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arXiv:0711.0820 · Physical Review A

Anyon statistics with continuous variables

Jing Zhang, Changde Xie, Kunchi Peng, Peter van Loock

Published Nov 26, 2008• 25 citations
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Abstract

We describe a continuous-variable scheme for simulating the Kitaev lattice model and for detecting statistics of Abelian anyons. The corresponding quantum optical implementation is solely based upon Gaussian resource states and Gaussian operations, hence allowing for a highly efficient creation, manipulation, and detection of anyons. This approach extends our understanding of the control and application of anyons and it leads to the possibility for experimental proof-of-principle demonstrations of anyonic statistics using continuous-variable systems.

DOI
10.1103/physreva.78.052121

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We describe a continuous-variable scheme for simulating the Kitaev lattice model and for detecting statistics of Abelian anyons. The corresponding quantum optical implementation is solely based upon Gaussian resource states and Gaussian operations, hence allowing for a highly efficient creation, manipulation, and detection of anyons. This approach extends our understanding of the control and application of anyons and it leads to the possibility for experimental proof-of-principle demonstrations of anyonic statistics using continuous-variable 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 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.
We describe how continuous-variable Abelian anyons, created on the surface of a continuous-variable analog of Kitaev's lattice model can be utilized for quantum computation. In particular, we derive protocols for the implementation of quantum gates using topological operations. We find that the topological operations alone are insufficient for universal quantum computation, which leads us to study additional nontopological operations such as offline squeezing and single-mode measurements. It is shown that these in conjunction with a non-Gaussian element allow for universal quantum computation using continuous-variable Abelian anyons.
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.
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.