The search task is one of the most difficult when it comes to execution speed, and reducing the latter is important both when working with large data and with small samples, if they need to be processed frequently and in a limited time. Grover's algorithm gave hope to quantum computing and served as an excellent base for all possible implementations and modifications. In this paper, we propose a slightly different algorithm that increases the probability of finding the nearest value by reducing the probability of undesirable values in a controlled manner (in proportion to their difference from the desired value), as well as implementing an oracle that requires a single call without an additional ancilla qubit to redistribute the amplitudes.
The search task is one of the most difficult when it comes to execution speed, and reducing the latter is important both when working with large data and with small samples, if they need to be processed frequently and in a limited time. Grover's algorithm gave hope to quantum computing and served as an excellent base for all possible implementations and modifications. In this paper, we propose a slightly different algorithm that increases the probability of finding the nearest value by reducing the probability of undesirable values in a controlled manner (in proportion to their difference from the desired value), as well as implementing an oracle that requires a single call without an additional ancilla qubit to redistribute the amplitudes.
The search task is one of the most difficult when it comes to execution speed, and reducing the latter is important both when working with large data and with small samples, if they need to be processed frequently and in a limited time. Grover''s algorithm gave hope to quantum computing and served as an excellent base for all possible implementations and modifications. In this paper, we propose a slightly different algorithm that increases the probability of finding the nearest value by reducing the probability of undesirable values in a controlled manner (in proportion to their difference from the desired value), as well as implementing an oracle that requires a single call without an additional ancilla qubit to redistribute the amplitudes.
Grover’s algorithm is a well-known contribution to quantum computing. It searches one value within an unordered sequence. A fundamental part of this algorithm is the so-called oracle, a quantum circuit that marks the quantum state corresponding to the desired value. In this work we present a classical algorithm that builds an oracle that performs a less-than operation, marking states representing natural numbers smaller than a given one. Results are shown to prove its functionality. This lessthan oracle implementation works on any number of qubits and does not require any ancilla qubits. Regarding depth, the proposed implementation is compared with Qiskit automatic method, Diagonal. We show that the depth of our less-than oracle implementation is always lower. In addition, a comparison with another method for oracle generation in terms of gate count is also conducted to prove the efficiency of our method. The result presented here is part of a research work that aims to achieve reusable quantum operations that can be composed to perform more complex ones.
Grover's algorithm is a well-known contribution to quantum computing. It searches one value within an unordered sequence faster than any classical algorithm. A fundamental part of this algorithm is the so-called oracle, a quantum circuit that marks the quantum state corresponding to the desired value. A generalization of it is the oracle for Amplitude Amplification, that marks multiple desired states. In this work we present a classical algorithm that builds a phase-marking oracle for Amplitude Amplification. This oracle performs a less-than operation, marking states representing natural numbers smaller than a given one. Results of both simulations and experiments are shown to prove its functionality. This less-than oracle implementation works on any number of qubits and does not require any ancilla qubits. Regarding depth, the proposed implementation is compared with the one generated by Qiskit automatic method, UnitaryGate. We show that the depth of our less-than oracle implementation is always lower. This difference is significant enough for our method to outperform UnitaryGate on real quantum hardware.
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.
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 is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2 n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024