Low-overhead code concatenation approaches for universal quantum computation

Fault-tolerant quantum circuit design can be done by using a set of transversal gates. However, as there is no error correction code with universal gates, several approaches have been proposed which, in combination make fault-tolerant computation possible. Magic state distillation, gauge fixing, switching, concatenation and pieceable fault tolerance are well-known examples such approaches. the overhead these one main bottlenecks for large-scale computation. In this paper, two computation, mainly based on concatenation, proposed. The first approach combines or magic distillation second extends nonuniformity concatenated codes allowing to apply CNOT gates between different codes. outperform terms both number qubits distance also significantly less resource than distillation. This achieved at cost reducing effective implementing non-transversal