes an ion, and unlike atoms, which are electrically neutral, ions have a charge, making it easier to control their position using an electric field. Using this principle, dozens of atoms are arranged in a vacuum, and then lasers are used to control the state of the ions, perform gate operations, and do some measurements. Ion-Q, co-founded Kim Jung-Sang at Duke University, and quantinuum, a subsidiary of the industrial giant Honeywell, are among the companies that have succeeded in creating ion-based quantum computers. Both approaches have resulted in quantum computers operating on the order of a few tens of qubits. These days the various systems used to build a quantum computer is called a quantum platforms. Josephson junctions, ions, and Rydberg atoms are among the most popular quantum platforms now in use and under rapid development these days.
The next step is to create non-abelian anyons on a quantum platform. As early as 1997, Kitaev proved that creating a state called a toric code on a quantum platform would create an abelian anyon as a quasiparticle, and creating a state called a quantum double would create non-abelian anyons as quasiparticles. This was a landmark discovery. Recall that the discovery of the abelian anyons in the fractional quantum Hall system in 1982 was, after all, a miraculous gift from nature totally unexpected to human ingenuity. Now, Kitaev’s proof showed us how to engineer and artificially create anyons, whether abelian or non-abelian, through pure human ingenuity. In 2022, using Josephson junctions and Rydberg atoms as quantum platforms, scientists have created the toric code in the lab and confirmed the existence of abelian anyons.
Unfortunately, the abelian anyons produced by toric code are insufficient to perform topological quantum operations. Because of their “commutativity” – the fact that you can swap the positions of two abelian anyons and still get the same quantum state – they are not capable of producing all of the rich Hilbert space that quantum computation requires. For the purposes of a topological quantum computer, we need to find a way to create non-abelian anyons. However, implementing a non-abelian anyon state on a quantum platform is much more challenging than their abelian counterpart, so much so that one has to find a shortcut to reach that lofty goal.
One conceivable shortcut is to create an abelian anyon and then apply additional manipulations to turn it into a non-abelian anyon. There is a mathematical proof that an abelian anyon behaves like a non-abelian anyon if the qubits are intentionally arranged on a squashed lattice instead of a perfectly regular grid of qubits. An experimental implementation of this proof was published by Google in 2023 6.
Another shortcut is to reduce the time required to create non-abelian anyons. To create an abelian anyon, the initial unentangled quantum state must be converted to a topological state, and the number of quantum gate operations involved in this conversion process grows with how many qubits there are in a given quantum platform. The more qubits you have, the more gate operations you need to perform, and the greater the chance of errors in the computation. Instead of tackling this challenge head-on, there is a clever way around it. It’s a hybrid method that involves making strategically planned measurements on a quantum platform and then using the results of those measurements to determine how to perform the next phase of quantum gate operations. If this hybrid method can be implemented, the number of quantum gate operations required to create a topologically ordered state can be dramatically reduced.
If you want to build non-abelian anyons from scratch, instead of building first the abelian anyons and then converting them to non-abelian anyons as Google did, you need to use this technique of measurement. It was commonly believed that quantum states collapse once they have been measured, and that measurement is detrimental to quantum computation. However, a methodology for measurement-based quantum computation that directly contradicts this common conception has been around for 20 years. Strategically applying a well-intended measurement to a quantum state creates a new quantum state. And then the next round of measurements gives us another quantum state. The proposal was that these ever-changing quantum landscape painted by measurements could eventually perform all the quantum operations we ever wanted. One of the valuable byproducts of this proposal is that it is now possible to create topological states from non-topological states through measurement. It’s not just possible, it’s easy – at least in theory.
In 2022, Ashvin Vishwanath, a brilliant theoretical condensed matter physicist at Harvard in collaboration with a formidable team of young collaborators, developed a sophisticated theory of how to build non-abelian anyons by strategically using measurements. Fortunately for the theorists, quantum engineers at Quantinuum had just finished building an ion-based quantum computer powerful enough to create those non-abelian anyons using Vishwanath’s theory. As soon as the Quantinuum mustered the ability to manipulate 27 ions in their device, they put the Harvard researchers’ proposal to the test, and in 2023 they finally succeeded in creating three non-abelian anyons and braiding them 7.
It’s been almost a century since 1925, when the Schrödinger equation, the foundation of quantum mechanics, was created. For a very long time, quantum wave functions were deemed a tool for describing atoms and the world beneath the atomic scale, in short the natural world. In quantum computation, we have a new perspective on these wave functions. Here they serve as devices for computation. The quantum wave functions of superconducting qubits, or atomic qubits are not the naturally occurring wave functions like those of a hydrogen atom, but are created and manipulated by human intent and control. Their intent is not to understand nature, but to manipulate certain quantum operation in a quantum computer. This is a beautiful instance of great physical laws discovered in one era gradually become the powerful tool of engineering for the next.
The successful creation and exchange of a non-abelian anyons in 2023 is nothing more than a baby’s first step toward the grand goal of topological quantum computation. However, as a thousand-mile journey must begin with a single step, it’s exciting to think about how a baby who has started walking now will grow up in 40 years. One can draw inspiration from the first experiment demonstrating the quantum entanglement of two particles that took place in 1980, and how it took some 40 years for the work to be recognized with Nobel prize and the foundation of what we now call the quantum engineering. If all the promises that we see these days of the quantum computer are to be fulfilled some 40 years from now, such world would be imaginatio transcendit (beyond recognition).
[You can find the Korean version of this article.]
1. Magnetic flux, angular momentum, and statistics, Frank Wilczek, Physical Review Letters 48, 1144 (1982)
2. Nonabelions in the fractional quantum Hall effect, Gregory Moore and Nicholas Read, Nuclear Physics B 360, 362 (1991)
3. Fault-tolerant quantum computation by anyons, Alexei Kitaev, Annals of Physics, 303, 2 (2003)
4. Anyons in an exactly solvable model and beyond, Alexei Kitaev, Annals of Physics, 321, 2 (2006)
5. Non-Abelian anyons and topological quantum computation, Chetan Nayak, Steven Simon, Ady Stern, Michael Freedman, Sankar das Sarma, Reviews of Modern Physics 80, 1083 (2008)
6. Non-Abelian braiding of graph vertices in a superconducting processor Authors: Google Quantum AI and Collaborators, Nature 618, 264 (2023).
7. Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor, Mohsin Iqbal et al. arXiv:2305.03766