Locality-preserving logical operators in topological stabilizer codes

Abstract

Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabiliser codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including colour codes and toric codes, as well as more general codes such as abelian quantum double models and codes with fermionic excitations in more than two dimensions.

Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates

“…Since at most c elements of A intersect at any subset of qudits Q i , we have the inequality To get (i) from Eqs. (13)(14)(15), note that they each hold for all c, and so we can replace ∆ c (G) with max c≥1 ∆ c (G) in all three equations. Since Eq.…”

“…The latter result has an important implication -one cannot achieve a universal gate set with constant-depth local circuits on two-dimensional (2D) topological codes such as those in [11,12]. We also remark that one can consider more general models beyond stabilizer codes, such as 2D topological quantum field theories, and characterize the set of gates implementable by locality-preserving unitaries [13,14].…”

Tensor networks with a twist: Anyon-permuting domain walls and defects in projected entangled pair states

“…Recently, a connection has been made between faulttolerant logical gates, locality-preserving symmetries and anyon-permuting domain walls. In particular, an equivalence was established between such logical gates and domain walls for topological stabilizer codes [17][18][19][20] .…”

Three-dimensional surface codes: Transversal gates and fault-tolerant architectures

“…We are particularly interested in two types of logical operators in topological codes: locality-preserving logical operators (LP-LOs) and transversal logical operators. LPLOs are operators that map errors in some region of a code R to errors in a region R which is at most a constant size C bigger than R [26]. A transversal logical operator is a logical operator realized by a quantum circuit of depth one which does not couple physical qubits in the same code (block).…”