Deixis Online

The Quantum Systems Accelerator at Berkeley Lab. DOE CSGF recipient Alex Ballow studies math that informs what happens in quantum systems. (Photo: Thor Swift/Berkeley Lab.)

Alex Ballow works as a translator, not of human tongues but of the languages of math and physics. “I have refused to choose between math, physics and computer science during my career,” she says. “These fields are really doing the same thing but talking about it in such different ways with different word choices.”

As a Ph.D. student studying algebraic topology and quantum mechanics at Montana State University in Bozeman, Ballow has bridged those fields through a form of universal mathematical language known as category theory. This shared lexicon could allow researchers with different perspectives to find new patterns in physical phenomena, discoveries that could lead to advances in quantum computing, condensed matter physics and other areas.

Ballow decided at age 10 she wanted to work at a university when her uncle, a professor, told her he could take free classes in any subject. “I just thought that was the coolest thing ever that he could just learn things for free forever,” she says. When it came time to choose an undergraduate major at Youngstown State University in her home state of Ohio, she doubled up in math and physics. “I didn’t really like the idea of being in an ivory tower and never applying anything.”

That interest in applied research led her to publish undergraduate research of all sorts: biomass, public health modeling and computer science, to name just a few. Ballow also had an opportunity to work in physics during an internship at Lawrence Berkeley National Laboratory in 2018. But she kept finding herself drawn to pure math, specifically abstract algebra. “Your whole life, you see real numbers, addition, multiplication, all these rules, and you never really know why those are there. Abstract algebra just kind of explains it. It’s really cool.”

Category theory — a recent branch of abstract algebra — doesn’t describe shapes or concepts such as topology. Rather, it looks at things in relation to other things of a similar type. Sometimes called the “mathematics of mathematics,” category theory provides a bird’s-eye view of an entire mathematical fields and reveals similarities that might not be obvious at a granular level. “You can take just about any field of math and represent it as a category,” Ballow says. That’s not to say categories are a magical fix, she adds; researchers still must consider objects and analyze what they do. “But I really like having such a general framework to discuss so many different types of objects with one definition.”

Ballow’s Ph.D. project focuses on applying category theory to entities that arise from interactions between other particles. These so-called quasiparticles, known as anyons, behave like particles but exist only in two-dimensional space. In this flat existence, anyons treat time as their third dimension: splitting, merging, and braiding around one another to trace out tangled knots as they move through time. Working with her advisor, mathematician David Ayala, she is trying to categorize these knots’ characteristics and describe how they exist in a larger system. If anyons’ behavior can be described with category theory, she says, physicists may be able to identify individual anyons and better understand patterns in overall anyon systems.

This wide-view approach allows researchers to not only identify the visible particles but also those they can infer for an entire topological system, or phantom particles. These theoretical doppelgangers of standard particles can’t be detected from looking at only one patch of the system.

There are practical applications too. Anyons’ ability to braid with one another enables encoding information, possibly for quantum computing. The ability to describe the quasiparticles using a universal mathematical language could one day aid efforts to design hardware for anyon-based computers.

Ballow studied a different form of mathematical relationship during her 2022 and 2023 practicums at Oak Ridge National Laboratory. Working under computational scientist Kathleen Hamilton, she surveyed quantum machine learning algorithms for patterns in the complex quantum states that arise in these systems. In quantum computers, the basic units of information, qubits, can entangle into states which can be mathematically described as graphs. These graphs are made of nodes that represent qubits, which encode 0, 1 and states in between, and edges representing entanglement between them. Ballow’s research tested the idea that related graphs would be connected through paths of quantum states that all perform similarly well or poorly in an algorithm. Knowing this would give software developers more options for optimizing algorithms.

After Ballow graduates in 2026 or 2027, she plans to keep working in research, either at a university or a national lab. She’s particularly interested in a subfield called higher category theory, which studies the intricacies of how two objects relate.