Figure 1

(a) Example of a trivalent lattice (in this case a honeycomb lattice) on which the Levin-Wen model can be defined. For the Fibonacci Levin-Wen model, a qubit is associated with each edge. A particular state which satisfies the vertex constraint $Q_{\mathbf{v}}=1$ on each vertex for this model is shown. Thick edges indicate qubits in the state $|1\rangle$, thin edges indicate qubits in the state $|0\rangle$. (b) Action of the vertex operator $Q_{\mathbf{v}}$ on the three qubits on the edges connected to a trivalent vertex, and of the plaquette operators $B_{\mathbf{p}}^s$ on the $2n$ qubits associated with an $n$-sided plaquette.Reuse & Permissions

Figure 2

Quantum circuit which can be used to measure $Q_{\mathbf{v}}$ for the Fibonacci code.Reuse & Permissions

Figure 3

(a) An $F$-move, a five-qubit unitary operation defined in terms of the tensor $F_{cd{e}^{\prime }}^{abe}$. (b) Action of an $F$-move on the abstract trivalent lattice of the Fibonacci code, which illustrates the decoupling of this lattice from the physical qubits. In this example, the qubits (open circles) are arranged in a kagome lattice and lie on the edges of an initial trivalent (hexagonal) lattice. After the $F$-move, the edges of the new trivalent lattice must be distorted if they are forced to coincide with the physical qubit lattice.Reuse & Permissions

Figure 4

$F$-move for Fibonacci anyons. Under this $F$–move, a unitary transformation is performed on one qubit—the one associated with the edge which goes from horizontal to vertical—conditioned on the state of the qubits on the other four edges. As in Fig. 1, thick lines indicate edges in the state $|1\rangle$ and thin lines indicate edges in the state $|0\rangle$. Only those states which satisfy the $Q_{\mathbf{v}}=1$ constraint are shown.Reuse & Permissions

Figure 5

(a) Quantum circuit which carries out an $F$-move for the Fibonacci code [the $2\times 2$ matrix $F$ is given in Eq. (5)]. The labels $abcde$ refer to the same labels in Fig. 3a. (b) Five-qubit controlled-$F$ gate expressed in terms of a five-qubit Toffoli gate. Here, $R(\pm \theta \widehat{y})=e^{\pm i\theta {\sigma }_y/2}$ are single-qubit rotations about the $y$ axis with $\theta ={\tan }^{-1}{\phi }^{-1/2}$ for which $R\left(\theta \widehat{y}\right)XR(-\theta \widehat{y})=F$.Reuse & Permissions

Figure 6

(a) The pentagon equation, a self-consistency condition which the $F$-move must satisfy. As shown here, the pentagon equation corresponds to a series of $F$-moves which take a particular seven-edged lattice (upper left) back to an identical lattice (lower left) while two of the qubits associated with the lattice edges are swapped. Here and in subsequent figures, the edges associated with the initial state before each $F$-move are color coded as in Fig. 3. (b) The pentagon equation as a quantum circuit identity. The sequence of $F$-moves shown in (a) are carried out by repeatedly applying the $F$ circuit defined in Fig. 5. The labels $abcde$ in each green box refer to the labels in Fig. 5. The circuit equality holds provided the vertex constraint $Q_{\mathbf{v}}=1$ is satisfied on all three vertices in the initial lattice. In the figure, the triplets of numbers given below “$Q_{\mathbf{v}}=1$” in the red box indicate the qubits which meet at these vertices.Reuse & Permissions

Figure 7

Simple two-qubit circuit identity obtained by setting the five effective control qubits (qubits 1, 2, 3, 4, and 7) in the pentagon circuit identity shown in Fig. 6b to the state $|1\rangle$.Reuse & Permissions

Figure 8

Reduction of a hexagonal plaquette to a tadpole through a sequence of $F$-moves.Reuse & Permissions

Figure 9

Reduced four-qubit $F$-move obtained by identifying the qubits labeled $a$ and $d$ in Fig. 5.Reuse & Permissions

Figure 10

(a) $S$ transformation acting on a two-qubit tadpole. The tensor $S_{b{b}^{\prime }}^a$ is defined in the text. (b) $S$ circuit which carries out an $S$ transformation. The $2\times 2$ matrix $S$ is given in Eq. (9). Here, $R(\pm \rho \widehat{y})=e^{\pm i\rho {\sigma }_y/2}$ are single-qubit rotations about the $y$ axis with $\rho ={\tan }^{-1}{\phi }^{-1}$ for which $R\left(\rho \widehat{y}\right)XR(-\rho \widehat{y})=S$. (c) Quantum circuit which uses the $S$ circuit to measure $B_{\mathbf{p}}$ for a two-qubit tadpole.Reuse & Permissions

Figure 11

Quantum circuit which can be used to measure $B_{\mathbf{p}}$ for the Fibonacci code on a hexagonal plaquette based on the plaquette reduction shown in Fig. 8. It must be verified that $Q_{\mathrm{v}}=1$ on each of the six vertices of the plaquette before carrying out the circuit.Reuse & Permissions

Figure 12

(a) Sequence of $F$-moves which pulls a tadpole through a line. (b) Four-qubit quantum circuit which initializes a tadpole with an $S$ circuit, carries out the sequence of two $F$-moves shown in (a), and then performs another $S$ circuit so that measuring qubit 3 would yield $B_{\mathbf{p}}$ for the new tadpole. The tadpole is initialized to a state with $B_{\mathbf{p}}=1$ or 0 depending on whether the initial state of qubit 4 is $|0\rangle$ or $|1\rangle$, respectively. The circuit equality holds provided $Q_{\mathbf{v}}=1$ on the vertices of the initial lattice. As in Fig. 6b, these vertices are labeled inside the red box. The $2\times 2$ matrix $U$ is given by Eq. (10) in the text.Reuse & Permissions

Figure 13

(a) Simplified two-qubit circuit identity obtained by setting qubits 1 and 2 to the state $|1\rangle$ in the circuit identity shown in Fig. 12b. (b) Equivalent circuit identity obtained by moving the two NOT gates from the left side of the identity shown in (a) to the right side.Reuse & Permissions