Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor

Abstract: Non-Abelian topological order (TO) is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged. These anyonic excitations are promising building blocks of fault-tolerant quantum computers. However, despite extensive efforts, non-Abelian TO and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian TO. In this work, we present the first unambiguous realization of non-Abelian TO and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum's H2 trapped-ion quantum processor, we create the ground state wavefunction of $D_4$ TO on a kagome lattice of 27 qubits, with fidelity per site exceeding $98.4\%$. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunneling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon -- a peculiar feature of non-Abelian TO. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices.

• The paper demonstrates the experimental realization of non-Abelian topological order on a 27-qubit trapped-ion processor with fidelity exceeding 98.4%.
• It employs adaptive quantum circuits with mid-circuit measurements and feed-forward techniques to engineer anyon braiding and detect non-Abelian properties.
• The study identifies 22 distinct ground states and highlights the potential for fault-tolerant quantum computing through robust logical qubit manipulation.

Non-Abelian Topological Order in Quantum Computing

The paper presents a significant advancement in the field of quantum computing by demonstrating the implementation of non-Abelian topological order on a trapped-ion quantum processor. This study elucidates distinct properties of non-Abelian anyons and their potential in realizing robust quantum information processing systems.

Overview of Non-Abelian Topological Order

Non-Abelian topological order represents a complex state of matter characterized by quasiparticles known as non-Abelian anyons. Unlike Abelian anyons, which only acquire phase factors upon braiding, non-Abelian anyons can induce unitary operations on the manifold of degenerate states, making them promising for fault-tolerant quantum computation. However, achieving controlled preparation and manipulation of non-Abelian anyons has been challenging due to their elusive nature.

Experimental Implementation

Using Quantinuum's H2 trapped-ion quantum processor, the authors successfully created the ground state wavefunction exhibiting $D_4$ topological order on a kagome lattice of 27 qubits. The fidelity per site exceeded 98.4%, showcasing the precision and efficiency of their experimental setup. This accomplishment involved an adaptive quantum circuit that employed mid-circuit measurements and feed-forward strategies to entangle ions on the lattice, thereby realizing non-Abelian topological phenomena.

Non-Abelian Anyon Braiding

A central focus of the study was the demonstration of non-Abelian braiding processes. The authors utilized anyon interferometry to detect the inherently non-Abelian braiding, specifically through moving anyons along paths that intersect Borromean rings—a configuration invisible to Abelian particles. Such braiding processes not only alter the fusion channel of the anyons but can result in excited states with unique topological characteristics.

Ground State Degeneracy and Logical Operators

Through meticulous tunneling experiments, the paper identifies the 22 distinct ground states associated with non-Abelian topological order, a peculiarity stemming from the non-square degeneracy that differentiates non-Abelian topological phases from their Abelian counterparts. The manipulation of these states entails using logical operators that span the torus, facilitated by braiding processes of non-Abelian anyons.

Practical and Theoretical Implications

The research illustrates the effectiveness of using adaptive quantum circuits to study complex forms of entanglement, revealing the potential of quantum processors to explore exotic phases of matter. As non-Abelian anyons carry the promise of new forms of quantum information processing, these findings could enable more robust platforms for quantum computation.

Speculation on Future Developments

The experimental realization of non-Abelian topological order paves the way for further exploration of non-Abelian anyons in fault-tolerant quantum computing. Future research may focus on whether these states can enhance error tolerance compared to traditional methods or implement universal quantum computation. Exploration of other entangled phases and development of quantum algorithms that leverage non-Abelian anyon braiding could provide additional insights into topological quantum computation.

To conclude, the paper successfully demonstrates non-Abelian topological order in a controlled quantum setting, enriching the understanding of how complex quantum states can be prepared and manipulated for practical quantum computing applications.