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International Conference on Information and Communication Technology Convergence (ICTC)

Analysis of Quantum Search Algorithm for Weighted Solutions: Simple Case

Youngjin Seo, Jun Heo

Published Oct 19, 2022• 0 citations
Publisher

Abstract

Quantum search algorithm proposed by Grover(Grover's algorithm) finds target data with equal probability for all target data. Y. Seo et al. [IEEE Access, vol. 10, pp.16209-16224 (2022)] proposed the quantum search algorithm for weighted solutions to discriminate the importance of target data. In the simple case, we represent the analysis of the algorithm. The low bound of the number of applying Grover operator$t$is described using amplitude expression. If we choose an appropriate$t$value, we provide the maximum ratio of the amplitude of high and low-importance target data. In addition, we arrange the maximum probability of high-importance target data using the bound.

DOI
10.1109/ictc55196.2022.9952459

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Quantum search algorithm proposed by Grover(Grover's algorithm) finds target data with equal probability for all target data. Y. Seo et al. [IEEE Access, vol. 10, pp.16209-16224 (2022)] proposed the quantum search algorithm for weighted solutions to discriminate the importance of target data. In the simple case, we represent the analysis of the algorithm. The low bound of the number of applying Grover operatortttis described using amplitude expression. If we choose an appropriatetttvalue, we provide the maximum ratio of the amplitude of high and low-importance target data. In addition, we arrange the maximum probability of high-importance target data using the bound.
The current Grover quantum searching algorithm cannot identify the difference in importance of the search targets when it is applied to an unsorted quantum database, and the probability for each search target is equal. To solve this problem, a Grover searching algorithm based on weighted targets is proposed. First, each target is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the targets are represented as quantum superposition states. Second, the novel Grover searching algorithm based on the quantum superposition of the weighted targets is constructed. Using this algorithm, the probability of getting each target can be approximated to the corresponding weight coefficient, which shows the flexibility of this algorithm. Finally, the validity of the algorithm is proved by a simple searching example.
The current Grover quantum searching algorithm can not identify the difference in importance of the marked items when it is applied to an unsorted quantum database, and the probability for each marked item is equal. To solve the problem, a Grover searching algorithm based on the weighted marked items is proposed. Firstly, each marked item is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the marked items are represented as a quantum superposition state. Secondly, the novel Grover searching algorithm based on the quantum superposition of the weighting marked items is constructed. Using this algorithm the probability of getting each marked item can be approximated to the corresponding weight coefficient, which the flexibility of this algorithm is shown. Finally, the validity of the algorithm is validated by a simple searching example.
As searching targets in an unordered database with Grover's algorithm,difference in marked items is not taken into consideration.If fraction of marked items is greater than 1/4,success probability of search decreases rapidly with increase of marked items.When the fraction of marked items is greater than 1/2,the algorithm is disabled.Aiming at above problems,an improved Grover's algorithm with weighted targets is proposed in which every target is assigned a weight coefficient according to its significance.With these weight coefficients,targets are represented as quantum superpositions which make probability getting target equal to its weight coefficient.An adaptive phase matching method is proposed based on weighted targets.The directions of phase rotations are contrary,and amplitudes of the two phase rotations are determined by inner-product of target quantum superposition and initial state of the system.As the inner-product is greater than(((3-5)/8)~(1/2),) success probability is equal to 1 with two steps of Grover iteration at most.The improved quantum searching algorithm and the new phase matching are verified by an example.
Grover search algorithm is applied to the current number of objectives in quantum unsorted database, the meaning of each goal difference is not taken into account. When the target is more than a quarter of the total items, the probability of the target falls quickly identify and target increases, and when the target is more than half of the total items, the algorithm will fail. To solve these problems, first of all, based on a weighted index improvement is proposed in which each target in accordance with the right given to the importance of the weight coefficient. These different weight coefficients, we give all the target state, which may make the probability of each target equal weights of the quantum superposition. Secondly, an improved phase matching is proposed the phase rotation in both directions is the same, and the inner product of two states by the target amplitude superposition and initial state of the system is determined. When this inner product is 1/2 or more, when using the improved phase matching, the target may be 100% probability and by only one Grover iteration. Finally, the effectiveness of these measures by a simple search instance validation.
We provide a tight analysis of Grover's algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Finally, we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover's algorithm comes within 2.62% of being optimal.
Grover's algorithm, which was discovered in 1996, is faster than any classical algorithm can ever be. Quantum systems are in a superposition of states and can examine multiple data simultaneously. This work conducted a comprehensive analysis to understand how Grover's algorithm works. Grover's search provides quadratic speed in comparison to the classical one; it explores target data in the unstructured database. Grover search starts with the superposition of all Qubits, which results in the same amplitude for all data points. To achieve superposition, the Hadamard gate was applied to all Qubits. Further, a black box function named Oracle is applied to mark the desired state. In the next phase, amplitude amplification is done, which enhances the probability of the desired data point. Thus, Grover's search yields results with an O (√ N) complexity, while its counterpart classical algorithm searches the desired data point with an O (N) complexity. Experiments have been performed on the Qiskit simulator using 2 and 3 qubits for the detailed analysis of Grover's search.