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

TL;DR

This paper introduces the SIC unitary propagators within the HSSS quantum search process, emphasizing their role in leveraging quantum symmetry to enhance quantum search efficiency through a novel duality-character approach.

Contribution

It develops a method to construct SIC unitary propagators from basic operators and establishes a theoretical framework using path integrals and energy eigenfunctions for their calculation.

Findings

SIC unitary propagators reflect quantum symmetry, which can be used as a speedup resource.

A concrete method for preparing SIC propagators in single-atom systems is provided.

A general theory for calculating SIC propagators using path integrals is established.

Abstract

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.