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International Journal of Quantum Information

A family of system Hamiltonian in quantum adiabatic search type problem

Jie Sun, Songfeng Lu

Published Mar 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749919500151

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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.
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.
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.
We apply the transitionless driving on the local adiabatic quantum search algorithm to speed up the adiabatic process. By studying quantum dynamics of the adiabatic search algorithm with the equivalent two-level system, we derive the transitionless driving Hamiltonian for the local adiabatic quantum search algorithm. We found that when adding a transitionless quantum driving term H D ( t ) on the local adiabatic quantum search algorithm, the success rate is 1 exactly with arbitrary evolution time by solving the time-dependent Schr¨odinger equation in eigen-picture. Moreover, we show the reason for the drastic decrease of the evolution time is that the driving Hamiltonian increases the lowest eigenvalues to a maximum of O ( N ) .
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 adiaba tic 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.
In our recent work we showed, by investigating the initialization of some unusual forms of assisted driving Hamiltonians, that the addition of an assisted driving Hamiltonian is not always useful in quantum adiabatic evolution. These unusual forms are those that are not the relatively fixed ones that are widely used in the literature. In this paper, we continue this study, providing further evidence for the validity of the conclusion above by researching some relatively more complex forms of assisted driving scheme, which generalize the ones studied in our previous work.
The early work of Wei et al. has shown that it is possible to do the computation in constant time for quantum search in quantum local adiabatic evolution framework if the system Hamiltonian has an extra enlarged factor proportional to the square-root of the size of the problem. Also in our prior work, we have shown that it is possible to apply a linear interpolating path in the system Hamiltonian to achieve an O(1) time complexity for a quantum global adiabatic computation. In this paper, we discuss the issues when implementing these two quantum algorithms on the quantum circuit model, and find that to our surprise, the time slices produced does not equal to the time complexity of the algorithm in each case. This is in contrast to the previous related results, because there these two quantities always coincide. When taking into account of the whole energy-level increase of the system compared with that of the usual quantum adiabatic evolutions and the corresponding time complexity simultaneously, we find that the time slices needed always consist of the two quantities multiplying together. This result is new, and may be hopeful to find application in designing quantum adiabatic algorithm for problems beyond quantum search.