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We design forward and backward fault-tolerant conversion circuits, whichconvert between the Steane code and the 15-qubit Reed-Muller quantum code so asto provide a universal transversal gate set. In our method, only 7 out of total14 code stabilizers need to be measured, and we further enhance the circuit bysimplifying some stabilizers; thus, we need only to measure eight weight-4stabilizers for one round of forward conversion and seven weight-4 stabilizersfor one round of backward conversion. For conversion, we treat randomsingle-qubit errors and their influence on syndromes of gauge operators, andour novel single-step process enables more efficient fault-tolerant conversionbetween these two codes. We make our method quite general by showing how toconvert between any two adjacent Reed-Muller quantum codes$\overline{\textsf{RM}}(1,m)$ and $\overline{\textsf{RM}}\left(1,m+1\right)$,for which we need only measure stabilizers whose number scales linearly with mrather than exponentially with m obtained in previous work. We provide theexplicit mathematical expression for the necessary stabilizers and theconcomitant resources required.

Fault-tolerant conversion between adjacent Reed-Muller quantum codes based on gauge fixing