General scheme for preparation of different topological states on cluster states

Although it is well-known that all quantum states can be produced by single-qubit measurements on the cluster states, it is not a simple task to explicitly find which measurement patterns on the cluster states can generate different quantum states. In this paper, we introduce a general scheme to find measurement patterns corresponding to Calderbank–Shor–Steane (CSS) topological states containing Kitaev's toric code states and color code states on different lattices and in different dimensions. Furthermore, we find a measurement pattern for generating non-Abelian anyons where measurement-induced defects on a toric code state play the role of Ising anyons. We also support our scheme by a graphical notation where, by following a few simple graphical transformations, one will be able to convert a CSS topological state to a cluster state. Our scheme can also be used for experimental realization of anyons on cluster states.

Figure 1

A stabilizer corresponding to central qubit on a graph state with five qubits.

Figure 2

The plaquette and vertex operators $A_s$ and $B_p$ of the TC state on a square lattice. There are two noncontractible loops corresponding to each direction of the lattice on a torus. For each direction $\sigma =\{0,1\}$, two loop operators $L_x^{\sigma }$ and $L_z^{\sigma }$ are constructed by $Z$ and $X$ operators.

Figure 3

A pair of charge (flux) anyons is created by applying an operator $X$ $\left(Z\right)$ on one of the qubits. A pair of $\epsilon =e\times m$ anyons is created by applying an operator $Y$ on a qubit.

Figure 4

A pair of charge (flux) anyons is created by applying an operator $X$ $\left(Z\right)$ on one of the qubits. A charge (flux) anyon can move on the lattice (dual lattice) with applying a string of operators $X$. Dotted line shows the winding of a flux anyon around a charge anyon; it leads to a minus sign to the wave function.

Figure 5

TC on a 3D lattice where qubits live in edges of the lattice. $Z$-type operators are defined corresponding to each plaquette of the lattice and $X$-type operators are defined corresponding to each vertex of the lattice.

Figure 6

Plaquettes of a hexagonal lattice are colored by three different colors. There are three noncontractible loops in two directions of the lattice corresponding to three colors of the plaquettes. A string of $X$ or $Z$ operators can generate two excitations in two plaquettes corresponding to two end points of that string.

Figure 7

CC on a 3-colex where qubits live in the vertices of the lattice. All cells and edges of the lattice are colored by four different colors where an edge with a specific color connects two cells with the same color. $X$-type operators are defined corresponding to each cell of the lattice and $Z$-type operators are defined corresponding to each plaquette of the lattice. A string of $Z$ operators on edges with a specific color generates excitations in two cells corresponding to end points of that string.

Figure 8

On the left-hand side, we show a CSS quantum state where each closed curve shows a $z$ cell of the model. On the right-hand side, we show a bipartite graph state corresponding to the CSS state in the left-hand side. White qubits should be measured in the $X$ basis.

Figure 9

On the left-hand side, there is a graph state where the white qubits are measured in the $X$ basis. It leads to a new graph state on unmeasured qubits as shown on the right-hand side, where two vertices of the initial graph are merged to each other.

Figure 10

(a) On the left-hand side, there is a graph state where the blue qubit is measured in the $Z$ basis. On the right-hand side, there is a new graph state on unmeasured qubits, where the blue qubit of the initial graph is removed. (b) On the left-hand side, there is a graph state where the yellow qubit is measured in the $Y$ basis. It leads to a new graph state on unmeasured qubits as shown on the right-hand side. Furthermore, after measurement of the yellow qubit on the left-hand side, it is necessary to apply an operator $\sqrt{Z}$ on unmeasured qubits to generate a graph state.

Figure 11

On the left-hand side, there is a graph state where the yellow qubits are measured in the $Y$ basis, successively. It leads to a new graph state with a crossing between the edges on unmeasured qubits.

Figure 12

On the left-hand side, we have a dual of TC on a 3D lattice where $X$-type operators are defined corresponding to plaquettes of the lattice and $Z$-type operators are defined corresponding to vertices of the lattice. On the right-hand side, we show a bipartite graph state corresponding to the CSS state on the left-hand side, where white qubits should be measured in the $X$ basis.

Figure 13

On the right-hand side, we use the uniformizing rule to reduce degrees of all vertices in the left-hand lattice.

Figure 14

On the left-hand side, we deform all edges of the lattice in Fig. (13 b) in order to convert the lattice to a square structure. On the right-hand side, we use the flattening rule to remove all crossings in the left-hand lattice.

Figure 15

A 3-colex with three cells. The central cell involves 32 qubits and qubic cells involve 8 qubits.

Figure 16

A bipartite graph state corresponding to CC model on 3-colex in Fig. 15, where white qubits should be measured in the $X$ basis.

Figure 17

We use the uniformizing rule to reduce degrees of vertices. For the central cell in Fig. 16, we have used the uniformizing rule for 16 times.

Figure 18

(a) We show a 3-colex with six cells where dual of a CC has been defined. (b) Similar to the previous example in Fig. 17, we use the uniformizing rule to reduce degrees of vertices. Furthermore, there are also a few crossings between some edges that can be removed by the flattening rule.

Figure 19

On the left-hand side, there is a square lattice with qubits living on the edges of the lattice. On the right-hand side, we have the same lattice where we attach a chess pattern on that lattice. A plaquette (vertex) operator $B_p$ in the left-hand side corresponds to a dark (white) plaquette in the right-hand side. A string operator in white (black) plaquettes generates two flux (charge) anyons in two end points of that string.

Figure 20

(a) We perform a measurement on the qubit $a$ in the $Y$ basis. (b) After measuring the qubit $a$, there are two new stabilizers corresponding to two deformed plaquettes. (c) We transport one of the deformed plaquettes by measuring another qubit, that we have denoted by 6, in the $Y$ basis.

Figure 21

Red curve: a flux particle is converted to a charge particle when transported around a twist. Green loop: if a charge particle turns around both of twists, it can generate a closed string.

Figure 22

On the right-hand side, we have shown a pair of twists which have been generated by dislocation in the lattice and have been proposed by Bombin [27]. On the left-hand side, we have shown a pair of twists which have been generated by measurement. In both models, a charge anyon converts to a flux anyon when it transports around a twist.

Figure 23

On the left-hand side, there is a TC state on a triangular lattice. The gray triangle shows a hole in the lattice where the corresponding plaquette operator is absent. On the right-hand side, there is a graph state on a bipartite graph where white qubits should be measured in the basis of the Pauli operator $X$.

Figure 24

On the left-hand side, there is a CC state on a hexagonal lattice. On the right-hand side, there is a graph state on a bipartite graph where white qubits should be measured in the basis of the Pauli operator $X$.

Figure 25

On the left-hand side, there is a graph state on a bipartite graph corresponding to the TC state on a triangular lattice, where the white qubits should be measured in the basis $X$. In the center, by deforming the edges, the initial graph is matched on a square lattice where the white qubits should be measured in basis $X$. On the right-hand side, by the face insertion the empty points are filled by blue qubits that should be measured in the $Z$ basis.

Figure 26

On the left-hand side, there is a graph state on a bipartite graph corresponding to the CC state on a hexagonal lattice, where the white qubits should be measured in the basis $X$. In the center, by merging rule, the initial graph is matched on a square lattice where the white qubits should be measured in basis $X$. On the right-hand side, by the face insertion the empty points are filled by blue qubits that should be measured in the $Z$ basis.

Figure 27

Each twist has been denoted by a star. Left: a closed string of a fermion $\epsilon$ can not characterize entering a fermion into the region involved by the red loop. Right: green and blue closed strings corresponding to winding charge and flux anyons around a pair of twists can characterize entering a fermion into the region involved by the blue and green loops. A red closed string corresponding to winding fermions $\epsilon$ around one of the twists can not characterize entering a fermion into the region involved by the red loop.