Aaron Lauda has been exploring an area of mathematics that most physicists have seen little use for, wondering if it might have practical applications. In a twist even he didn’t expect, it turns out that this kind of math could be the key to overcoming a long-standing obstacle in quantum computing—and maybe even for understanding the quantum world in a whole new way.
Quantum computers, which harness the peculiarities of quantum physics for gains in speed and computing ability over classical machines, may one day revolutionize technology. For now, though, that dream is out of reach. One reason is that qubits, the building blocks of quantum computers, are unstable and can easily be disturbed by environmental noise. In theory, a sturdier option exists: topological qubits spread information out over a wider area than regular qubits. Yet in practice, they’ve been difficult to realize. So far, the machines that do manage to use them aren’t universal, meaning they cannot do everything full-scale quantum computers can do. “It’s like trying to type a message on a keyboard with only half the keys,” Lauda says. “Our work fills in the missing keys.” He and his group at the University of Southern California published their findings in a new paper in the journal Nature Communications.
Lauda and his colleagues solve some of the problems with topological qubits by using a class of theoretical particles they call neglectons, named for how they were derived from overlooked theoretical math. These particles could open a new pathway toward experimentally realizing universal topological quantum computers.
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Unlike ordinary qubits, which store information in the state of a single particle, topological qubits store it in the arrangement of several particles—which is a global property, not a local one, making them far more robust.
Take, for example, braided hair. The type and number of braids that a person has are global properties that remain the same regardless of how they shake their head. In contrast, the position of an individual hair strand is a local property that can shift with the slightest movement.
Aaron Lauda’s mathematical notation for his research study “Universal quantum computation using Ising anyons from a non-semisimple topological quantum field theory” on a chalkboard.
Topological qubits work on a similar principle known as anyon braiding. Anyons are quasiparticles—not actual particles like protons, for instance, but rather emergent phenomena from the collective behavior of many particles, like ripples in a pond. They appear in two-dimensional quantum systems.
In our three-dimensional world, swapping two particles is like weaving one string over or under the other. You can always unweave them back to their original structure. When you swap particles in two dimensions, however, you cannot go over or under; you have to make the strings go through each other, which permanently changes the structure of the strings.
Because of this property, swapping two anyons can completely transform the state of a system. These swaps can be repeated among multiple anyons—a process called anyon braiding. The final state depends on the order in which the swaps, or braids, are formed, much like the way the pattern of a braid depends on the sequence of its strands.
Because braiding anyons changes the quantum state of the qubit, the procedure can be used as a quantum gate. Just as a logical gate in a regular computer changes bits from 0 to 1 to allow computation, quantum gates manipulate qubits. This braid-based logic is the foundation of how topological quantum computers compute.
Theoretically, many types of anyons exist. One variety, called Ising anyons, “are our best chance for quantum computing in real systems,” Lauda says. “However, by themselves, they are not universal for quantum computation.”
Picture a qubit as a number on a calculator display and the quantum gates as the buttons on the calculator. A nonuniversal computer is like a calculator that only has buttons for doubling or halving. You can reach plenty of numbers—but not all of them, which limits your computing power. A universal quantum computer would be able to reach all numbers.
Most experimentalists make Ising computers universal by using a special state of Ising anyons. But this state, like a single unbraided hair strand, isn’t protected by global topological properties, making it vulnerable to errors and therefore undermining the main advantage of using Ising anyons.
Lauda’s team found a different way to make an Ising computer universal by introducing a new kind of anyon, the neglecton. It emerges from a broader mathematical framework called nonsemisimple topological quantum field theory, which changes how certain “negligible” components are counted. For years, these components were discarded because they could cause nonsensical behavior, resulting in probabilities that sum to more than one or dip below zero, or other outcomes that make no physical sense. By finding a way to make sense of them instead of discarding them, Lauda’s team unlocked an unexplored area of quantum theory.
It’s a shift that evokes the early days of imaginary numbers, which are numbers built on negative square roots. They were originally just a mathematical trick with no physical meaning—until Erwin Schrödinger used them in the wave equation that became a cornerstone of quantum mechanics. “This is similar,” says Eric Rowell, a mathematician at Texas A&M University, who was not involved in the work. “It’s like there’s another door we hadn’t pursued because we couldn’t see it as physical. Maybe it needs to be opened now.”
“In the world of topology, this idea turned out to be very powerful,” Lauda says. It was like looking into quantum theory with a magnifying glass. In Lauda’s design, the neglecton stays stationary while the other anyons braid around it. This setup introduces a new gate that makes the quantum computer universal. In the calculator picture of qubit states, this gate acts like adding or subtracting 1; over time, the process can arrive at all numbers, unlike the nonuniversal version of the calculator.
The catch is that adding a neglecton risks pushing everything into unphysical territory, in which probabilities stop adding up the way they should. “There’s this much larger theory,” Lauda says, “and sitting inside it, there’s a place where everything physically makes sense.” It’s like when you wander off the map in a video game—the game starts glitching, you can walk through walls, and all the rules break down. The trick is to build an algorithm that keeps the player safely inside the map. That job fell to Lauda’s graduate student, Filippo Iulianelli, who reworked an algorithm he’d encountered in a recent class.
The next hurdle is finding a real-world version of this system; the neglecton remains entirely hypothetical for now. Lauda is optimistic. In the 1930s physicists used mathematical symmetries to predict the existence of a strange subatomic particle—the meson—years before experiments confirmed it. “We’re not claiming we’re in the same situation,” he says, “but our work gives experimentalists a target to look for in the same systems that are realizing Ising anyons.”
Shawn Cui, a mathematician at Purdue University who peer-reviewed the new paper, calls the research “very exciting theoretical progress” and hopes to see studies exploring physical systems where such anyons might emerge. Rowell agrees, and he suggests that the neglecton could arise from some interaction between an Ising system and its environment. “Maybe there’s just a little bit of extra engineering needed to construct this neglecton,” he says.
For Lauda, the implementation is only part of the excitement. “My goal is to make as compelling a case as possible to other researchers that the nonsemisimple framework is not just valid but an exciting approach to better understanding quantum theory,” he says. The neglecton is unlikely to be neglected for much longer.
