The problem of searching for an element within a dictionary was considered. Over the past decades, various approaches, including classical and quantum algorithms, have been proposed to solve it. One possible solution is the method of quantum amplitude amplification, which underpins the well-known Grover algorithm and enables a quadratic speedup in the search process. In this article, a new approach to searching for an element w of length m in a dictionary V of size n with the use of the quantum fingerprinting function was introduced. The developed algorithm has a query complexity of O(√n) and requires O(logn+logm) qubits.
The problem of searching for an element within a dictionary was considered. Over the past decades, various approaches, including classical and quantum algorithms, have been proposed to solve it. One possible solution is the method of quantum amplitude amplification, which underpins the well-known Grover algorithm and enables a quadratic speedup in the search process. In this article, a new approach to searching for an element w of length m in a dictionary V of size n with the use of the quantum fingerprinting function was introduced. The developed algorithm has a query complexity of O(√n) and requires O(logn+logm) qubits.
In this work, we present a quantum query algorithm for searching a word of length m in an unsorted dictionary of size n. The algorithm uses O(n) queries (Grover operators), like previously known algorithms. What is new is that the algorithm is based on the quantum fingerprinting-hashing technique, which (a) provides a first level of amplitude amplification before applying the sequence of Grover amplitude amplification operators and (b) makes the algorithm more efficient in terms of memory use -- it requires O(logn+logm) qubits. Note that previously developed algorithms by other researchers without hashing require O(logn+m) qubits.
Abstract In this paper, we consider the problem of searching for an element in a dictionary. Various approaches to solving this problem have been proposed in recent decades: classical and quantum algorithms. We present two algorithms: a hybrid classical-quantum algorithm [1], that implements Grover’s search, and a “pure” quantum algorithm based on the quantum fingerprinting technique. Both algorithms work (a) with a high probability of obtaining the correct result and (b) with a quadratic query acceleration compared to the classical one. Our algorithms are much more memory efficient in terms of the number of qubits used compared to previously known quantum algorithms.
We consider the problem of searching for an element in a dictionary. Various approaches to solving this problem have been proposed in recent decades: classical and quantum algorithms. We present two algorithms: a hybrid classical-quantum algorithm, which implements Grover’s search, and a “pure” quantum algorithm based on the quantum fingerprinting technique. Both algorithms work with a high probability of obtaining the correct result and with a quadratic query acceleration compared to the classical one. Our algorithms are much more memory efficient in terms of the number of qubits used compared to previously known quantum algorithms.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.
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.
Abstract. Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. Classical search procedures for an N element database require at most O(N) time complexity. Grover’s algorithm is able to find a solution with high prob-ability in O( N) time through an amplitude amplification scheme. In this work we draw elements from both classical and quantum computation to develop an alternative search proposal based on quantum entanglement detection schemes. In 2002, Horodecki and Ekert proposed an efficient method for direct detection of quantum entanglement. Our proposition to quantum search combines quantum entanglement detection along-side entanglement inducing operators. Grover’s quantum search relies on measuring a quantum superposition after having applied a unitary evolution. We deviate from the standard method by focusing on fine-tuning a unitary operator in order to infer the solution with certainty. Our proposal sacrifices space for speed and depends on the mathematical properties of linear positive maps Λ which have not been operationally characterized. Whether such a Λ can be easily determined remains an open question.