A graph-based formalism for surface codes and twists

Abstract

Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to any graph $G$ embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce some new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width.

Figures (18)

• Figure 1: We show a graph embedded on the projective plane. A general rotation system is defined by a set of flags (gray triangles) and three permutations: $\lambda$ swaps flags along the same side of an edge, $\rho$ swaps them within a face adjacent to a vertex, and $\tau$ swaps them across an edge.
• Figure 2: (a) An example section of a Majorana surface code. Majorana operators are placed on each half-edge (blue circles) and at each odd-degree vertex (green circles). The stabilizer associated to an even-degree vertex is the product of Majoranas around that vertex. The stabilizer associated to an odd-degree vertex is the product of Majoranas around that vertex and the Majorana located at that vertex. Gray lines cordon off the Majoronas involved in these stabilizers. The stabilizer associated to the pentagonal face is the product of the ten blue Majoranas on edges around that face. (b) A Majorana surface code of reference vijay2015majorana. In our framework, it arises from a graph triangulating the torus. Here opposite sides of the hexagon are identified (directionally, as indicated by arrows) along with Majoranas on those edges. A total of 72 Majoranas survive this identification. All stabilizers associated to both vertices and faces are the product of six Majoranas.
• Figure 4: Qubit versions of the Majorana codes in Fig. \ref{['fig:Maj_code_example']}. Place a single qubit on vertices with degrees three or four and a pair of qubits on vertices with degrees five or six. Around each vertex write a cyclically anticommuting list of Paulis (acting on qubits at that vertex). Each face represents a stabilizer defined as the product of all Paulis written within that face. Note that (b) depicts a toric code with $X^{\otimes4}$ and $Z^{\otimes4}$ stabilizers but on a lattice different from Kitaev's square lattice kitaev2003fault. In this case, the code is $\llbracket24,2,4\rrbracket$. The number of encoded qubits is clear by Corollary \ref{['cor:number_encoded_qubits']} as this graph embedding is checkerboardable.
• Figure 5: (a) The triangular surface code yoder2017surface, (b) the rotated surface code bombin2007optimal and (c) a stellated surface code kesselring2018boundaries with even greater symmetry. Each code fits in our framework -- in this case, by Definition \ref{['def:qubit_surface_code']} applied to these planar graphs. In (b), we show explicitly the assignment of CALs that gives the familiar rotated surface code. Notice that the outer face also gets assigned a stabilizer. Because of Lemma \ref{['lem:stabilizer_dependence']}(e), the product of stabilizers on all non-outer faces is proportional to the stabilizer on the outer face.
• Figure 6: (a) The smallest error-correcting quantum code, the $\llbracket5,1,3\rrbracket$ code, is a surface code defined by an embedding of the 5-vertex complete graph $K_5$ in the torus. There are different embeddings of $K_5$ in the torus, but no other gives a 5-qubit code with distance more than two. Parts (b-f) show members of the cyclic toric code family with more qubits. See Theorem \ref{['thm:cyclic_toric_codes']} and the paragraph above it for the code definition and parameters. A demonstrative CAL is written around one vertex in each graph. The codes are cyclic when this same CAL is orientated similarly around each vertex.

Theorems & Definitions (75)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• Definition 3.2