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arXiv:0811.0663

Complete Adiabatic Quantum Search in Unsorted Databases

Nanyang Xu, Xinhua Peng, Mingjun Shi, Jiangfeng Du

Published Nov 5, 2008• 0 citations
arXivHTMLPDF

Abstract

We propose a new adiabatic algorithm for the unsorted database search problem. This algorithm saves two thirds of qubits than Grover's algorithm in realizations. Meanwhile, we analyze the time complexity of the algorithm by both perturbative method and numerical simulation. The results show it provides a better speedup than the previous adiabatic search algorithm.

DOI
10.48550/arxiv.0811.0663

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We propose a new adiabatic algorithm for the unsorted database search problem. This algorithm saves two thirds of qubits than Grover's algorithm in realizations. Meanwhile, we analyze the time complexity of the algorithm by both perturbative method and numerical simulation. The results show it provides a better speedup than the previous adiabatic search algorithm.
We propose a non-Hermitian quantum annealing algorithm which can be useful for solving complex optimization problems. We demonstrate our approach on Grover's problem [L. K. Grover, Phys. Rev. Lett. 79, 325 (1997)] of finding a marked item inside of an unsorted database. We show that the energy gap between the ground and excited states depends on the relaxation parameters and is not exponentially small. This allows a significant reduction of the searching time, which is proportional to the number of qubits. We discuss the relations between the probabilities of finding the ground state and the survival of a quantum computer in a dissipative environment, and we discuss alternate ways to solve NPNPNP-complete problems.
Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.
Recently, Zhang and Lu provided a quantum search algorithm based on partial adiabatic evolution, which beats the time bound of local adiabatic search when the number of marked items in the unsorted database is larger than one. Later, they found that the above two adiabatic search algorithms had the same time complexity when there is only one marked item in the database. In the present paper, following the idea of Roland and Cerf [Roland J and Cerf N J 2002 Phys. Rev. A 65 042308 ], if within the small symmetric evolution interval defined by Zhang et al., a local adiabatic evolution is performed instead of the original "global" one, this "new" algorithm exhibits slightly better performance, although they are progressively equivalent with M increasing. In addition, the proof of the optimality for this partial evolution based local adiabatic search when M = 1 is also presented. Two other special cases of the adiabatic algorithm obtained by appropriately tuning the evolution interval of partial adiabatic evolution based quantum search, which are found to have the same phenomenon above, are also discussed.
In this summary we discuss two new algorithms for Grover's unsorted database search problem that claimed to have reached exponential speedup over Grover's original algorithm. One is in the quantum setting with "power queries" that allow for exponential reduction in the number of queries over Grover's original algorithm with "bit queries". The other is to use "dubit queries" on a duality computer - a new computing model uses a quantum system's wave-particle duality, which is able to achieve even greater computing power and better capability than existent quantum computers we have been discussing. We discuss the shortages and difficulties of both schemes as well.
Grover's algorithm has achieved great success. But quantum search algorithms still are not complete algorithms because of Grover's Oracle. We concerned on this problem and present a new quantum search algorithm in adiabatic model without Oracle. We analyze the general difficulties in quantum search algorithms and show how to solve them in the present algorithm. As well this algorithm could deal with both single-solution and multi-solution searches without modification. We also implement this algorithm on NMR quantum computer. It is the first experiment which perform a real quantum database search rather than a marked-state search.