We consider a two-dimensional spin system that exhibits Abelian anyonic excitations. Manipulations of these excitations enable the construction of a quantum computational model. While the one-qubit gates are performed dynamically, the model offers the advantage of having a two-qubit gate that is of topological nature. The transport and braiding of anyons on the lattice can be performed adiabatically enjoying the robust characteristics of geometrical evolutions. The same control procedures can be used when dealing with non-Abelian anyons. A possible implementation of the manipulations with optical lattices is developed.
We consider a two-dimensional spin system that exhibits Abelian anyonic excitations. Manipulations of these excitations enable the construction of a quantum computational model. While the one-qubit gates are performed dynamically, the model offers the advantage of having a two-qubit gate that is of topological nature. The transport and braiding of anyons on the lattice can be performed adiabatically enjoying the robust characteristics of geometrical evolutions. The same control procedures can be used when dealing with non-Abelian anyons. A possible implementation of the manipulations with optical lattices is developed.
Quantum computation provides a unique opportunity to explore new regimes of physical systems through the creation of non-trivial quantum states far outside of the classical limit. However, such computation is remarkably sensitive to noise and undergoes rapid dephasing in most cases. One potential solution to these prosaic concerns is to encode and process the information using topological manipulations of so-called anyons, particles in two dimensions with non-Abelian statistics. Unfortunately, practical implementation of such a topological system remains far from complete, both in terms of physical methods but also in terms of connecting the underlying topological field theory with a specific physical model, including the imperfections expected in any realistic device. Here we develop a complete picture of such topological quantum computation using a variation of the Kitaev honeycomb Hamiltonian as the basis for our approach. We show the robustness of this system against noise, confirm the non-Abelian statistics of the quasi-particles to be Ising anyons, and develop new techniques for turning topological information into measurable spin quantities.
Anyons are quasiparticles that may be realized in two dimensional systems. They come in two types, the simpler Abelian anyons and the more complex non-Abelian anyons. Both of these have been considered as a means for quantum computation, but non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. First the means to perform quantum computation with Abelian anyon models is considered. These models, like many non-Abelian models, cannot realize universal quantum computation by braiding alone. Non-topological operations must be used in addition, whose complexity depends on the physical means by which the anyons are realized. Here we consider anyons based on spin lattice models, with single spin measurements playing the role of non-topological operations. The computational power achieved by various kinds of measurement is explored and the requirements for universality are determined. The possibility to simulate non-Abelian anyons using Abelian ones is then considered. Finally, a non-Abelian quantum memory is dissected in order to determine the means by which it provides fault-tolerant storage of information. This understanding is then employed to build equivalent quantum memories with Abelian anyon models. The methodology provides with the means to demonstrate that Abelian models have the capability to simulate non-Abelian anyons, and to realize the same computational power and fault-tolerance as non Abelian models. Apart from the intellectual interest in relating topological models with each other, and of understanding the properties of non-Abelian anyons in terms of the simpler Abelian ones, these results can also be applied in the lab. The simpler structure of Abelian anyons means that their physical realization is more straightforward. The demonstration of non-Abelian properties with Abelian models therefore allows features of non-Abelian anyons to be realized with present and near future technology. Based on this possibility, proposals are made here for proof of principle experiments.
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.
Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons, i.e., the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.
Anyons are particlelike excitations of strongly correlated phases of matter with fractional statistics, characterized by nontrivial changes in the wave function, generalizing Bose and Fermi statistics, when two of them are interchanged. This can be used to perform quantum computations [A. Yu. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003)]. We show how to simulate the creation and manipulation of Abelian and non-Abelian anyons in topological lattice models using trapped atoms in optical lattices. Our proposal, feasible with present technology, requires an ancilla particle which can undergo single-particle gates, be moved close to each constituent of the lattice and undergo a simple quantum gate, and be detected.
Abstract A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons.