Universal transversal gates with color codes: A simplified approach

We provide a simplified yet rigorous presentation of the ideas from Bombín's paper (arXiv:1311.0879v3). Our presentation is self-contained, and assumes only basic concepts from quantum error correction. We provide an explicit construction of a family of color codes in arbitrary dimensions and describe some of their crucial properties. Within this framework, we explicitly show how to transversally implement the generalized phase gate $R_n=\text{diag}(1,e^{2\pi i/2^n})$, which deviates from the method in the aforementioned paper, allowing an arguably simpler proof. We describe how to implement the Hadamard gate $H$ fault tolerantly using code switching. In three dimensions, this yields, together with the transversal controlled-not (cnot), a fault-tolerant universal gate set $\{H,cnot,R_3\}$ without state distillation.

Figure 1

Construction of color codes in two dimensions. In (a) and (b), qubits are placed at vertices, and $X$- and $Z$-type stabilizer generators are associated with faces. In (c) and (d) (the dual picture), qubits are placed on faces, and $X$- and $Z$-type stabilizer generators are associated with vertices. (a) Take a lattice ${\mathcal{L}}_0^{*}$, which is a tilling of the 2-sphere with 3-colorable faces and 3-valent vertices. The surrounding circle is identified with a vertex $v$. The color code on ${\mathcal{L}}_0^{*}$ encodes no logical qubits. (b) To obtain ${\mathcal{L}}^{*}$, remove from ${\mathcal{L}}_0^{*}$ the vertex $v$, together with the three edges and three faces containing it. The color code on ${\mathcal{L}}^{*}$ encodes one logical qubit. (c) (Dual) lattice ${\mathcal{L}}_0$ is obtained from ${\mathcal{L}}_0^{*}$ by replacing faces, edges, and vertices by vertices, edges, and faces, respectively. All faces are triangles, and the vertices are 3-colorable. The color code on ${\mathcal{L}}_0$ encodes no logical qubits. (d) Lattice $\mathcal{L}$ formed from ${\mathcal{L}}_0$ by removing a single face. No stabilizer generators are associated with those vertices belonging to the boundary of the removed face. The color code on $\mathcal{L}$ encodes one logical qubit.

Figure 2

(a) The set of vertices of ${\mathcal{L}}^{*}$, the lattice used to define the color code, is bipartite; it can be split into two subsets: $T$ (hollow circles), and its complement $T^c$ (filled circles). Vertices in $T$ are only connected to vertices in $T^c$ and vice versa. The logical gate ${\overline{R}}_2$ can implemented by applying $R_2^k$ to qubits in $T$, and $R_2^{-k}$ to qubits in $T^c$, where $k\equiv \left|T\right|-|T^c|mod4$. (b) The dual lattice $\mathcal{L}$. Faces are bipartite.

Figure 3

Family of color codes. For a given lattice $\mathcal{L}$, only color codes below the $d\text{th}$ diagonal line can be realized, where $d=dim\mathcal{L}$ and the point $(x,z)$ corresponds to the color code $C{C}_{\mathcal{L}}(x,z)$. This constraint holds since $x$ and $z$ have to satisfy $x+z\le d-2$. An arrow from code $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$ indicates partial order between them, $\mathcal{C}\succ {\mathcal{C}}^{\prime }$. The number placed at $(x,z)$ indicates the maximum gate ${\overline{R}}_n$ which can be implemented transversally with the stabilizer color code $C{C}_d(x,z)$, with $d=x+z+2$, resulting in $n=\lfloor d/(d-1-z)\rfloor$.

Figure 4

Construction of a lattice for a color code in 2D. (a) Take a 2-simplex $\delta$, with vertices colored in red, green, and blue. (b) Divide $\delta$ into “smaller” simplices with matching colors. This is a 3-colorable homogeneous simplicial 2-complex $\mathcal{K}$. (c) Place $\mathcal{K}$ inside a 2-simplex $\tau$ and attach 2-simplices between $\tau$ and $\mathcal{K}$. The resulting homogeneous simplicial 2-complex $\mathcal{L}$ is 3-colorable, and thus we can define a color code on the lattice $\mathcal{L}$.

Figure 5

The family of (fractal) color codes in two dimensions. The first three members of the family: two-dimensional color codes encoding one logical qubits using (a) 7, (b) 13, and (c) 19 physical qubits.

Figure 6

Quantum Reed-Muller code $\mathrm{QRM}\left(m\right)$ as a special case of a (stabilizer) color code $C{C}_{m-1}(0,m-3)$ for (a) $m=3$ (Steane's 7-qubit code), and (b) $m=4$ (the 15-qubit Reed-Muller code). Steane's code with all the possible transversal gates have recently been implemented experimentally [36].