Universal quantum computation in the surface code using non-Abelian islands

The surface code is currently the primary proposed method for performing quantum error correction. However, despite its many advantages, it has no native method to fault-tolerantly apply non-Clifford gates. Additional techniques are therefore required to achieve universal quantum computation. Here we propose a hybrid scheme which uses small islands of a qudit variant of the surface code to enhance the computational power of the standard surface code. This allows the non-trivial action of the non-Abelian anyons in the former to process information stored in the latter. Specifically, we show that a nonstabilizer state can be prepared, which allows universality to be achieved.

Figure 1

The $D\left(S_3\right)$ quantum double model is defined on an oriented square lattice. Six-level qudits, shown as dots, are situated at each edge. For each vertex and each plaquette, we define operators $A_v^g$ and $B_{p,v^{\prime }}^h$ acting on the lattice qudits according to Eqs. (5) and (6).

Figure 2

(a) The domain wall $\mathfrak{D}$ separates the lattice into two regions. Region ${\mathfrak{R}}_1$ (${\mathfrak{R}}_2$) is associated with the $D\left({\mathbb{Z}}_2\right)\left[D\left(S_3\right)\right]$ quantum double model. Qubits (six-level qudits) are indicated as red (blue) dots. The corresponding stabilizer operators act on the vertices and plaquettes as indicated. (b) A domain-wall vertex consists of two “halves” $v_l$ and $v_r$, where $v_l$ ($v_r$) belongs to ${\mathfrak{R}}_1$ (${\mathfrak{R}}_2$). (c) A domain-wall plaquette consists of a single qubit and a single six-level spin.

Figure 3

Let ${\mathfrak{R}}_1$ (${\mathfrak{R}}_2$) be associated with the $D\left({\mathbb{Z}}_2\right)\left[D\left(S_3\right)\right]$ quantum double model. (a) An $A+B+2C$ hole living in ${\mathfrak{R}}_2$ is moved towards the domain wall. Upon removing the domain-wall stabilizer $A_{v_d}^K$, the condensation rules ensure that the logical information is delocalized both in the $D\left({\mathbb{Z}}_2\right)$ as well as in the $D\left(S_3\right)$ part of the hole, and that no mixing between basis states occurs. (b) In the intermediate stage of the transfer, the hole has support in both ${\mathfrak{R}}_1$ and ${\mathfrak{R}}_2$. (c) The transfer is completed by measuring $A_{v_d}^K$ and applying appropriate error correction procedures, leaving us with a $1+e$ hole living in ${\mathfrak{R}}_1$.

Figure 4

Examples of ribbons. $\tau$ is a dual triangle with end points $s_0\text{and}{s}_1$. ${\tau }^{\prime }$ is a direct triangle with endpoints $t_0$ and $t_1$. $\rho$ is an arbitrary ribbon composed of seven triangles and with end points given by $r_0\text{and}{r}_1$.

Figure 5

Definition of controlled multiplications for qudits taking values in the group algebra $\mathbb{C}\left[S_3\right]$. (a) Definition of the controlled left multiplication. (b) Definition of the controlled right multiplication. Note that by a slight abuse of notation, the gate $R$ is defined as applying the operator $R^{g^{-1}}$ (and not $R^g$) for a control qubit in state $|g\rangle$.

Figure 6

(a) The convention for labeling the data qudits around vertices. The vertex syndrome qudit is denoted as $v$ by a slight abuse of notation, and the data qudits in $\mathrm{star}\left(v\right)$ are labeled $a$ to $d$. The zig-zag sequence is crucial in order for neighboring circuits not to interfere with each other. (b) The convention for labeling the data qudits around plaquettes. The plaquette syndrome qudit is denoted as $p$. Again particular attention must be paid to the specific ordering of the labels. (c) The stabilizer circuit measuring the irreducible representations of $S_3$ according to which the qudits in $\mathrm{star}\left(v\right)$ transform under the vertex operations $A_v^g$. The vertex syndrome qudit is denoted as $v$, and the data qudits in $\mathrm{star}\left(v\right)$ are labeled $a$ to $d$ in the order indicated in (a). Information on Fourier transforms on non-Abelian groups can be found in Ref. [30]. (d) The stabilizer circuit measuring the flux at plaquette $p$ with respect to the top right neighboring vertex $v$. The plaquette syndrome qudit is denoted as $p$ and the data qudits in $\partial p$ are labeled $a$ to $d$ in the order indicated in (b).

Figure 7

The $B$ occupancy of a pair of $G$ particles is transferred to an $A+B+2C$ hole. In the notations introduced in Appendix pp5, this maps the state ${|\psi \rangle }_f={a|A\rangle }_f+b{|B\rangle }_f$ to the state ${|\psi \rangle }_h={a|A\rangle }_h+b{|B\rangle }_h$. The dashed triangles indicate the support of the ribbon operators used to perform the movement of the $G$ particles. The shaded region denotes the hole, i.e., the region where the vertex stabilizers are turned off and replaced by single qudit measurements in the computational basis. (a) A pair of $G$ particles in the state ${|\psi \rangle }_f={a|A\rangle }_f+b{|B\rangle }_f$ and an $A+B+2C$ hole with vacuum anyonic occupancy. (b) One of the $G$ particles is moved until its charge part is located inside the hole. (c) The second $G$ particle is moved into the hole as well and its charge part fused with the charge part of the first particle. (d) The flux parts of the two $G$ particles are fused, removing the $G$ particles from the lattice and leaving us with the hole in the state ${|\psi \rangle }_h={a|A\rangle }_h+b{|B\rangle }_h$.