  Quantum search algorithms offer a remarkable advantage of quadratic reduction
in query complexity using quantum superposition principle. However, how an
actual architecture may access and handle the database in a quantum superposed
state has been largely unexplored so far; the quantum state of data was simply
assumed to be prepared and accessed by a black-box operation -- so-called
oracle, even though this process, if not appropriately designed, may adversely
diminish the quantum query advantage. Here, we introduce an efficient quantum
data-access process, dubbed as quantum data-access machine (QDAM), and present
a general architecture for quantum search algorithm. We analyze the runtime of
our algorithm in view of the fault-tolerant quantum computation (FTQC)
consisting of logical qubits within an effective quantum error correction code.
Specifically, we introduce a measure involving two computational complexities,
i.e. quantum query and $T$-depth complexities, which can be critical to assess
performance since the logical non-Clifford gates, such as the $T$ (i.e.,
$\pi/8$ rotation) gate, are known to be costliest to implement in FTQC. Our
analysis shows that for $N$ searching data, a QDAM model exhibiting a
logarithmic, i.e., $O(\log{N})$, growth of the $T$-depth complexity can be
constructed. Further analysis reveals that our QDAM-embedded quantum search
requires $O(\sqrt{N} \times \log{N})$ runtime cost. Our study thus demonstrates
that the quantum data search algorithm can truly speed up over classical
approaches with the logarithmic $T$-depth QDAM as a key component.
