| Summary: | In this paper, we present the fault-tolerant conversion between quantum Reed–Muller (QRM)(2, 5) and QRM(2, 7), and also the conversion between QBCH(15, 7) and QRM(2, 7). Either of the two schemes provides a method to realize universal fault-tolerant quantum computation. In particular, the gate overhead and logical error rate of a logical T gate are provided, as well as the comparison with magic state distillation scheme. In addition, we propose two other fault-tolerant conversion schemes based on $({\boldsymbol{u}}| {\boldsymbol{u}}+{\boldsymbol{v}})$ and $({\boldsymbol{a}}+{\boldsymbol{x}}| {\boldsymbol{b}}+{\boldsymbol{x}}| {\boldsymbol{a}}+{\boldsymbol{b}}-{\boldsymbol{x}})$ constructions.
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