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The HSSS quantum search process owns the dual character that it obeys boththe unitary quantum dynamics and the mathematical-logical principle of theunstructured search problem. It is essentially different from a conventionalquantum search algorithm. It is constructed with the duality-character oracleoperations of unstructured search problem. It consists of the two consecutivesteps: (1) the search-space dynamical reduction and (2) the dynamicalquantum-state-difference amplification (QUANSDAM). The QUANSDAM process isdirectly constructed with the SIC unitary propagators, while the latter eachare prepared with the basic SIC unitary operators. Here the preparation for theSIC unitary propagators of a single-atom system is concretely carried out bystarting from the basic SIC unitary operators. The SIC unitary propagator of aquantum system may reflect the quantum symmetry of the quantum system, whilethe basic SIC unitary operators may not. The quantum symmetry is considered asthe fundamental quantum-computing-speedup resource in the quantum-computingspeedup theory. The purpose for the preparation is ultimately to employ thequantum symmetry to speed up the QUANSDAM process. The preparation is asolution-information transfer process. It is unitary and deterministic. Itobeys the information conservation law. In methodology it is based on theenergy eigenfunction expansion and the multiple-quantum operator algebra space.Furthermore, a general theory mainly based on the Feynman path integrationtechnique and also the energy eigenfunction expansion method is established totreat theoretically and calculate a SIC unitary propagator of any quantumsystem in the coordinate representation, which may be further used to constructtheoretically an exponential QUANSDAM process in future.

The duality-character Solution-Information-Carrying (SIC) unitary propagators