An N/4 fixed-point duality quantum search algorithm

Abstract

Here a fixed-point duality quantum search algorithm is proposed. This algorithm uses iteratively non-unitary operations and measurements to search an unsorted database. Once the marked item is found, the algorithm stops automatically. This algorithm uses a constant non-unitary operator, and requires N/4 steps on average (N is the number of data from the database) to locate the marked state. The implementation of this algorithm in a usual quantum computer is also demonstrated. fixed-point search algorithm, duality quantum computing, the N4 duality search algorithm PACS: 03.67.Lx, 03.67.Dd, 03.65.TaA quantum computer possesses quantum parallelism, and it offers more computing power than a classical computer. Primarily, a quantum computer offers efficient simulation of quantum systems [1][2][3][4][5], and greatly speeds up the factorization problem [6] and unsorted database search problem [7]. Finding a marked item from an unsorted database is common but difficult. If the dimension of the database is N, it usually requires N/2 iterations on average for a classical computer to find the desired marked item by examining items one by one. By contrast, a quantum computer uses O( √ N) iterations under the Grover's quantum search algorithm [7]. The Grover algorithm has attracted much attention and been further developed and applied to various problems [8][9][10][11][12][13][14][15][16][17][18].In recent years, the fixed-point search algorithm, which solves the over-cooking and under-cooking problem, has attracted much attention. The over-cooking and under-cooking problems mean that the probability of success will be small if the number of iterations is greater than or smaller than the optimal number of steps. For instance, the success probability of the Grover algorithm is sin 2 (2 j + 1)β , where β = arcsin(1/ √ N). It is the maximum when j is the optimal number j opt = π √ N/4. If the number is both greater and smaller than the optimal number (and not near the next optimal num-*Corresponding author (email: gllong@tsinghua.edu.cn) †Contributed by LONG GuiLu ber), the probability of success will decrease. The fixedpoint search algorithm solves this problem. The fixed-point search algorithm [19] (fp-Grover algorithm) was proposed by Grover and the probability of success is always increasing. However the algorithm costs the loss in speed of the original Grover algorithm and it requires O(3 n ) number of iterations. Other fixed-point search algorithms [20][21][22][23][24] have been developed along the lines of ref.[19] but with different choices of phases.In another development, a fixed-point quantum search algorithm (N1-algorithm) was proposed in ref. [25] where O(2 n ), namely O(N) number of queries is required using the idea of duality quantum computing [26]. On average, N number of queries is required. This duality quantum algorithm uses fewer number of queries than the fp-Grover algorithm in ref. [9]. The peculiar feature of duality quantum computer is the non-unitary operations it allows for. In this paper, we give a ne...

Duality Quantum Computing and Duality Quantum Information Processing

“…There have been intensive interests in the theory of duality computer in recent years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Here in this review article we will briefly describe the main results.…”

Deleting a marked state in quantum database in a duality computing mode

“…Later it is experimentally demonstrated [20]. An improved fixed-point duality quantum algorithm is given in [21]. Duality quantum computing has been generalized into more broad range, duality quantum information processing [22,23].…”

A quantum algorithm that deletes marked states from an arbitrary database

“…When the number of marked states is no more than 3N 4 , this algorithm requires just a single query, and this achieves exponential speedup over classical algorithm. [6][7][8][9][10], recent interests have been focused on quantum algorithms using duality mode in a quantum register [11,12]. Deleting an item from a database is a routine task in database processing.…”

“…A quantum large integer factoring algorithm [2] and Grover search algorithm [3] can complete tasks exponentially faster than classical counterparts. Quantum algorithms have attracted much attention and been further developed and applied to various problems [6][7][8][9][10], recent interests have been focused on quantum algorithms using duality mode in a quantum register [11,12].…”