Search an Unsorted Database with Quantum Mechanics
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http://www.paper.edu.cn Front. Comput. Sci. China 2007, 1(3): 247−271 DOI 10.1007/s11704-007-0026-z REVIEW ARTICLE Search an unsorted database with quantum mechanics LONG Guilu ( ), LIU Yang Key Laboratory for Atomic and Molecular Nanosciences of Ministry of Eduation, Department of Physics, Tsinghua University, Beijing 100084, China Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China © Higher Education Press and Springer-Verlag 2007 Abstract In this article, we review quantum search algo- markably, there is a quantum search algorithm which en- rithms for unsorted database search problem. Unsorted da- ables this search problem to be speeded up substantially, tabase search is a very important problem in science and requiring only ON() operations. Moreover, the quan- technology. In a quantum computer, a marked state can be tum search algorithm is general in the sense that they can found with very high probability using the Grover's algo- be applied far beyond the route-finding example just de- rithm, or exactly with the Long algorithm. We review the scribed to speed up many (though not all) classical algo- Grover algorithm and related generalizations. In particular, rithms that use search heuristics. It plays very important we review the phase matching conditions in quantum roles in the fields of information and computation. For search algorithm. Several issues that may cause confusion example, finding the owner of a phone number, deciphering about the quantum search algorithm are also clarified. DES like code [1], solving the Simon problem [2], quantum counting problem [3]. Spaces can be efficiently searched by Keywords quantum search algorithm, Grover algorithm, a quantum robot using quantum search algorithm [4]. It can Long algorithm, Phase matching, quantum amplitude am- also speed up in a square-root manner difficult problems plification, duality computer such as the hidden shift problem [5], the Hamiltonian cir- cuit problem [6] and NP-complete problem in general [7]. 1 Introduction In this review article, we will describe various aspects of quantum search algorithms. In Section 2, we give a brief in- Finding a marked item from an unsorted database (UDB), troduction of the Grover search algorithm. In Section 3, before the so-called unsorted database search (UDS) problem is a going to several important generalizations, we give a clear very common and difficult problem. Many scientific prob- separation of the basic elements in a quantum searching algo- lems can be reduced to UDS problem and the UDS has rithm. We divide the quantum searching problem into two wide applications in science and technology. For instance, basic elements: the search engine and the database. The ob- suppose you are given a map containing many cities, and scurity between a quantum search engine and a quantum da- want to find the shortest route passing through all cities on tabase often misleads people to err. Most impor- the map. A simple algorithm to find this route is to search tantly, the unitary operators U' in ||0ψ0 〉=U' 〉, the database, all possible routes through the cities and keep a record of −1 and U in the search engine G = −UR U Rτ is which route has the shortest length. On a classical computer, 0 not the same in general. However these two are often the same if there are N possible routes, it is obvious that O(N) opera- in many quantum search algorithms. Separating them clears tions are required to determine the shortest route. Re- many misunderstanding. In Section 4 we describe several generalizations of the Grover algorithm, including finding Received July 20, 2007 multiple marked items, replacing the Walsh-Hadamard trans- E-mail: [email protected] formation with an arbitrary unitary transformation, namely the 转载 中国科技论文在线 http://www.paper.edu.cn 248 quantum amplitude amplification. In Section 5, we concen- trate on one important generalization of replacing the two phase inversions with general phase rotations. Contrary to Excellent work have been done for classical algorithms on anticipation, arbitrary phase rotations are not allowed in a searching problem [8, 9]. Grover algorithm and Shor al- good quantum search algorithm. A good quantum search al- gorithm are the two most successful algorithms in quantum gorithm can find the marked state with high probability. It computing [10, 11]. Grover proposed a quantum mechani- is only when the two phase rotation angles satisfy a phase cal search algorithm [12] which is polynomially faster than matching condition that a good quantum search algorithm classical algorithm. The investigated problem is: there is can be built. It depends both on the quantum search engine an unsorted database containing N items of which only one and on the quantum database. In Section 6, we review an item is to be retrieved. This is the UDS problem. exact quantum search algorithm, the Long algorithm. The The classical algorithm for this problem is to examine standard Grover algorithm finds a marked state with very items one by one. Thus the algorithm needs N/2 iterations high probability in general for very large database. How- in average for finding a desired marked item. Quantum me- ever in some applications, the quantum database size is lim- chanics systems can be in a superposition of the basis states. ited, or the number of marked states is large, then Grover By properly adjusting the various operations, Grover search algorithm runs into difficulty. For instance, Grover algo- algorithm enables√ this method to speed up substantially, re- rithm fails for a database with only two items or half of the quiring only O( N) operations. data is the marked states. The application of the Grover al- Now we examine the Grover algorithm. Let a quantum n gorithm does not increase the finding probability at all. In system have N =2 states which are labelled as |i(i = this case, the Long algorithm is very suited for. It involves 1, 2,...,N), there is a unique marked state |τ that satisfies replacing the two phase inversions with two phase angles a query function C(i)=1, whereas any other state satis- satisfying the phase matching condition and dependent on fies function C(i)=0. The search problem is to identify the size of the database. An analytic expression is explicitly the state τ. To fulfill Grover algorithm, two types of ele- given for the angles. In Section 7, we review the other im- mentary unitary operations are required in the processing: portant generalization of the Grover algorithm, the quantum 1) the Walsh-Hadamard transformation [13], denoted as W ; amplitude amplification (QAA). Standard Grover algorithm 2) selective inversions of the phase of states. The strategy of uses evenly distributed database. In QAA, the items in the the Grover algorithm is to begin with an evenly distributed database may have different coefficients. In Section 8, we state to perform successive above-mentioned operations so briefly review the newly proposed duality computer and the as to increase sufficiently large probability of obtaining the quantum search algorithm in it. A duality computer exploits marked state. both the particle and wave nature of quantum systems. In The W operator performed on single qubit is represented by the matrix addition to quantum parallelism, it also possesses the duality ⎡ ⎤ parallelism. The duality search algorithm provides a way of 1 ⎢ 11⎥ running classical search algorithms in a quantum computer. H = √ ⎣ ⎦ . (1) In Section 9, we give a brief analysis of misunderstanding 2 1 −1 about the quantum search algorithm. This includes: the H transforms a qubit in |0 state to a superposition state relation between speed of search and the initial amplitude 1 1 √ (|0+|1) and |1 state to a superposition state √ (|0− of the marked state in the database, the relation between the 2 2 search speed and the quantum search engine. In Section 10, |1). In a quantum system of n-qubit, H transformation is we give an error tolerance analysis of quantum search algo- performed on each of the n qubit independently to realize rithms. In Section 11, we briefly review a classical parallel the W transformation. Then the total unitary operation is implementation of the quantum search algorithm. By using denoted as classical parallelism, additional speedup can be obtained at W = H⊗n the expense of more qubit resources. In Section 12, a sum- and the elements of matrix are mary and discussion is given about the experimental imple- −n/2 ¯i·¯j mentations of the quantum search algorithm and other re- Wij =2 (−1) , lated issues. where ¯i and ¯j are the binary representations of i and j re- 中国科技论文在线 http://www.paper.edu.cn 249 spectively, ¯i · ¯j denotes the bitwise inner product of the bit gorithm [11, 12] and its later version [22]. In the first version strings ¯i and ¯j. [11, 12], the D operation was implemented by two Walsh- Specifically, the Grover algorithm has the following pro- Hadamard gates with the operation −I0 whose action is to cedures. inverse the signs of all components except the |0 basis state. D Database construction Initialize√ the√ quantum√ register In its later version, the minus sign in was ignored, and it / N, / N, / N,..., I to√ the evenly distributed state (1 1 1 was implemented by two Walsh-Hadamard and the 0 oper- 1/ N). This superposition state requires O(log N) steps. ator whose action is to inverse the sign of |0 component and The initial state can be written as leaves other amplitudes untouched. In this review article, we adopt the later version. The diffusion transform D can be |ψ W | 1 |i β|τ β|c, 0 = 0 = N =sin +cos (2) i expressed as ⎧ where ⎪ 2 ⎨⎪ ,i= j 1 1 N |c = |i,β=arcsin .