While the development of quantum computers promises a myriad of advantages over their classical counterparts, care must be taken when designing algorithms that substitute a classical technique with a potentially advantageous quantum method. The probabilistic nature of many quantum algorithms may result in new behavior that could negatively impact the performance of the larger algorithm. The purpose of this work is to preserve the advantages of applying quantum search methods for generalized pattern search algorithms (GPSs) without violating the convergence criteria. It is well known that quantum search methods are able to reduce the expected number of oracle calls needed for finding the solution to a search problem from O(N) to O(N) However, the number of oracle calls needed to determine that no solution exists with certainty is exceedingly high and potentially infinite. In the case of GPS, this is a significant problem since overlooking a solution during an iteration will violate a needed assumption for convergence. Here, we overcome this problem by introducing the quantum improved point search (QIPS), a classical–quantum hybrid variant of the quantum search algorithm QSearch. QIPS retains the O(N) oracle query complexity of QSearch when a solution exists. However, it is able to determine when no solution exists, with certainty, using only O(N) oracle calls.
While the development of quantum computers promises a myriad of advantages over their classical counterparts, care must be taken when designing algorithms that substitute a classical technique with a potentially advantageous quantum method. The probabilistic nature of many quantum algorithms may result in new behavior that could negatively impact the performance of the larger algorithm. The purpose of this work is to preserve the advantages of applying quantum search methods for generalized pattern search algorithms (GPSs) without violating the convergence criteria. It is well known that quantum search methods are able to reduce the expected number of oracle calls needed for finding the solution to a search problem from O(N) to O(N) However, the number of oracle calls needed to determine that no solution exists with certainty is exceedingly high and potentially infinite. In the case of GPS, this is a significant problem since overlooking a solution during an iteration will violate a needed assumption for convergence. Here, we overcome this problem by introducing the quantum improved point search (QIPS), a classical–quantum hybrid variant of the quantum search algorithm QSearch. QIPS retains the O(N) oracle query complexity of QSearch when a solution exists. However, it is able to determine when no solution exists, with certainty, using only O(N) oracle calls.
Large-scale optimization problems, prevalent in diverse fields such as machine learning, logistics, and engineering design, pose significant challenges to classical optimization algorithms. These algorithms often suffer from slow convergence rates, premature convergence, and difficulty in exploring complex solution spaces. This paper explores the application of quantum-inspired search algorithms to address these limitations. We introduce a novel approach that draws inspiration from quantum measurement theory, specifically superposition and interference, to enhance the exploration capabilities of classical optimization techniques. The core idea is to mimic the probabilistic nature of quantum measurements to navigate the solution space more effectively. This approach allows for a more efficient exploration of potential solutions, leading to improved convergence rates and the ability to tackle larger and more complex problems. The algorithms presented are designed to be implementable within standard computational environments without requiring specialized quantum hardware. The performance of these algorithms is evaluated through simulations, demonstrating their potential to outperform traditional optimization methods in specific scenarios.
Abstract The Quantum-Based Optimisation Method (QBOM) is a novel optimization approach based on quantum computing concepts. The novel optimization method's durability is studied using its capacity to conjoin with existing optimization techniques. This study uses The QBOM with the Pattern Search (PS) technique to solve engineering optimization problems. The first strategy, Hybrid I, uses QBOM for global search optimization, followed by PS searching in the nearby region for the optimum solution. The second strategy, Hybrid II, uses QBOM as a local search optimization within Pattern Search. In each iteration, QBOM starts searching inside PS for a better solution than the one detected at that stage, which is labelled as PS's new search point. These two hybrid techniques attempt to expand the possibilities of QBOM's local search mechanism while demonstrating its resilience. The hybridised methodologies are used to solve benchmark optimization problems and six real-world engineering optimization problems. The study revealed that the two hybrid techniques worked brilliantly, producing solutions that exceeded previous methods described in the literature for certain benchmark optimization problems. Not only did the hybridised methods produce better results in less computational time, but they also demonstrated that QBOM could be used to improve the search mechanism and accelerate the performance of the evolutionary algorithm in the local search to match its execution in the global search.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
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.
Too often, quantum computer scientists seek to create new algorithms entirely fresh from new cloth when there are extensive and optimized classical algorithms that can be generalized wholesale. At the same time, one may seek to maintain classical advantages of performance and runtime bounds, while enabling potential quantum improvement. Hybrid quantum algorithms tap into this potential, and here we explore a class of hybrid quantum algorithms called Iterative Quantum Algorithms (IQA) that are closely related to classical greedy or local search algorithms, employing a structure where the quantum computer provides information that leads to a simplified problem for future iterations. Specifically, we extend these algorithms beyond past results that considered primarily quadratic problems to arbitrary k-local Hamiltonians, proposing a general framework that incorporates logical inference in a fundamental way. As an application we develop a hybrid quantum version of the well-known classical Davis-Putnam-Logemann-Loveland (DPLL) algorithm for satisfiability problems, which embeds IQAs within a complete backtracking based tree search framework. Our results also provide a general framework for handling problems with hard constraints in IQAs. We further show limiting cases of the algorithms where they reduce to classical algorithms, and provide evidence for regimes of quantum improvement.
Quantum-enhanced constraint propagation offers a novel approach to algorithm design, leveraging the principles of quantum mechanics to accelerate the discovery of new algorithmic patterns and optimize algorithm efficiency. This paper introduces a quantum-driven constraint propagation algorithm designed to explore the vast search space of possible algorithm designs, identifying previously unseen patterns and dramatically improving algorithm performance. We explore the potential of quantum superposition and entanglement to enhance the search process, leading to faster convergence and the discovery of innovative algorithmic solutions. The core mechanism focuses on dynamically adjusting the constraint propagation strategy based on quantum-enhanced measurements, ensuring a focused exploration of the relevant search space. The results demonstrate significant improvements in algorithm efficiency compared to classical constraint propagation methods, particularly in complex search spaces.