In this paper, we will define a quantum operator that performs the inversion about the mean only on a subspace of the system (partial diffusion operator). This operator is used in a quantum search algorithm that runs in O(N/M) for searching an unstructured list of size N with M matches such that, 1 ⩽ M ⩽ N. We will show that the performance of the algorithm is more reliable than known fixed operator quantum search algorithms especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than one‐third of the search space. A performance comparison with Grover’s algorithm will be provided.
In this paper, we will define a quantum operator that performs the inversion about the mean only on a subspace of the system (partial diffusion operator). This operator is used in a quantum search algorithm that runs in O(N/M) for searching an unstructured list of size N with M matches such that, 1 ⩽ M ⩽ N. We will show that the performance of the algorithm is more reliable than known fixed operator quantum search algorithms especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than one‐third of the search space. A performance comparison with Grover’s algorithm will be provided.
Abstract Grover's algorithm solves the unstructured search problem. Grover's algorithm can find the target state with certainty only if searching one out of four. Designing the deterministic search algorithm can avoid any repetition of the algorithm, especially when Grover's algorithm is a subroutine in other algorithms. Grover's algorithm can be deterministic if the phase of the oracle or the diffusion operator is delicately designed. The precision of the phases could be a problem. A near‐deterministic quantum search algorithm without the phase design is proposed. The algorithm has the same oracle and diffusion operators as Grover's algorithm. One additional component is the rescaled diffusion operator. It acts partially on the database. The success probability of Grover's algorithm is improved by the partial diffusion operator in two different ways. The possible cost is one or two more queries to the oracle. The deterministic search algorithm is also designed when searching one out of eight, sixteen, and thirty‐two.
Amplitude amplification techniques are the basic building blocks in many quantum algorithms. Amplifying the probability of success to the highest possible probability to find solutions in the search space is an essential aim. To simplify the construction of the hardware, the search operators with fixed phases must be used to handle the whole search space. This paper presents more reliable fixed-phase quantum search algorithm, where the phase shifts that used in the oracle and the diffusion operator are not matched, in contrary to the wide belief in literature that the matching condition between the phase shifts is the best choice. This paper shows that there are many phase shifts which can be used for the search operators to give better results when the number of solutions is known in advance. The minimum probability of success of the proposed algorithm is 99.61% in quadratic speed-up which is better than any other known fixed operator quantum search algorithms.
Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of Is, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator Ds with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of O(B), where B is a characteristic of the eigenspectrum of Ds and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still O(B) times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.
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.
Building quantum devices using fixed operators is a must to simplify hardware construction of a quantum computer. Quan- tum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured list of size N will be proposed. Fixing phase shifts to 1.91684π in the standard amplitude amplification will make the minimum probability of success is 99.58% in O N/M for 0 <M ≤ N better than any know fixed operator quantum search algorithm. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O N/M whether the number of matches is known or not in advance.
This paper introduces a new approach to improve Grover's search algorithm by utilizing variational learning, with a specific focus on incorporating a partial diffusion operator. Grover's search algorithm is a well-known quantum algorithm that provides a quadratic speedup for searching an unsorted database. However, its performance can be further enhanced by modifying its diffusion operator. In this work, we aim to identify a parameterized quantum circuit that can effectively learn and optimize Grover's search algorithm, incorporating a partial diffusion operator. The key idea behind this approach is to use variational learning, a technique that employs parameterized quantum circuits to optimize the algorithm's performance. By adjusting the parameters of the quantum circuit, the algorithm can be tailored to better solve the search problem. Variational learning is employed to determine the optimal parameters for the partial diffusion operator, enabling the quantum circuit to adapt to different problem instances. Our experimental results demonstrate that the optimized quantum circuit outperforms the traditional Grover’s algorithm that uses the standard partial diffusion operator. The results suggest that the proposed approach offers significant improvements in terms of algorithmic efficiency and performance. This advancement could have important implications for the development of quantum algorithms, particularly in applications related to database search and optimization problems. In conclusion, this paper presents a promising new method to enhance Grover's search algorithm, utilizing variational learning and a partial diffusion operator, with results showing improved performance over the original algorithm.