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arXiv:quant-ph/0702031 · Physical Review Letters

Scheme for Demonstration of Fractional Statistics of Anyons in an Exactly Solvable Model

Yong‐Jian Han, Robert Raussendorf, Luming Duan

Published Apr 12, 2007• 96 citations
arXivHTMLPDF

Abstract

We propose a scheme to demonstrate fractional statistics of anyons in an exactly solvable lattice model proposed by Kitaev that involves four-body interactions. The required many-body ground state, as well as the anyon excitations and their braiding operations, can be conveniently realized through dynamic laser manipulation of cold atoms in an optical lattice. Because of the perfect localization of anyons in this model, we show that a quantum circuit with only six qubits is enough for demonstration of the basic braiding statistics of anyons. This opens up the immediate possibility of proof-of-principle experiments with trapped ions, photons, or nuclear magnetic resonance systems.

DOI
10.1103/physrevlett.98.150404

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We propose a scheme to demonstrate fractional statistics of anyons in an exactly solvable lattice model proposed by Kitaev that involves four-body interactions. The required many-body ground state, as well as the anyon excitations and their braiding operations, can be conveniently realized through dynamic laser manipulation of cold atoms in an optical lattice. Because of the perfect localization of anyons in this model, we show that a quantum circuit with only six qubits is enough for demonstration of the basic braiding statistics of anyons. This opens up the immediate possibility of proof-of-principle experiments with trapped ions, photons, or nuclear magnetic resonance systems.
In two-dimensions, the laws of physics even permit the existence of anyons which exhibit fractional statistics ranging continuously from bosonic to fermionic behaviour. They have been responsible for the fractional quantum Hall effect and proposed as candidates for naturally fault-tolerant quantum computation. Despite these remarkable properties, the fractional statistics of anyons has never been observed in nature directly. Here we report the demonstration of fractional statistics of anyons by simulation of the first Kitaev lattice-spin model on a nuclear magnetic resonance system. We encode four-body interactions of the lattice-spin model into two-body interactions of an Ising spin chain in molecules. It can thus efficiently prepare and operate the ground state and excitations of the model Hamiltonian. This quantum system with convenience of manipulation and detection of abelian anyons reveals anyonic statistical properties distinctly. Our experiment with interacted Hamiltonian could also prove useful in the long run to the control and application of anyons.
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 have exotic statistical properties, fractional statistics, differing from bosons and fermions. They can be created as excitations of some Hamiltonian models. Here, we present an experimental demonstration of anyonic fractional statistics by simulating a version of the Kitaev spin-lattice model proposed by Han et al. [Phys. Rev. Lett. 98, 150404 (2007)] using an NMR quantum-information processor. We use a seven-qubit system to prepare a six-qubit pseudopure state to implement the ground-state preparation and realize anyonic manipulations, including creation, braiding and anyon fusion. The anyonic braiding process is equivalent to two successive particle exchanges. We obtain a phase difference of $(0.52\ifmmode\pm\else\textpm\fi{}0.01)\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}2$ between the states with and without anyon braiding, which is different from the $\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}2$ and $2\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}2$ phase changes for fermions and bosons after two successive particle exchanges, and agrees with the prediction of the anyonic fractional statistics.
Since the anyonic excitations in the Kitaev toric model are perfectly localized quasiparticles, it is possible to generate dynamically the ground state and the excitations of the model Hamiltonian to simulate the anyonic interferometry. We propose a scheme in circuit QED to simulate the interferometry. The qubit-cavity interaction can be engineered to realize effective state control as well as the controlled dynamics of qubits, which are sufficient to prepare the ground states, create and remove the anyonic excitation, and simulate the anyonic interferometry. The simplicity and high fidelity of the operations used open the very promising possibility of simulating fractional statistics of anyons in a macroscopic material in the near future.
Although Han et al. [Phys. Rev. Lett. 98, 150404 (2007)] proposed a quantum circuit with six qubits to demonstrate the basic braiding statistics of anyons, we have found that a four-qubit system is enough for realizing such a process. We call it the minimum unit for the fractional statistics of anyons. With some special order, the minimum unit can be used to construct two kinds of fractional charge excitation n/(n+1)n/(n+1)n/(n+1) [or (n+1)/n(n+1)/n(n+1)/n] and $({n}^{2}\ensuremath{-}\mathit{kn}+6n+2)/({n}^{2}\ensuremath{-}\mathit{kn}+5n+4)$ [or $({n}^{2}\ensuremath{-}\mathit{kn}+2n+k+1)/3({n}^{2}\ensuremath{-}\mathit{kn}+5n+4)$], where n,kn,kn,k are positive integers and $k\ensuremath{\leqslant}n$. This minimum unit can be realized by a photonic quantum simulator or by electrons in an optical lattice or quantum dot array. Its use will have some advantages in metrology.
Anyons are quasiparticles that obey fractional statistics, producing a phase intermediate to bosons and fermions under particle interchange. Anyons form the basis for topological quantum computation and error correction, where the topological aspect of anyonic braiding is one of the important features that gives rise to fault tolerance. A central model that exhibits anyons is Kitaev’s toric code, where the excited states involve anyons that exhibit topological behavior. Here, we experimentally create the ground state and anyonic excitations of a nine-qubit planar code using eight photons. Stabilizer measurements of the states of the planar code are performed to directly reveal the locations of the anyons in the system. We further perform braiding operations on the anyons, which gives rise to a topologically path-independent phase. Our work provides a platform for simulating the braiding operations with linear optics, opening up the possibility of further exploring the features of anyonic statistics.