Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.23382

Non-unitary extension of Grover's search algorithm

V. N. A. Lula-Rocha, M. A. S. Trindade

quant-phmath-phSubmitted Apr 25, 2026• Announced Apr 28, 2026• 0 citations
arXivHTMLPDF

Abstract

We have developed a non-unitary extension of Grover's search algorithm by changing the hidden geometry of Hilbert space carried by diffusion operator. Our algorithm finds the solution for search problem by performing a unique bigger rotation rather than small rotations in order polynomial times in the size $N$ of search space. We analyze the complexity of implementing the non-unitary operation and we observed that the price paid by performing this rotation is due the normalization. In Kraus operator approach we need $O(N)$ repetition of the algorithm to have a chance of measuring a solution in a post-selection, this is no better than the classical solution. However, the quantum singular value transform in addition with block encoding and Chebyshev polynomial approximation, we got complexity $O(\sqrt{N})$ and reach the Grover's bound with an extra resource of one single qubit, compared with the standard Grover's algorithm.

Related papers

Faster quantum searching with almost any diffusion operator

Avatar Tulsi

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 ${I}_{s}$, 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 ${D}_{s}$ 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 ${D}_{s}$ 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.

arXiv:1503.05712•Physical Review A•May 7, 2015•10 citations

A Grover-compatible manifold optimization algorithm for quantum search

Zhijian Lai, Dong An, Jiang Hu +1 more

Grover's algorithm is a fundamental quantum algorithm that offers a quadratic speedup for the unstructured search problem by alternately applying physically implementable oracle and diffusion operators. In this paper, we reformulate the unstructured search as a maximization problem on the unitary manifold and solve it via the Riemannian gradient ascent (RGA) method. To overcome the difficulty that generic RGA updates do not, in general, correspond to physically implementable quantum operators, we introduce Grover-compatible retractions to restrict RGA updates to valid oracle and diffusion operators. Theoretically, we establish a local Riemannian $μ$-Polyak-Łojasiewicz (PL) inequality with $μ= \tfrac{1}{2}$, which yields a linear convergence rate of $1 - κ^{-1}$ toward the global solution. Here, the condition number $κ= L_{\mathrm{Rie}} / μ$, where $L_{\mathrm{Rie}}$ denotes the Riemannian Lipschitz constant of the gradient. Taking into account both the geometry of the unitary manifold and the special structure of the cost function, we show that $L_{\mathrm{Rie}} = O(\sqrt{N})$ for problem size $N = 2^n$. Consequently, the resulting iteration complexity is $O(\sqrt{N} \log(1/\varepsilon))$ for attaining an $\varepsilon$-accurate solution, which matches the quadratic speedup of $O(\sqrt{N})$ achieved by Grover's algorithm. These results demonstrate that an optimization-based viewpoint can offer fresh conceptual insights and lead to new advances in the design of quantum algorithms.

quant-pharXiv:2512.08432•Dec 10, 2025•0 citations

Quantum Search using Grover’s Algorithm: Principles, Performance, and Applications

Anggit Gusti Nugraheni, Heri Mahyuzar, Sri Hastuti

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.

Asian journal of mathematics and computer research•Jun 23, 2026•0 citations

Grover-Diffusion Operator Unicity for the Grover Quantum-Search Algorithm

Mihai Octavian Dima, Mihai Octavian Dima, M. Mihailescu

Abstract The repetition of Grover-diffusion operator the order of √N times is the essence of the Grover quantum selection algorithm. For large N the coherence time T2 of the qubits limits the application of the algorithm. We explore what other operators could be devised in its place to accelerate convergence. We present a C++ SU(2) model of the Grover-diffusion operator implemented using our SU2 package.

Physics of Particles and Nuclei•Aug 1, 2025•0 citations

Improved Performance of Quantum Search via Quantum Singular Value Transformation

Zheng Zhang, Minzhong Luo

Quantum singular value transformation (QSVT) is a powerful framework that unifies many quantum algorithms, including Grover’s search, by enabling polynomial transformations of the singular values of linear operators embedded in unitary matrices. We present an improved quantum search algorithm within this framework. Instead of amplifying the target state amplitude, our method suppresses the non-target amplitude using a modified Chebyshev polynomial. This approach achieves the optimal \(\mathcal {O}(\sqrt {N}\log (1/\delta))\) query complexity while requiring lower-degree polynomials. For N = 512, our method reduces the polynomial degree from 71 to 43 compared to prior QSVT-based fixed-point search. We also demonstrate how QSVT enables exact search construction.

International Conference on Computer Information and Big Data Applications (CIBDA)•Mar 20, 2026•0 citations

A Quantum Walk Enhanced Grover Search Algorithm for Global Optimization

Yan Wang

One of the significant breakthroughs in quantum computation is Grover's algorithm for unsorted database search. Recently, the applications of Grover's algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed so that the search is accelerated with improved threshold values. By taking advantage of the quantum tunneling effect, better threshold values can be found at the early stage of the search process so that the sharpness of probability improves. The results between the new algorithm, existing Grover search, and classical heuristic algorithms are compared.

arXiv:1711.07825•Nov 18, 2017•4 citations
We have developed a non-unitary extension of Grover's search algorithm by changing the hidden geometry of Hilbert space carried by diffusion operator. Our algorithm finds the solution for search problem by performing a unique bigger rotation rather than small rotations in order polynomial times in the size NNN of search space. We analyze the complexity of implementing the non-unitary operation and we observed that the price paid by performing this rotation is due the normalization. In Kraus operator approach we need O(N)O(N)O(N) repetition of the algorithm to have a chance of measuring a solution in a post-selection, this is no better than the classical solution. However, the quantum singular value transform in addition with block encoding and Chebyshev polynomial approximation, we got complexity O(N)O(\sqrt{N})O(N​) and reach the Grover's bound with an extra resource of one single qubit, compared with the standard Grover's algorithm.
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{I}_{s}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{D}_{s}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)O(B)O(B), where BBB is a characteristic of the eigenspectrum of Ds{D}_{s}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)O(B)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 a fundamental quantum algorithm that offers a quadratic speedup for the unstructured search problem by alternately applying physically implementable oracle and diffusion operators. In this paper, we reformulate the unstructured search as a maximization problem on the unitary manifold and solve it via the Riemannian gradient ascent (RGA) method. To overcome the difficulty that generic RGA updates do not, in general, correspond to physically implementable quantum operators, we introduce Grover-compatible retractions to restrict RGA updates to valid oracle and diffusion operators. Theoretically, we establish a local Riemannian μμμ-Polyak-Łojasiewicz (PL) inequality with μ=12μ= \tfrac{1}{2}μ=21​, which yields a linear convergence rate of 1−κ−11 - κ^{-1}1−κ−1 toward the global solution. Here, the condition number κ=LRie/μκ= L_{\mathrm{Rie}} / μκ=LRie​/μ, where LRieL_{\mathrm{Rie}}LRie​ denotes the Riemannian Lipschitz constant of the gradient. Taking into account both the geometry of the unitary manifold and the special structure of the cost function, we show that LRie=O(N)L_{\mathrm{Rie}} = O(\sqrt{N})LRie​=O(N​) for problem size N=2nN = 2^nN=2n. Consequently, the resulting iteration complexity is O(Nlog⁡(1/ε))O(\sqrt{N} \log(1/\varepsilon))O(N​log(1/ε)) for attaining an ε\varepsilonε-accurate solution, which matches the quadratic speedup of O(N)O(\sqrt{N})O(N​) achieved by Grover's algorithm. These results demonstrate that an optimization-based viewpoint can offer fresh conceptual insights and lead to new advances in the design of 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.
Quantum singular value transformation (QSVT) is a powerful framework that unifies many quantum algorithms, including Grover’s search, by enabling polynomial transformations of the singular values of linear operators embedded in unitary matrices. We present an improved quantum search algorithm within this framework. Instead of amplifying the target state amplitude, our method suppresses the non-target amplitude using a modified Chebyshev polynomial. This approach achieves the optimal O(Nlog⁡(1/δ))\mathcal {O}(\sqrt {N}\log (1/\delta))O(N​log(1/δ)) query complexity while requiring lower-degree polynomials. For N = 512, our method reduces the polynomial degree from 71 to 43 compared to prior QSVT-based fixed-point search. We also demonstrate how QSVT enables exact search construction.
One of the significant breakthroughs in quantum computation is Grover's algorithm for unsorted database search. Recently, the applications of Grover's algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed so that the search is accelerated with improved threshold values. By taking advantage of the quantum tunneling effect, better threshold values can be found at the early stage of the search process so that the sharpness of probability improves. The results between the new algorithm, existing Grover search, and classical heuristic algorithms are compared.