Topological order, a concept that revolutionizes the understanding of quantum matter beyond the Landau-Ginzburg symmetry-breaking paradigm, is characterized by long-range entanglement and anyonic quasiparticles exhibiting unique braiding statistics. These anyons hold significant promise for fault-tolerant quantum computing, offering topologically protected logical code spaces immune to local noise. Logical operations are performed by creating, braiding, and fusing anyons. While Abelian anyons produce a phase factor upon braiding, non-Abelian anyons act with a unitary matrix on the degenerate state manifold, enabling universal quantum computation. However, realizing non-Abelian topological orders is challenging. Recent progress involved digital quantum simulation to observe signatures of non-Abelian statistics, but not all topological orders support universal quantum computation. This research addresses the crucial gap by utilizing a superconducting quantum processor to simulate the Fibonacci string-net model, a minimal model supporting universal topological quantum computation. The study aims to demonstrate the braiding of Fibonacci anyons within this model, verifying their non-Abelian nature and highlighting the potential for practical applications in fault-tolerant quantum computing.

The discovery of topological order fundamentally shifted the understanding of quantum matter, moving beyond the traditional symmetry-breaking paradigm. Various topologically ordered phases can share the same symmetries but differ topologically, characterized by long-range entanglement and anyonic quasiparticles. These anyons, with their Abelian or non-Abelian braiding statistics, have attracted significant interest for their potential in fault-tolerant quantum computing. The non-local nature of topological degrees of freedom offers inherent protection against local perturbations. Non-Abelian anyons, in particular, can be used to encode and manipulate quantum information in a topologically protected manner. Previous work has explored digital quantum simulation techniques to observe signatures of non-Abelian statistics; however, realizing universal quantum computation necessitates specific forms of topological order. The Fibonacci string-net model is a promising candidate, being the simplest model capable of supporting universal topological quantum computation via anyon braiding.

The researchers employed the Fibonacci string-net model, a Levin-Wen model defined on a honeycomb lattice, for their simulation. The Hamiltonian of this model consists of vertex and plaquette operators (Qv and Bp), which are three-body and twelve-body projectors respectively. The ground state of this Hamiltonian exhibits topological order, and its quasiparticle excitations are Fibonacci anyons, obeying the fusion rule τ × τ = 1 + τ (where 1 and τ represent the vacuum and Fibonacci anyon). Experimentally, a flip-chip superconducting quantum processor with frequency-tunable transmon qubits arranged in a square lattice was used. A subset of 27 qubits formed a honeycomb lattice, with qubits on edges representing spins in the string-net Hamiltonian. The high qubit coherence times (median 117 µs) and gate fidelities (around 99.96% and 99.50% for single and two-qubit gates) enabled the preparation of the ground state and implementation of complex braiding circuits. Ground-state preparation was achieved using a variational unitary synthesis technique, optimizing the preparation of the ground state by targeting the final state, reducing the circuit depth to 53. Topological entanglement entropy was measured using a randomized measurement method, allowing for the estimation of the topological order. Braiding of Fibonacci anyons involved creating two pairs, braiding them using string operators (F and R moves), and then measuring the fusion outcomes to determine their braiding statistics. Logical qubits were encoded into the four Fibonacci anyons, and braiding operations were represented as unitary logical gates. The monodromy matrix and quantum dimension of the Fibonacci anyon were extracted from the fusion results, further validating the experiment.

The researchers successfully simulated the Fibonacci string-net model on a superconducting quantum processor and demonstrated the braiding of Fibonacci anyons. They verified the non-trivial topological nature of the prepared ground state by measuring the topological entanglement entropy, obtaining a value of -0.82, significantly lower than zero (for a topologically trivial state) and -ln2 (for a Z2 topologically ordered state). This result is consistent with the theoretical value for the Fibonacci topological order. The creation, braiding, and fusion of Fibonacci anyons were experimentally demonstrated. Five different braiding sequences were implemented, with the results (fusion probabilities) matching theoretical predictions for the non-Abelian braiding statistics of Fibonacci anyons. The measured probabilities matched the theoretical predictions for the non-Abelian fusion rule (τ × τ = 1 + τ). From the experimental results, the monodromy matrix and the quantum dimension of the Fibonacci anyon were extracted. The experimental results yielded a quantum dimension of approximately 1.60, close to the theoretical value of φ (the golden ratio, approximately 1.618). This confirms that the quasiparticles created in the experiment are indeed Fibonacci anyons. The use of variational methods allowed the efficient implementation of the complex multi-qubit operations necessary for the simulations.

The successful simulation of the braiding of Fibonacci anyons on a superconducting quantum processor represents a significant advancement in the field of topological quantum computing. This work demonstrates the feasibility of using current noisy intermediate-scale quantum processors to explore non-Abelian topological states and their associated braiding statistics. While the braiding in this experiment lacked the topological protection afforded by an energy gap between ground and excited states (it is more of a quantum simulation of braiding), the results clearly show the non-Abelian nature of Fibonacci anyons and their potential for quantum computation. The fidelity of the results, despite experimental imperfections, underscores the potential of this approach for further development and refinement. The methodology used, particularly the variational techniques for circuit optimization and the randomized measurement approach for topological entanglement entropy, are valuable for future experiments aiming to simulate complex topological systems.

This research successfully demonstrated the braiding of Fibonacci anyons using a superconducting quantum processor, a crucial step towards realizing topological quantum computation. The experimental results strongly support the non-Abelian nature of Fibonacci anyons and the validity of the Fibonacci string-net model simulation. Further research could focus on improving the coherence times and gate fidelities of the qubits to enhance the precision of the braiding operations and achieve greater topological protection. Exploring more sophisticated error correction techniques, such as the Fibonacci Turaev-Viro code, would be essential for building robust and fault-tolerant topological quantum computers.

The current experiment does not incorporate active error correction, meaning the braiding operations are not inherently topologically protected. Experimental imperfections, such as gate errors and decoherence, contribute to the observed deviations from theoretical predictions. Although variational methods were used to reduce circuit depth, the scalability of this approach to larger systems remains to be thoroughly investigated. The system size used in this study is relatively small. Further scaling up to larger systems would require significant advancements in quantum processor technology.