  The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged
as a key technique in achieving fault-tolerant quantum computation using
photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502
(2019)] showed that experimentally tractable Gaussian operations combined with
preparing a GKP codeword $\lvert 0\rangle$ suffice to implement universal
quantum computation, this implementation scheme involves a distillation of a
logical magic state $\lvert H\rangle$ of the GKP code, which inevitably imposes
a trade-off between implementation cost and fidelity. In contrast, we propose a
scheme of preparing $\lvert H\rangle$ directly and combining Gaussian
operations only with $\lvert H\rangle$ to achieve the universality without this
magic state distillation. In addition, we develop an analytical method to
obtain bounds of fundamental limit on transformation between $\lvert H\rangle$
and $\lvert 0\rangle$, finding an application of quantum resource theories to
cost analysis of quantum computation with the GKP code. Our results lead to an
essential reduction of required non-Gaussian resources for photonic
fault-tolerant quantum computation compared to the previous scheme.
