Low Overhead Universal Quantum Computation with Triorthogonal Codes

Abstract: We explore the use of triorthogonal codes for universal fault-tolerant quantum computation and propose two methods to circumvent the Eastin-Knill theorem, which prohibits any single quantum error-correcting code from supporting both universality and a transversal gate set. First, we simplify the implementation of the logical Hadamard gate for triorthogonal codes by exploiting the fact that they have transversal controlled-Z (CZ) gates, resulting in a circuit with reduced overhead. Then, we introduce procedure for generating a symmetric Calderbank-Shor-Steane code paired with a triorthogonal code, which allows CNOT and CZ gate transversality across the pair of codes. We also present a circuit for realizing a logical state teleportation protocol between the two codes, enabling all logical operations to be performed transversally. Finally, we demonstrate how these methods can be integrated into the Steane error correction framework without incurring additional resource cost.