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Journal of Intelligent & Fuzzy Systems

Can adiabatic algorithms with extra items always be efficient in quantum computation?

Zhigang Zhang, Songfeng Lu, Xin Cong Zhou

Published Feb 27, 2018• 1 citation

Abstract

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.

DOI: 10.3233/jifs-169400

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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.

In this paper, we discuss the adiabatic evolution of one-dimensional projector Hamiltonians. Three kinds of adiabatic algorithms for this problem are shown, in which two of them provide a quadratic speedup over the other one. But when the ground state of the initial Hamiltonian and that of the final Hamiltonian have a zero overlap, the algorithms above all fail, in the sense of infinite time complexity. A corresponding revised method for this phenomenon through adding a driving Hamiltonian is also shown, from which a constant time complexity can be gained. But by a simple analysis, we find that the original time complexity for the adiabatic evolution is shifted to implement the driving Hamiltonian, which is supported by several early works in the literature.

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