While some classical computers have built-in error correction using bits as their memory, quantum computers will also require error correction mechanisms using the vastly more sensitive qubits in order to function in the future.

The construction by Cornell researchers of a straightforward model with novel particles known as non-Abelian anyons that is compact and useful enough to operate on contemporary quantum hardware is a recent step toward fault-tolerant quantum computing. This is a step toward putting it into practice in the real world because these particles can only exist in two dimensions.

Yuri Lensky, a former Bethe/Wilkins/Kavli Institute at Cornell (KIC) postdoctoral fellow in physics in the College of Arts and Sciences (AandS), and Eun-Ah Kim, professor of physics (A), were able to work together thanks to some innovative thinking.

Yuri Lensky, a former Bethe/Wilkins/Kavli Institute at Cornell (KIC) .

“This two-dimensional state is intriguing from the perspectives of quantum information and quantum condensed matter physics because it exhibits several unexpected traits that are quite unique to two dimensions.”

Their article, “Graph Gauge Theory of Mobile Non-Abelian Anyons in a Qubit Stabilizer Code,” which they co-wrote with Google Quantum AI theorists, appeared in the Annals of Physics on March 24. In a successful experiment described in the preprint publication “Observation of Non-Abelian Exchange Statistics on a Superconducting Processor” on the research-sharing website arXiv, researchers from Google Quantum AI, Lensky, and Kim have demonstrated the theory.

According to Lensky, both from a quantum information and a quantum condensed matter physics standpoint, this two-dimensional state is intriguing. It exhibits some novel properties that are very unique to 2D physics. It’s something that is truly quantum, but it could also be helpful for quantum computation. Our protocol enables us to compute with these bits of protected quantum information while it stores them non-locally.”.

Kim demonstrated the idea that drives non-Abelian anyone by holding out two identical one-pound barbells. The identical barbells shift positions when she crosses her arms, but because they are classically defined objects, their state doesn’t change. They can both be used.

Incredibly, in some 2D systems, the trails those barbells leave behind in space-time can produce a quantifiable record of the change (imagine the crossed arms) if they represent two identical quantum particles. This exchange procedure is known as a braid because of the patterns left by the particle trails.

The wave function, which is a solution to the Schrödinger equation describing quantum mechanical motion, can be multiplied by a phase factor or it can transform into something very different when you move one particle around the other, Kim explained, holding one weight still and moving the other in a circle around it.

An Abelian anyon is a global sign that the wave function acquires that can only be measured through the interference of waves in interferometry. It’s a non-Abelian anyon, she said, when the wave function starts to differ noticeably.

Nonlocally encoded qubits could be made by using non-Abelian anyons to define them on a pair of identical quantum particles rather than a single particle.

“Whatever happens locally to one of these anyons, the zero state will remain if I put the shared qubit between these particles in a zero state and separate them. The qubit that has been set to zero is uncorruptible, according to Kim. Non-Abelian anyons might be employed as a platform for secure qubits.”.

However, even though physicists have theorized about these exotic particles for years—Alexei Kitaev proposed operating on protected bits of quantum memory by braiding non-Abelian anyons back in around 2001, Lensky said—they have never been observed in a physical system prior to this.

To realize the surface code and braiding of Abelian anyons in a physical system, Google Quantum AI developed the quantum processor platform. According to Lensky, “This was [our] inspiration to look for a way to realize the physics of non-Abelian anyons as soon as possible.”. “.

Kim said, “We knew they had the necessary components, but they didn’t have a recipe. The experimentalists were given instructions after we figured out how to move these non-Abelian anyons. Because Yuri and I had flexible, original, and open-minded thinking, it was possible.”.

Theoretical studies in the past found non-Abelian properties but failed to explain how to move them, a crucial step. The decision by Lensky and Kim to forgo the grid’s regularity in favor of an almost-manually drawn arrangement of qubits was a key insight.

The protocol for taking this picture and implementing it on a chip in an effective manner was developed using gauge theory after this straightforward geometrical insight, according to Kim. “With this 10-qubit system, we were able to encode multiple non-Abelian anyons, leading to multiple logical information-carrying qubits and a detailed guide for what the experimenters must do at each stage.”.

Although the theory and experiment’s primary goal is to simply realize non-Abelian anyons in the real world, Lensky noted that this can also be seen as the first modest step toward putting computation by braiding into practice.

**More information:** Yuri D. Lensky et al, Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code, *Annals of Physics* (2023). DOI: 10.1016/j.aop.2023.169286