Anyon statistics with continuous variables

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.

Figure 1

(Color online) Surface code model and anyonic braiding with continuous variables. (a) The measurement pattern to prepare the anyonic ground state $\mid \psi ⟩$ from a two-dimensional CV cluster state, where $x$ and $p$ denote a single-mode measurement in the position and momentum basis, respectively, $F$ is the Fourier transform for all remaining modes. (b) The CV surface code model consists of the star operators $A_s\left(\xi \right)$ and plaquette operators $B_f\left(\eta \right)$. (c) Application of an $X\left(t\right)$ position translation on a single mode in the horizontal edge yields a pair of $m\left(t\right)$-type anyons placed at the neighboring plaquettes. (d) Application of $X\left(t\right)$ on a single mode in the vertical edge yields a pair of $m(-t)$-type anyons. (e) Application of $Z\left(s\right)$ on a single mode yields a pair of $e\left(s\right)$-type anyons at the neighboring stars. (f) Application of $X\left(t\right)$ on the modes along a string would create $m$-type anyons, where two $m$-type anyons are created on the same plaquettes. These two anyons annihilate each other according to the fusion rule $m\left(t\right)\times m(-t)=1$. Thus, the two single strings are glued together with two anyons at the endpoints. When a part of a string forms a loop, it cancels (dashed). (g) An $m$-type anyon follows a closed loop around an $e$-type anyon.

Figure 2

(Color online) Surface code state with four modes used for demonstration of anyonic statistics. (a) The anyonic ground state ${\mid \psi ⟩}_4$, forming one star and four plaquettes, is given by a four-mode GHZ state with the modes labeled 1–4. (b) The four-mode cluster state which after Fourier transforms on modes 1, 3, and 4, is equivalent to the anyonic ground state ${\mid \psi ⟩}_4$. (c) The quantum circuit for creation, braiding, and annihilation of the anyons, and for the final state detection. (d) The correlation diagram of the momentum linear combination operator ${\widehat{p}}_1+{\widehat{p}}_2+{\widehat{p}}_3+{\widehat{p}}_4$ with (red) and without a closed loop of anyon $m$ around $e$ (black) in the case of finite squeezing.