Figure 1

(a) The first Kitaev model [5]. The qubits live on the edges, and the stabilizer operators $A_v$, $B_f$ (1) represent the four-body interactions as illustrated. (b) Creation of quasiparticles and braid of an $m$ particle around an $e$ particle. A pair of $e$ particles ($m$ particles) are created on vertices (faces) by applying a $Z$ ($X$) operation on the edge qubit. The quasiparticles can be moved horizontally and vertically by repeated applications of $Z$ or $X$ operations. The figure shows an example of how an $m$ particle forms a closed loop around an $e$ particle through a series of moves.Reuse & Permissions

Figure 2

(a) The small Kitaev spin lattice system with six qubits used for demonstration of anyonic braiding operations. (b) The six-qubit graph state which, after Hadamard (H) transformations on qubit 2, 3, 4, and 5, is equivalent to the ground state of the system in Fig. 2a. The graph state is associated with a graph, where each vertex denotes a qubit and each edge represents a controlled phase gate having been applied between the two connected qubits [18, 19]. The graph state is a common eigenstate of the stabilizer operators $g_i$, that is, $g_i|\psi {⟩}_6=|\psi {⟩}_6$, which describe the correlation in the state, and the graph state is the unique state fulfilling this. Here, we use the same label as Figs. 2a-b in Ref. 14. (c) Experimental setup. A ultraviolet laser successively passes through three $\beta$-barium borate (BBO) crystals to generate three pairs of entangled photons [20]. The photons $a$, $b$, $c$, $d$, and $f$ are combined on the three polarizing beam splitters (PBSs) step by step [27]. To achieve good spatial and temporal overlap, all photons are spectrally filtered ($\Delta {\lambda }_{\mathrm{FWHW}}=3.2\mathrm{nm}$) and detected by fiber-coupled single-photon detectors ($D_1,\cdots ,D_6$). The detector labels correspond to the qubit labels in the text and in Figs. 2a, 2b. The coincidence events are registered by a programmable multichannel coincidence unit. For single-qubit rotations and polarization analysis, quarter wave plates (QWPs), half wave plates (HWPs), together with polarizers or PBSs are used.Reuse & Permissions

Figure 3

The measured expectation values of the operators $A_1,\cdots ,B_4$ of the ground state $|\psi {⟩}_6$ (a) and the excited state $|{\psi }_{em}{⟩}_6$ (b). The excited state $|{\psi }_{em}{⟩}_6$ has a pair of $e$ particles on $v_1$, $v_2$ and a pair of $m$ particles on $f_1$, $f_3$; thus, the values for $A_1$, $A_2$, $B_1$, $B_3$ become negative. Each expectation value is derived from a complete set of 64 sixfold coincidence events in $15h$ in measurement basis $Z^{\otimes 6}$ or $X^{\otimes 6}$. The error bars represent 1 standard deviation, deduced from propagated Poissonian statistics of the raw detection events.Reuse & Permissions

Figure 4

(a). Measured fringes for the state $|{\psi }_s{⟩}_6$ and $|{\psi }_f{⟩}_6$. Two sinusoid curves are fitted to the data with visibilities of 0.61. The measurements in basis ($|+⟩+e^{i\alpha }|-⟩$) are done using a combination of HWPs, QWP, and PBS. (b). The expectation values of the operators $A_1,\cdots ,B_4$ of the state $|{\psi }_f{⟩}_6$ after the $\sqrt{Z_3}$ transformation.Reuse & Permissions