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arXiv:quant-ph/0412117

Quantum search algorithm by adiabatic evolution under a priori probability

Zhaohui Wei, Mingsheng Ying

Published Dec 15, 2004• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0412117

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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.
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.
A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of a database search and is known to be optimal when no a priori information about the elements of the database is provided. In this work we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis, and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.
We present two continuous-time quantum search algorithms similar to the adiabatic search algorithm, but now without the requirement of adiabatic evolution. Both algorithms can find the marked state in a time proportional to N\sqrt{N}N​. The behavior of the first algorithm is, essentially, similar to Grover's algorithm, but the second model possesses the important property that one does not need to single out a given time in order to find the searched state. After a well-defined transition time, this second algorithm will converge towards the marked state with a high probability, provided the parameters of the Hamiltonian are chosen appropriately. This convergence shows a resemblance to quantum search algorithms with a fixed point [L. K. Grover, Phys. Rev. Lett. 95, 150501 (2005)].
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
Fixed-point quantum search algorithms succeed at finding one of MMM target items among NNN total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property---the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.