The question of which resources drive the advantages in quantum algorithms has long been a fundamental challenge. While entanglement and coherence are critical to many quantum algorithms, our results indicate that they do not fully explain the quantum advantage achieved by the Grover search algorithm. By introducing a generalized Grover search algorithm, we demonstrate that the success probability depends not only on the querying number of oracles but also on the coherence fraction, which quantifies the fidelity between an arbitrary initial quantum state and the equal superposition state. Additionally, we explore the role of the coherence fraction in the quantum minimization algorithm, which offers a framework for solving complex problems in quantum machine learning. These findings offer insights into the origins of quantum advantage and open pathways for the development of new quantum algorithms.
The question of which resources drive the advantages in quantum algorithms has long been a fundamental challenge. While entanglement and coherence are critical to many quantum algorithms, our results indicate that they do not fully explain the quantum advantage achieved by the Grover search algorithm. By introducing a generalized Grover search algorithm, we demonstrate that the success probability depends not only on the querying number of oracles but also on the coherence fraction, which quantifies the fidelity between an arbitrary initial quantum state and the equal superposition state. Additionally, we explore the role of the coherence fraction in the quantum minimization algorithm, which offers a framework for solving complex problems in quantum machine learning. These findings offer insights into the origins of quantum advantage and open pathways for the development of new quantum algorithms.
Quantum algorithms have demonstrated provable speedups over classical counterparts, yet establishing a comprehensive theoretical framework to understand the quantum advantage remains a core challenge. In this work, we decode the quantum search advantage by investigating the critical role of quantum state properties in random-walk-based algorithms. We propose three distinct variants of quantum random-walk search algorithms and derive exact analytical expressions for their success probabilities. These probabilities are fundamentally determined by specific initial state properties: the coherence fraction governs the first algorithm's performance, while entanglement and coherence dominate the outcomes of the second and third algorithms, respectively. We show that increased coherence fraction enhances success probability, but greater entanglement and coherence reduce it in the latter two cases. These findings reveal fundamental insights into harnessing quantum properties for advantage and guide algorithm design. Our searches achieve Grover-like speedups and show significant potential for quantum-enhanced machine learning.
The Grover algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems, requiring O(√N) queries instead of O(N) classically. It works by repeatedly applying an oracle and a diffusion operator to amplify the probability of marked states. This advantage makes it relevant to cryptography, optimization, constraint satisfaction, and as a general primitive via amplitude amplification in areas like quantum machine learning and simulation. However, practical implementations are severely constrained by current Noisy Intermediate Scale Qualtum machines with limited coherence, deep oracle circuits and lack of scalable QRAM, restricting demonstrations to small-scale experiments with reproducibility challenges.
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.
Coherence is one of the most basic concepts and resources in quantum information. To clear the role coherence plays on the essential operator level in Grover's search algorithm, here we discuss the coherence dynamics of the state after each basic operator is applyied. As it is known, Grover's search algorithm repeats the application of Grover operator G, which can be decomposed into $G={H}^{\ensuremath{\bigotimes}n}P{H}^{\ensuremath{\bigotimes}n}O$, where H is Hadamard operator, P is the condition phase-shift operator, and O is the oracle operator. First, we show that O and P are incoherent operators while ${H}^{\ensuremath{\bigotimes}n}$ is coherent. Second, we prove that the amount of the operator coherence of the first ${H}^{\ensuremath{\bigotimes}n}$ and the operator coherence produced or depleted by ${H}^{\ensuremath{\bigotimes}n}$ depends not only on the size of the database and the success probability, but also on target states. Moreover, the amount of operator coherence is larger when the superposition state of targets is entangled rather than product. Third, we show that the two ${H}^{\ensuremath{\bigotimes}n}$ have different effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence is in depletion.
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.
Grover’s algorithm is widely celebrated as providing quadratic quantum speedup for unsorted database search, forming the theoretical foundation for numerous claimed quantum advantages in machine learning, optimization, and computational applications. We demonstrate that Grover’s algorithm and oracle-based quantum search algorithms face three fundamental barriers that severely limit their practical applicability. First, we identify the Grover Dilemma: when the computational space exceeds the valid data space, Grover’s algorithm must choose between 1) creating uniform superposition over all computational states—including invalid states that lead to incorrect results, 2) restricting superposition to only valid states containing solutions—requiring prior knowledge that reduces the problem to trivial O(1) complexity, or 3) constructing superposition over valid states through classical preprocessing—requiring O(N) cost that eliminates the claimed O( N ) quantum advantage. We extend this analysis to establish a unified framework of three independent barriers affecting oracle-based quantum search: 1) the Grover Dilemma (superposition construction for structured problems), 2) the Setup Cost Dilemma (oracle construction and data loading costs), and 3) Oracle Circularity (oracle specification requiring solution of the target problem). These barriers are logically independent—a quantum algorithm must avoid all three to achieve genuine advantage. Through systematic analysis of major problem classes and specific quantum algorithms, we found that oracle-based quantum search provides genuine computational advantage only in an exceptionally narrow class of problems. Our contributions are formalized in ten theorems: Theorems 6.1-6.5 establish the general framework (Grover Dilemma Barrier, Setup Cost Barrier, Oracle Circularity Barrier, Composite Barrier, and general conditions for no quantum advantage); Theorems 6.6-6.8 prove no quantum advantage for Deutsch’s Algorithm, Deutsch-Jozsa Algorithm, and Simon’s Algorithm in practical scenarios; Theorems 6.9-6.10 prove no quantum advantage for NP-complete problems and learning/optimization problems subject to Oracle Circularity. The three-barrier framework provides a systematic method for evaluating quantum search algorithm claims and distinguishing genuine advantages from artifacts of incomplete cost analysis.