The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T junction, and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.
The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T junction, and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.
Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined plays a fundamental role in defining the non-Abelian anyons. We study the nonadiabatic effects in braidings of Ising-type anyons, namely, Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider nonadiabatic corrections to non-Abelian statistics from (1) tunneling splitting of anyons imposing an additional dynamical phase to the transformation of ground states and (2) transitions to excited states that are potentially destructive to non-Abelian statistics since the nonlocal fermion occupation can be spoiled by such processes. However, if the bound states are localized and being braided together with the anyons, non-Abelian statistics can be recovered once the definition of Majorana operators is appropriately generalized taking into account the fermion parity in these states. On the other hand, if the excited states are extended over the whole system and form a continuum, the notion of local fermion parity no longer holds. We then quantitatively characterize the errors introduced in this situation.
Indistinguishability of particles is a fundamental principle of quantum mechanics. For all elementary and quasiparticles observed to date - including fermions, bosons, and Abelian anyons - this principle guarantees that the braiding of identical particles leaves the system unchanged. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well developed mathematical description of non-Abelian anyons and numerous theoretical proposals, the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. While efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasi-particles, superconducting quantum processors allow for directly manipulating the many-body wavefunction via unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons, we implement a generalized stabilizer code and unitary protocol to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of employing the anyons for quantum computation and utilize braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and - through the future inclusion of error correction to achieve topological protection - could open a path toward fault-tolerant quantum computing.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
Abstract Indistinguishability of particles is a fundamental principle of quantum mechanics 1 . For all elementary and quasiparticles observed to date—including fermions, bosons and Abelian anyons—this principle guarantees that the braiding of identical particles leaves the system unchanged 2,3 . However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions 4–8 . Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well-developed mathematical description of non-Abelian anyons and numerous theoretical proposals 9–22 , the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons 9,10 , we implement a generalized stabilizer code and unitary protocol 23 to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of using the anyons for quantum computation and use braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards fault-tolerant quantum computing.
Topological quantum states of matter, both Abelian and non-Abelian, are characterized by excitations whose wavefunctions undergo nontrivial statistical transformations as one excitation is moved (braided) around another. Topological quantum computation proposes to use the topological protection and the braiding statistics of a non-Abelian topological state to perform quantum computation. The enormous technological prospect of topological quantum computation provides new motivation for experimentally observing a topological state. Here, we explicitly work out a realistic experimental scheme to create and braid the Abelian topological excitations in the Kitaev model built on a tunable robust system, a cold atom optical lattice. We also demonstrate how to detect the key feature of these excitations: their braiding statistics. Observation of this statistics would directly establish the existence of anyons, quantum particles that are neither fermions nor bosons. In addition to establishing topological matter, the experimental scheme we develop here can also be adapted to a non-Abelian topological state, supported by the same Kitaev model but in a different parameter regime, to eventually build topologically protected quantum gates.
The manipulation of non-Abelian anyons is a foundational ingredient for topological quantum computing and underlies topologically protected quantum memories. Here we analyze a quantum-impurity-based protocol for braiding non-Abelian anyons. Exploiting mobile impurities with addressable internal spin states, we provide a concrete protocol to experimentally measure the matrix elements of the R and F matrices that characterize the non-Abelian statistics of anyons. In particular, focusing on the case of Ising and Fibonacci anyons, we illustrate how the anyonic nature of the excitations emerges directly in the Ramsey interferometer we propose. Building on this framework, we further introduce a conventional-to-topological qubit encoder to transfer quantum information from conventional flying qubits into a topologically protected quantum memory.