Classification of Topologically Protected Gates for Local Stabilizer Codes

Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be “turned off” for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate $U$ is implementable by a constant-depth circuit only if $UP{U}^{†}$ is in the Clifford group for any Pauli operator $P$. This class of gates includes some non-Clifford gates such as the $\pi /8$ rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.

Figure 1

Noncontractible closed strips ${\gamma }_1$, ${\gamma }_2$ and ${\delta }_1$, ${\delta }_2$ on the torus.

Figure 2

Simplicial partition of the lattice $\Lambda =A\cup B\cup C$. Starting from a triangulation with regular triangles having sides of length $R$, let $C$ be the union of discs of radius $R/4$ centered on the vertices of the triangulation. Let $B\subset \Lambda \setminus C$ be the union of the $R/8$ neighborhoods of each edge in the remaining surface. Finally, let $A=\Lambda \setminus (B\cup C)$ be the union of the remaining capped triangles. A similar but rectangular partition has been used in Ref. 23 to derive upper bounds on parameters of TSCs but is less suitable for generalization to $D>2$.