Logical gates by code deformation in topological quantum codes

Quantum error correcting codes (QECCs) allow us to protect qubits from noise and are expected to be essential features of any kind of scalable, fault-tolerant quantum computer. By encoding information in a QECC we make unintentional modification of that information less likely, but also make intentional modification more difficult. Operations that perform such modifications are referred to as “logical operations” or “logical gates” and a common, fault-tolerant approach to performing these operations is the use of “transversal” logical gates. However, a fundamental theorem of quantum error correction is that no QECC can possess a universal set of transversal gates. An alternate approach to performing logical gates is the technique of code deformation, which involves a sequence of modifications (deformations) of the code which transform the encoded information. In the class of QECCs called topological codes these deformations have natural mathematical interpretations in terms of transformations of a manifold, and physical interpretations in terms of the motions of quasiparticles in certain condensed matter systems. Here we examine two different code deformation techniques. The first is the braiding of a certain type of defect (a twist defect) in multiple copies of the two- dimensional surface code. We classify the set of logical operations which can be performed in this fashion by drawing a connection to the braiding relations of a hierarchy of anyon models. The second example involves switching between two- and three-dimensional versions of a code and an unorthodox method of decoding called just-in-time (JIT) decoding. We numerically demonstrate the existence of a threshold for this decoding strategy in surface codes and then proceed to examine the errors that occur if partial transversal gates are interleaved with this procedure