Search

Recently, Kieu derived a new class of time–energy uncertainty relations, which is not only formal but also suitable for evaluating the speed limit of quantum dynamics. However, when this time–energy uncertainty relation was applied to study the quantum adiabatic Grover's search, some approximation had been done for reaching the right conclusion. That is, the final state being orthogonal to the initial state in this problem was assumed which may cause confusion, while the truth is that the overlap between these two states is indeed very small when the problem size is very big but not equal to zero exactly. In this paper, we modify the time–energy uncertainty relation of Kieu applied to two quantum evolutions of Grover's problem, and find that the resulting computational complexities match well with the previous works on this topic but without the need of making the assumption like that in the work of Kieu. [ABSTRACT FROM AUTHOR]

In this paper [J. Sun, S. Lu and F. Liu, Open Syst. Info. Dyn.23(3) (2016) 1650016], we have studied a general class of adiabatic evolutions in quantum adiabatic search type problem, and found another limitation on it, besides those reported in our early work. In this paper, we extend this study by considering adding an extra driving scheme to the system Hamiltonian of this type of quantum adiabatic evolution. It will be shown that, like the previous study in [J. Sun, S. Lu and F. Liu, Open Syst. Info. Dyn.23(3) (2016) 1650016], the limitation above can also possibly penetrate into the search type quantum-adiabatic computation even by this more general family of system Hamiltonian, which may shed light on what we should have in mind when solving practical problems by means of quantum adiabatic evolution. [ABSTRACT FROM AUTHOR]

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]

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]

In this paper, we propose a modified construction of the quantum adiabatic algorithm for Boolean functions studied by M. Andrecut et al. [13, 14]. Our algorithm has the time complexity for the evaluation of Boolean functions, without additional computational cost of implementing the driving Hamiltonian, which is required by the adiabatic evolution described in [13, 14]. [ABSTRACT FROM AUTHOR]