Transversality Across Two Distinct Quantum Codes and Its Application to Quantum Repeaters

Abstract

In this work, we generalize the concept of transversality in quantum error-correcting codes. Unlike conventional methods, we study transversal logical two-qubit gates between two different codes. We consider an application of this concept for quantum networks which consist of multiple intermediate stations equipped with quantum repeaters (QRs). The stations may experience different error models. For instance, in one station, a particular Pauli error may dominate, whereas in its neighboring stations, other types of Pauli errors are more likely. The standard approach of using the same CSS code Q in different stations does not allow for simultaneously adaptating Q to be optimized to the errors prevailing in various stations. Considering this fact, we suggest using different CSS codes in each station. In this work, we analyze CNOT and CZ transversality for pairs of CSS codes and provide a complete characterization. We formulate necessary and sufficient conditions for a pair of CSS codes to be CNOT and CZ transversal. In comparison to the conventional approach of having the same CSS code, we show that these conditions are less restrictive.