Pairs of CSS Error Correction Codes with Transversal Gate Implementation

The realization of universal fault-tolerant (FT) quantum computation remains one of the most challenging problems in the field. The Eastin-Knill theorem establishes that no single quantum error correction code (QECC) can support a logical gate set that is simultaneously transversal and universal. While existing bypass strategies such as magic state distillation and code switching provide a theoretical pathway to universality, they often introduce prohibitive resource overheads in terms of physical qubit counts and circuit depth. This thesis proposes a generalized framework for cross-code transversality as a high-efficiency alternative for achieving universal FT computation.

Moving beyond the traditional definition of transversality, we derive the mathematical conditions required for multi-qubit gates, specifically CNOT and CZ gates, to act transversally between distinct QECCs. This theoretical framework allows for the interaction of logical information stored in different codes. Furthermore, we introduce a novel hybrid system that supports a transversal Toffoli gate, which has already been proved no single-QECC system could have such transversality.

The practical utility of this cross-code framework is demonstrated through three main applications. First, we optimize second-generation (2G) quantum repeater systems by introducing a code design called mirrored code structure. By using structurally correlated but distinct codes, we improve communication fidelity and reduce the minimum number of intermediate stations required for reliable long-distance entanglement distribution. Second, we propose a transversal code switching protocol that minimizes the qubit and gate overhead by transferring logical states between triorthogonal and paired symmetric codes via one-way transversal interactions. Finally, we present a high-fidelity Toffoli state distillation scheme based on a hybrid-code architecture without the need of T-magic states.

Numerical simulations under realistic noise models indicate that these cross-code methods significantly reduce the qubit and gate overhead required for FT operations compared to traditional single-code protocols. By finding the conditions for interactions between diverse code families, this work provides a scalable roadmap for universal FT quantum computing that leverages the different features of multiple QECCs within a single, integrated quantum system.