Abstract Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.
We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the
Z3 parafermion fractional quantum Hall state in the second Landau level with filling factor
ν=12/5. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the
n-qubit topological quantum register in terms of
2n+2 Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in
Z3 parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.
This paper proposes a cryptographic framework whose core security foundationrests upon the physical process of anyon braiding in topological quantum computation. In two-dimensional topological materials, the fusion and braiding operationsof non-Abelian anyons are strictly protected by topological order: local perturbations cannot alter the fusion channels of anyon pairs, and braiding operationsdepend solely on the homotopy class of the braiding trajectory. This topologicalprotection mechanism provides a security resource for cryptography that is fundamentally different from computational complexity assumptions. The central ideaof this paper is to directly map anyon braiding operations into cryptographic primitives: the private key corresponds to the secret braiding sequences held by Aliceand Bob, respectively, and the public key consists of exchanged auxiliary anyonsand their partial observables. Security is rooted in the principle of topologicalprotection—any eavesdropper who has not executed the exact homotopy class ofbraiding operations cannot extract the information encoded in the fusion spacewithout destroying the ground-state degeneracy of the system. This paper constructs a non-interactive key exchange protocol based on anyon braiding, analyzesits inherent fault tolerance under physical noise, and discusses the prospects forimplementation on current physical platforms.