Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a depth optimization method for the quantum search algorithm. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for parallel running of the quantum search algorithm.
Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a depth optimization method for the quantum search algorithm. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for parallel running of the quantum search algorithm.
Quantum computing has noteworthy speedup over classical computing by taking advantage of quantum parallelism, i.e., the superposition of states. In particular, quantum search is widely used in various computationally hard problems. Grover's search algorithm finds the target element in an unsorted database with quadratic speedup than classical search and has been proved to be optimal in terms of the number of queries to the database. The challenge, however, is that Grover's search algorithm leads to high numbers of quantum gates, which make it infeasible for the Noise-Intermediate-Scale-Quantum (NISQ) computers. In this paper, we propose a novel hardware efficient quantum search algorithm to overcome this challenge. Our key idea is to replace the global diffusion operation with low-cost local diffusions. Our analysis shows that our algorithm has similar oracle complexity to the original Grover's search algorithm while significantly reduces the circuit depth and gate count. The circuit cost reduction leads to a remarkable improvement in the system success rates, paving the way for quantum search on NISQ machines.
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.
Grover's algorithm stands as a cornerstone of quantum computing, offering a theoretical quadratic speedup for unstructured search problems. This allows for finding a target item in an unsorted database of N entries in approximately the square root of N steps, a significant improvement over classical algorithms which require a number of operations on the order of N in the worst case. This speedup becomes particularly compelling for very large datasets, where classical exhaustive search becomes computationally intractable. The algorithm achieves this by leveraging fundamental quantum mechanical principles, primarily superposition and interference. It operates by preparing qubits in a uniform superposition of all possible states, then iteratively applying a quantum oracle to mark the desired state by flipping its phase, followed by a diffusion operator that amplifies the probability of the marked state. Despite its theoretical promise, real-world implementation faces substantial challenges stemming from Noisy Intermediate-Scale Quantum (NISQ) hardware, which is characterized by high error rates and short coherence times. Consequently, achieving a true, scalable quantum advantage remains a future endeavor, contingent on significant hardware advancements. Nevertheless, Grover's algorithm holds significant potential across diverse fields, including enhancing brute-force attacks on cryptographic systems, accelerating solutions for optimization problems, and speeding up search processes in machine learning and unstructured database queries.
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.
Abstract With the advent of powerful quantum computers, the quest for more efficient quantum algorithms becomes crucial in attaining quantum supremacy over classical counterparts in the noisy intermediate-scale quantum era. While Grover's search algorithm and its generalization, quantum amplitude amplification, offer quadratic speedup in solving various important scientific problems, their exponential time complexity limits scalability as the quantum circuit depths grow exponentially with the number of qubits. To overcome this challenge, we propose Variational Quantum Search (VQS), a novel algorithm based on variational quantum algorithms and parameterized quantum circuits. We show that a depth-10 Ansatz can amplify the total probability of k (k ≥ 1) good elements, out of 2n elements represented by n + 1 qubits, from k/2n to nearly 1, as verified for n up to 26, and that the maximum depth of quantum circuits in the VQS increases linearly with the number of qubits. Our experimental results have validated the efficacy of VQS and its exponential advantage over Grover's algorithm in circuit depth for up to 26 qubits. We demonstrate that a depth-56 circuit in VQS can replace a depth-270,989 circuit in Grover's algorithm. Envisioning its potential, VQS holds promise to accelerate solutions to critical problems.
Abstract Quantum search algorithm (also known as Grover's algorithm) lays the foundation for many other quantum algorithms. Although it is very simple, its implementation is limited on noisy intermediate-scale quantum (NISQ) processors. Grover's algorithm was designed without considering the physical resources, such as depth, in the real implementations. Therefore, Grover's algorithm can be improved for NISQ devices. In this paper, we demonstrate how to implement quantum search algorithms better on NISQ devices. We present detailed benchmarks of the five-qubit quantum search algorithm on different quantum processors, including IBMQ, IonQ, and Honeywell quantum devices. We report the highest success probability of the five-qubit search algorithm compared to previous works. Our results show that designing the error-aware quantum search algorithms is possible, which can maximally harness the power of NISQ computers.