  Magic state distillation (MSD) is an essential element for universal
fault-tolerant quantum computing, which distills a high fidelity magic state
from noisy magic states using ideal (error-corrected) Clifford operations. For
ideal Clifford operations, it needs to be performed on the logical qubits and
hence takes a large spatiotemporal overhead, which is one of the major
bottlenecks for the realization of fault-tolerant quantum computers (FTQC).
Here we propose zero-level distillation, which prepares a high fidelity logical
magic state using physical qubits on a square lattice using nearest-neighbor
two-qubit gates without using multiple logical qubits. The key idea behind is
using the Steane code to distill a logical magic state by using noisy Clifford
gates with error detection. Then the Steane code state is teleported or
converted to the surface codes. By carefully designing such circuits
fault-tolerantly, the error rate of the logical magic state scales $\sim 100
\times p^2$ in terms of the physical error rate $p$. For example, with a
physical error rate of $p=10^{-4}$ ($10^{-3}$), the logical error rate is
reduced to $p_L=10^{-6}$ ($10^{-4}$), resulting in an improvement of two (one)
orders of magnitude. This contributes to reducing both space and time overhead
for early FTQC as well as full-fledged FTQC combined with conventional
multi-level distillation protocols.
