Entangling Qubits by Heisenberg Spin Exchange and Anyon Braiding

Babatunde M. Ayeni, Robert N. C. Pfeifer, Gavin K. Brennen

Non-Abelian anyons can exist as pointlike particles in two-dimensional systems and have particle exchange statistics which are neither bosonic nor fermionic. Like in spin systems, the role of fusion (Heisenberg-like) interactions between anyons has been well studied. However, unlike our understanding of the role of bosonic and fermionic statistics in the formation of different quantum phases of matter, little is known concerning the effect of non-Abelian braid statistics. We explore this physics using an anyonic Hubbard model on a two-legged ladder which includes braiding and nearest-neighbor Heisenberg interactions among anyons. We study two of the most prominent non-Abelian anyon models: the Fibonacci and Ising types. We discover rich phase diagrams for both anyon models and show the different roles of their fusion and braid statistics.

Richard J. Fitzgerald

Guanru Feng, Guilu Long, Raymond Laflamme

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.

Michael Shulman

Lachezar S Georgiev

We give a rigorous and self-consistent derivation of the elementary braid matrices representing the exchanges of adjacent Ising anyons in the two inequivalent representations of the Pfaffian quantum Hall states with even and odd numbers of Majorana fermions. To this end we use the distinct operator product expansions of the chiral spin fields in the Neveu–Schwarz and Ramond sectors of the two-dimensional Ising conformal field theory. We find recursive relations for the generators of the irreducible representations of the braid group in terms of those for , as well as explicit formulae for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wavefunctions are completely equivalent to the spinor representations of SO(2 n + 2) and give the equivalence matrices explicitly. This actually proves that the correlation functions of 2 n chiral Ising spin fields σ do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2 n ) as conjectured by Nayak and Wilczek.