Utah State University mathematicians Nathan Geer and Matthew Young, along with their students and collaborators throughout the world, revel in exploring the exquisite and puzzling intricacies of mathematical theory. Some 14 years ago, Geer and colleagues developed non-semisimple topological quantum field theories (TQFTs), a mathematical framework to study a wide range of physical theories, which have blossomed into continuing research.
“We did this purely from a mathematical perspective — simply guided by the internal logic and beauty of the subject, following where the ideas led us and adding new collaborators along the way,” says Geer, professor in USU’s Department of Mathematics and Statistics. “Yet this abstract work has become directly relevant to quantum physics, connecting with mathematical physics and opening new doors for discovery.”
Remarkably, he says, it turns out non-semisimple theories are more powerful than the traditional semisimple ones and possess novel properties.
“The exciting part of this discovery is these new properties allow researchers to predict the existence of entirely new kinds of quantum particles — illustrating the beauty of mathematics,” Geer says. “What we’re finding could help to unlock some vexing obstacles to the widespread implementation of quantum computing.”
Geer and his team’s TQFT research is cited in recent articles in PhysicsWorld and Scientific American, which detail findings published in the journal Nature Communications by Geer and Young’s longtime peer and frequent collaborator, Aaron Lauda, professor and dean at the University of Southern California. Lauda and his team use Geer’s TQFTs to suggest a breakthrough path through current quantum computer barriers.
And what are those barriers? Thinking back to the beginnings of personal computer use helps to explain challenges posed by quantum computing.
People of a certain age will remember when the 1982 sci-fi movie TRON pioneered computer-generated imagery to illustrate binary digits — “bits” — as cranky, beleaguered polyhedral-shaped objects chirping “yes” and “no” as they sped, collided and bounced through cyberspace. The visual representation helped fledging users comprehend the inner workings of an emerging explosion of accessible, mass-market personal computers.
As mystifying as it was when introduced to the ordinary user, digital or “classical” computing, with its either black or white discrete states, is much simpler than leveraging quantum phenomena to crunch data.
In contrast, quantum computing, which promises exponentially faster and more accurate computational power and problem-solving uses quantum bits, or “qubits,” which can be “yes” and “no” simultaneously and linked with other qubits. Yet these fundamental units of quantum information — the building blocks of this new technology — are very fragile.
“Quantum computing relies on the ability to very accurately manipulate quantum systems – for example, moving a small particle to a specified point in space and keeping it there,” says Young, assistant professor in the Department of Mathematics and Statistics. “A challenge with this technology is preventing errors from entering the system, causing particles to, using the building block analogy, shift and topple out of place.”
And that’s where the TQFTs he, Geer, their students and collaborators are studying may offer a path forward.
“In theory, TQFTs provide a sturdier option, which spreads the information out over a stronger, more stable area, making qubits less vulnerable to environmental noise,” Geer says.
The different properties of non-semisimple TQFTs would, in theory, lead to better qubits, though they’re not yet physically realized, he says. “But this approach gives engineers and physicists a mathematical foundation to look for them.”
Following this idea, Lauda and his collaborators’ recent paper outlines a concrete proposal to use non-semisimple TQFTs to realize new qubits, which they call “neglectons.”
“The name neglecton was chosen because these quasi-particles are neglected in more traditional semisimple approaches,” Geer says.
Geer and Young, both National Science Foundation CAREER research award recipients, along with their students, continue to actively pursue non-semisimple TQFT and its mathematical physics applications, including its implications for the future of quantum computing.
“If neglectons truly exist, it would be a dream come true to see the theoretical mathematics we’ve worked on lead to a real-world discovery,” Geer says.