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Quantum computation has emerged as a powerful computational medium of ourtime, having demonstrated the remarkable efficiency in factoring a positiveinteger and searching databases faster than any currently known classicalcomputing algorithm. Adiabatic evolution of quantum systems have been studiedas a potential means that physically realizes quantum computation. Up to now,all the research on adiabatic quantum systems has dealt with polynomialtime-bounded computation and little attention has been paid to, for instance,adiabatic quantum systems consuming only constant memory space. Such quantumsystems can be modeled in a form similar to quantum finite automata. Thisexposition dares to ask a bold question of how to make adiabatic quantumcomputation fit into the rapidly progressing framework of quantum automatatheory. As our answer to this eminent but profound question, we first lay out afundamental platform to carry out adiabatic evolutionary quantum systems(AEQSs) with limited computational resources (in size, energy, spectral gap,etc.) and then establish how to construct such AEQSs by operating suitablefamilies of quantum finite automata. We further explore fundamental structuralproperties of decision problems (as well as promise problems) solved quickly bythe appropriately constructed AEQSs.

How Does Adiabatic Quantum Computation Fit into Quantum Automata Theory?