  Quantum computing has been studied over the past four decades based on two
computational models of quantum circuits and quantum Turing machines. To
capture quantum polynomial-time computability, a new recursion-theoretic
approach was taken lately by Yamakami [J. Symb. Logic 80, pp.~1546--1587, 2020]
by way of recursion schematic definition, which constitutes six initial quantum
functions and three construction schemes of composition, branching, and
multi-qubit quantum recursion. By taking a similar approach, we look into
quantum polylogarithmic-time computability and further explore the expressing
power of elementary schemes designed for such quantum computation. In
particular, we introduce an elementary form of the quantum recursion, called
the fast quantum recursion, and formulate $EQS$ (elementary quantum schemes) of
``elementary'' quantum functions. This class $EQS$ captures exactly quantum
polylogarithmic-time computability, which forms the complexity class
BQPOLYLOGTIME. We also demonstrate the separation of BQPOLYLOGTIME from
NLOGTIME and PPOLYLOGTIME. As a natural extension of $EQS$, we further consider
an algorithmic procedural scheme that implements the well-known
divide-and-conquer strategy. This divide-and-conquer scheme helps compute the
parity function but the scheme cannot be realized within our system $EQS$.
