Some classical computers have error correction built into their bit-based memories; quantum computers, to work in the future, will also need much more sensitive qubit-based error correction mechanisms.

Cornell researchers have recently taken a step toward fault-tolerant quantum computing: They have built a simple model containing exotic particles called non-abelian anyons that is compact and practical enough to run on modern quantum hardware. The realization of these particles, which can only exist in two dimensions, is a step towards their implementation in the real world.

Through creative thinking, Yuri Lensky, former Bethe/Wilkins/Kavli Institute of Cornell (KIC) Postdoctoral Fellow in Physics in the College of Arts and Sciences (A&S), collaborated with Eun-Ah Kim, Professor of Physics (A&S), proposed a simple “recipe” that could be used for robust computation with non-abelian anyons, including specific instructions for performing the effect experimentally on devices available today.

Their paper, “Graph Gauge Theory of Mobile Non-Abelian Anyons in a Qubit Stabilizer Code,” written in collaboration with Google Quantum AI theorists, published March 24 in *Annals of Physics*. Google Quantum AI researchers, in collaboration with Lensky and Kim, proved the theory with a successful experiment, as reported in a preprint post, “Observation of Non-Abelian Exchange Statistics on a Superconducting Processor,” on the sharing platform research *arXiv*.

“This two-dimensional state is interesting both from the perspective of quantum condensed matter physics – it has new properties very particular to 2D physics – and from the perspective of quantum information,” Lensky said. “It’s something really quantum, but it’s also potentially useful for quantum computing. It protects bits of quantum information by storing it non-locally, and our protocol allows us to compute with those bits.”

Kim explained the principle that drives non-abelian anyons by holding two identical one-pound dumbbells. When she crosses her arms, the identical dumbbells change position, but as objects defined by classical physics, their state remains the same. They are interchangeable.

If these dumbbells represent two identical quantum particles, remarkably in some 2D systems, their trails through spacetime can produce a measurable record of change (pictured with arms crossed). This swapping process is called a braid, after the particle shapes the trails.

“Quantum mechanics, when you move one particle around the other,” said Kim, holding a stationary weight and moving the other in a circle around it, “the wave function, which is a solution to the equation of Schrödinger describing quantum mechanical motion, can be multiplied by a phase factor or it can become something very different.”

When the wave function acquires a global sign that can only be observed by interferometry, a measure of wave interference, it is called an abelian anyon. When the wavefunction becomes noticeably different, it’s a non-abelian anyon, she said.

Non-abelian anyons could be exploited to create qubits defined not on a single particle, but on a pair of identical quantum particles: non-locally encoded.

“If I put the qubit shared between these particles in a zero state and separate them, then whatever happens locally to one of these anyons, the zero state will remain. The qubit set to zero is at the safe from corruption,” Kim said. “Non-abelian anyons could be used in a platform for shielded qubits.”

But while physicists have theorized about these exotic particles for years — Alexei Kitaev proposed operating on protected bits of quantum memory by braiding non-abelian anyons around 2001, Lensky said — they have never been observed in a physical system before.

When Google Quantum AI expanded the quantum processor platform’s capabilities to perform surface coding and braiding of abelian anyons in a physical system, Lensky said, “It was [our] the inspiration to look for a way to realize the physics of non-abelian anyons as soon as possible.”

“We knew they had the working ingredients, but they didn’t have a recipe,” Kim said. “We figured out how to move these non-abelian anyons, and then we told the experimenters what to do. It was possible because Yuri and I thought in a flexible, creative, and open-minded way.”

Past theoretical research has identified non-abelian properties, but failed to find how to move them, a necessary step. A key idea of Lensky and Kim was to abandon the regularity of a grid and organize the qubits in an almost hand-drawn way but backed by robust mathematics.

“After this simple geometric idea, using gauge theory, we were able to come up with the protocol for taking this image and implementing it on a chip in a robust and efficient way,” Kim said. “With this 10-qubit system, we were able to encode multiple non-abelian anyons, and therefore multiple information-carrying logical qubits, and a precise recipe for what experimenters should do at each step of the process.”

“Although the goal of the theory and experiment is simply to realize non-abelian anyons in the real world, it can also be seen as a first small step towards implementing braided computation,” said Lensky.

**More information:**

Yuri D. Lensky et al, Graphic gauge theory of non-abelian moving anyons in a qubit-stabilizing code, *Annals of Physics* (2023). DOI: 10.1016/j.aop.2023.169286

Trond I. Andersen et al, Observation of non-abelian exchange statistics on a superconducting processor, *arXiv* (2022). DOI: 10.48550/arxiv.2210.10255

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