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It is known that when the initial state has no overlap with the final state in an adiabatic computation, an extra assisted driving Hamiltonian has to be used to avoid the failure of quantum adiabatic algorithm. This assisted Hamiltonian has a relatively fixed form, which can be seen in the related literature but the mechanism behind it is still not clear enough. In this paper, we discuss the usefulness of a generalized form of assisted driving Hamiltonian in adiabatic evolution which may shed light on this problem. [ABSTRACT FROM AUTHOR]

It has been known that when an assisted driving item is added to the Hamiltonian of the main system, the efficiency of the resultant adiabatic evolution can be significantly improved. In some special cases, it can be seen that, only by adding an assisted driving Hamiltonian can the resultant adiabatic evolution work. So it seems that, the additional driving Hamiltonian plays an important role in adiabatic computing. In this paper, we show five kinds of possible forms of the driving Hamiltonian, and analyses the performance of the algorithms. We find that the former fourth kinds of driving Hamiltonian may still cause the adiabatic algorithm lose their efficiency when the overlap of the initial and final states equals some special value. More importantly, we find that only the final kinds of the driving Hamiltonian can be extended. By modifying its interpolating functions, the final kind of adiabatic algorithm can suit any fidelity of the systems to keep the adiabatic evolution always be valid. Finally, the time complexity of the extended algorithm is better than the original adiabatic algorithm, which owns a constant running time complexity. Although the time complexity is shifted to the complexity of implementing the driving Hamiltonian added. [ABSTRACT FROM AUTHOR]

The general class of models of adiabatic evolution was proposed to speed up the usual adiabatic computation in the case of quantum search problem. It was shown [8] that, by temporarily increasing the ground state energy of a time-dependent Hamiltonian to a suitable quantity, the quantum computation can perform the calculation in time complexity O(1). But it is also known that if the overlap between the initial and final states of the system is zero, then the computation based on the generalized models of adiabatic evolution can break down completely. In this paper, we find another severe limitation for this class of adiabatic evolution-based algorithms, which should be taken into account in applications. That is, it is still possible that this kind of evolution designed to deal with the quantum search problem fails completely if the interpolating paths in the system Hamiltonian are chosen inappropriately, while the usual adiabatic evolutions can do the same job relatively effectively. This implies that it is not always recommendable to use nonlinear paths in adiabatic computation. On the contrary, the usual simple adiabatic evolution may be sufficient for effective use. [ABSTRACT FROM AUTHOR]