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arXiv:quant-ph/0107015 · Physical Review A

Quantum search by local adiabatic evolution

Jérémie Roland, Nicolas J. Cerf

Published Mar 26, 2002• 629 citations
arXivHTMLPDF

Abstract

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order $\sqrt{N},$ where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.

DOI
10.1103/physreva.65.042308

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Quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving instances of NP-complete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one through the evolution. In this letter, we show a way of speedup in quantum adiabatic evolution algorithm,using the extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides the new point of view to reduce the running time. It is generally expected that quantum computation may solve the problems which are hard in classical context. Shor’s quantum factorization algorithm[1] and Grover’s quantum search algorithm[2] are the preludes to such expectation. Factorization problem, which is to factorize a large number, needs exponential time resource when a known classical algorithm is applied. However, Shor’s algorithm solves the problem with exponential speedup. Grover algorithm finds a target with quadratic speedup, compared to any known classical search algorithms. Quantum adiabatic evolution algorithm was proposed by Farhi et al. and applied to solve instances of NPcomplete

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The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order N,\sqrt{N},N​, where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.
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.
We report on a detailed analysis of generalization of the local adiabatic search algorithm. Instead of evolving directly from an initial ground state Hamiltonian to a solution Hamiltonian a different evolution path is introduced and is shown that the time required to find an item in a database of size NNN can be made to be independent of the size of the database by modifying the Hamiltonian used to evolve the system.
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.
We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a database of N items. In particular, we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time-dependent Hamiltonian temporarily to a maximum of ∝√ N , it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the √ N bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time-dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.
Quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving instances of NP-complete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one through the evolution. In this letter, we show a way of speedup in quantum adiabatic evolution algorithm,using the extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides the new point of view to reduce the running time. It is generally expected that quantum computation may solve the problems which are hard in classical context. Shor’s quantum factorization algorithm[1] and Grover’s quantum search algorithm[2] are the preludes to such expectation. Factorization problem, which is to factorize a large number, needs exponential time resource when a known classical algorithm is applied. However, Shor’s algorithm solves the problem with exponential speedup. Grover algorithm finds a target with quadratic speedup, compared to any known classical search algorithms. Quantum adiabatic evolution algorithm was proposed by Farhi et al. and applied to solve instances of NPcomplete