  The HSSS quantum search process owns the dual character that it obeys both
the unitary quantum dynamics and the mathematical-logical principle of the
unstructured search problem. It is essentially different from a conventional
quantum search algorithm. It is constructed with the duality-character oracle
operations of unstructured search problem. It consists of the two consecutive
steps: (1) the search-space dynamical reduction and (2) the dynamical
quantum-state-difference amplification (QUANSDAM). The QUANSDAM process is
directly constructed with the SIC unitary propagators, while the latter each
are prepared with the basic SIC unitary operators. Here the preparation for the
SIC unitary propagators of a single-atom system is concretely carried out by
starting from the basic SIC unitary operators. The SIC unitary propagator of a
quantum system may reflect the quantum symmetry of the quantum system, while
the basic SIC unitary operators may not. The quantum symmetry is considered as
the fundamental quantum-computing-speedup resource in the quantum-computing
speedup theory. The purpose for the preparation is ultimately to employ the
quantum symmetry to speed up the QUANSDAM process. The preparation is a
solution-information transfer process. It is unitary and deterministic. It
obeys the information conservation law. In methodology it is based on the
energy eigenfunction expansion and the multiple-quantum operator algebra space.
Furthermore, a general theory mainly based on the Feynman path integration
technique and also the energy eigenfunction expansion method is established to
treat theoretically and calculate a SIC unitary propagator of any quantum
system in the coordinate representation, which may be further used to construct
theoretically an exponential QUANSDAM process in future.
