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Communications in Theoretical Physics

A Scheme for Simulation of Quantum Gates by Abelian Anyons

Yao Shen, Qing Ai, Gui‐Lu Long

Published Nov 1, 2011• 6 citations
Publisher

Abstract

Anyons can be used to realize quantum computation, because they are two-level systems in two dimensions. In this paper, we propose a scheme to simulate single-qubit gates and CNOT gate using Abelian anyons in the Kitaev model. Two pairs of anyons (six spins) are used to realize single-qubit gates, while ten spins are needed for the CNOT gate. Based on these quantum gates, we show how to realize the Grover algorithm in a two-qubit system.

DOI
10.1088/0253-6102/56/5/13

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Anyons can be used to realize quantum computation, because they are two-level systems in two dimensions. In this paper, we propose a scheme to simulate single-qubit gates and CNOT gate using Abelian anyons in the Kitaev model. Two pairs of anyons (six spins) are used to realize single-qubit gates, while ten spins are needed for the CNOT gate. Based on these quantum gates, we show how to realize the Grover algorithm in a two-qubit system.
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.
Recently, a pair of experiments (Lu et al 2009 Phys. Rev. Lett. 102 030502; Pachos et al 2009 New J. Phys. 11 083010) demonstrated the simulation of Abelian anyons in a spin network of single photons. The experiments were based on the Abelian discrete gauge theory spin lattice model of Kitaev (Kitaev 2003 Ann. Phys., NY 303 2). Here, we propose an experiment using linear optics and single photons to simulate non-Abelian anyons. The scheme makes use of joint qutrit–qubit encoding of the spins, and the resources required are three pairs of parametric downconverted photons and 14 beam splitters.
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.
Recently, a pair of experiments (Lu et al 2009 Phys. Rev. Lett. 102 030502; Pachos et al 2009 New J. Phys. 11 083010) demonstrated the simulation of Abelian anyons in a spin network of single photons. The experiments were based on the Abelian discrete gauge theory spin lattice model of Kitaev (Kitaev 2003 Ann. Phys., NY 303 2). Here, we propose an experiment using linear optics and single photons to simulate non-Abelian anyons. The scheme makes use of joint qutrit–qubit encoding of the spins, and the resources required are three pairs of parametric downconverted photons and 14 beam splitters.
Anyons are exotic quasiparticles living in two dimensions that do not fit into the usual categories of fermions and bosons, but obey a new form of fractional statistics. Following a recent proposal [Phys. Rev. Lett. 98, 150404 (2007)], we present an experimental demonstration of the fractional statistics of anyons in the Kitaev spin lattice model using a photonic quantum simulator. We dynamically create the ground state and excited states (which are six-qubit graph states) of the Kitaev model Hamiltonian, and implement the anyonic braiding and fusion operations by single-qubit rotations. A phase shift of pi related to the anyon braiding is observed, confirming the prediction of the fractional statistics of Abelian 1/2 anyons.
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.