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IEEE Access

Quantum Search Algorithm for Weighted Solutions

Youngjin Seo, Yujin Kang, Jun Heo

Published Jan 1, 2022• 8 citations
Publisher

Abstract

This paper presents a quantum search algorithm (QSA) for weighted solutions. In a QSA, the final quantum state has an equal probability of solutions. However, solutions should have different probabilities in specific problems, such as dynamic spectrum management (DSM). We propose a method to make differences with probability in accordance with rewards for each solution, and present an application to DSM. The proposed algorithm has some error terms for the final quantum state, and we provide the error terms in accordance with the number of iterations. In addition, we present the simulation results of the proposed algorithm compared with a classical exact algorithm and QSA. The proposed algorithm has a lower complexity and relative difference than the classical exact algorithm. Furthermore, the proposed algorithm has a higher performance than the QSA in specific cases.

DOI
10.1109/access.2022.3149351

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This paper presents a quantum search algorithm (QSA) for weighted solutions. In a QSA, the final quantum state has an equal probability of solutions. However, solutions should have different probabilities in specific problems, such as dynamic spectrum management (DSM). We propose a method to make differences with probability in accordance with rewards for each solution, and present an application to DSM. The proposed algorithm has some error terms for the final quantum state, and we provide the error terms in accordance with the number of iterations. In addition, we present the simulation results of the proposed algorithm compared with a classical exact algorithm and QSA. The proposed algorithm has a lower complexity and relative difference than the classical exact algorithm. Furthermore, the proposed algorithm has a higher performance than the QSA in specific cases.
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.
High-dimensional optimization problems pose significant challenges to classical stochastic search algorithms. The "curse of dimensionality" dramatically reduces the effectiveness of these methods as the number of variables increases. This paper proposes a novel stochastic search algorithm, Quantum-Inspired Stochastic Search (QISS), that draws inspiration from quantum mechanics to address this issue. QISS leverages the concepts of superposition and interference to efficiently explore the solution space. Each node in the search represents a potential solution existing in a probabilistic superposition of states. The algorithm simulates interference by dynamically adjusting the weighting of solution candidates based on their "quantum potential," a measure of uncertainty and promisingness. We demonstrate that QISS, by directly mimicking quantum phenomena, can potentially outperform traditional stochastic methods in complex, high-dimensional landscapes without the need for quantum computing resources. The core of the algorithm is formulated as follows: Let *N* be the number of nodes in the search space. Let *Si* represent the state of node *i*, where *i* = 1, 2, ..., *N*. The quantum potential of node *i*, denoted as *Pi*, is calculated as: *Pi* = ∑j=1N |*sij* - *sk*|2 / (Σj=1N |*sij* - *sk|2 + ε) where *sij* is the value of the *j*th variable in node *i*, *sk* is the average value of all nodes in the search space, and ε is a small constant to avoid division by zero. The weighting factor for node *i* is then given by: *Wi* = exp(-*Pi*) The algorithm iteratively updates the node states based on these weighted probabilities, effectively simulating quantum interference to guide the search toward promising regions of the solution space. The algorithm's performance is evaluated based on convergence rate and solution quality.
This paper introduces a novel stochastic search algorithm, termed Quantum-Inspired Stochastic Search (QISS), which leverages principles derived from quantum mechanics to address the challenges of complex optimization problems. Traditional stochastic search algorithms often struggle with high dimensionality and non-convex landscapes, leading to inefficient exploration and premature convergence. QISS overcomes these limitations by incorporating a "quantum-inspired" operator that mimics the probabilistic nature of quantum measurement. This operator allows the algorithm to simultaneously explore multiple potential solutions based on weighted probabilities, effectively simulating a quantum state. The core of QISS lies in its ability to dynamically adjust these probabilities, fostering a more comprehensive and efficient search process. The algorithm's design directly addresses the inherent uncertainties associated with complex optimization, offering a fundamentally different approach compared to conventional stochastic methods. We demonstrate the algorithm's potential through theoretical analysis and outline its key components and operational principles. The results suggest that QISS can significantly enhance the efficiency of stochastic search algorithms, particularly for high-dimensional, non-convex optimization problems.
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.
Abstract A binary constant weight code is a type of error-correcting code with a wide range of applications. The problem of finding a binary constant weight code has long been studied as a combinatorial optimization problem in coding theory. In this paper, we propose a quantum search algorithm for binary constant weight codes. Specifically, the search problem is formulated as a polynomial binary optimization problem and Grover adaptive search is used for providing the quadratic speedup. Focusing on the inherent structure of the problem, we derive an upper bound on the minimum of the objective function value and a lower bound on the exact number of solutions. By exploiting these two bounds, we successfully reduced the constant overhead of the algorithm, although the overall query complexity remains exponential due to the NP-complete nature of the problem. In our algebraic analysis, it was found that this proposed algorithm is capable of reducing the number of required qubits, thus enhancing the feasibility. Additionally, our simulations demonstrated that it reduces the average number of classical iterations by 63% as well as the average number of total Grover rotations by 31%. The proposed approach may be useful for other quantum search algorithms and optimization problems.
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.