© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2610.06364

Adaptive complete quantum search with unknown number of solutions

Raj Alexandru Guţoiu, Bogdan-Călin Ciobanu, Pantelimon George Popescu

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations

Abstract

Unstructured database search is a fundamental problem that requires finding one or more elements satisfying a specific property within a set of candidates. While Grover's algorithm provides a quadratic speedup for finding a marked element when the number of solutions is known, finding all solutions when their number is unknown presents additional challenges. In this paper, we thoroughly analyze this problem by considering all the different regimes of combinations of classical, Grover and generalized fixed-point quantum search instances. We unify these into an adaptive Bayesian approach that maximizes the expected success probability per unit execution time at each searching instance. For a problem with a known number of solutions, much lower than the database size, we prove that our proposed approach is asymptotically optimal within the considered search model. Furthermore, we prove that for any probability distribution characterizing the likely number of solutions, in regimes where the classical verification time is at most 25% of the quantum amplitude amplification time, hybrid classical and Grover approaches are preferred to generalized fixed-point. Outside of this regime, we derive an upper bound on the maximal improvement achievable by fixed-point search compared to the best Grover-based score. Additionally, we determine the optimal transition point between classical sampling and quantum searching using an integral approximation of the probability distribution of the number of solutions. The analytical results are also supported by numerical simulations of the proposed algorithm and by comparisons between the generalized fixed-point and Grover-based search policies.

Comments: 35 pages, 8 figures

Related papers

Quantum Search with Prior Knowledge

Xiaoyu He, Jialin Zhang, Xiaoming Sun

Search-base algorithms have widespread applications in different scenarios. Grover's quantum search algorithms and its generalization, amplitude amplification, provide a quadratic speedup over classical search algorithms for unstructured search. We consider the problem of searching with prior knowledge. More preciously, search for the solution among N items with a prior probability distribution. This letter proposes a new generalization of Grover's search algorithm which performs better than the standard Grover algorithm in average under this setting. We prove that our new algorithm achieves the optimal expected success probability of finding the solution if the number of queries is fixed.

arXiv:2009.08721•Sep 18, 2020•1 citation

Asymptotic bounds on quantum partial search algorithm and its applications to parallel search

Yan-Bo Jiang, Xiao-Hui Wang, Kun Zhang +1 more

Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is the standard and most extensively studied protocol for this task. In this work, we provide systematic numerical evidence that the GRK operator sequence gives the highest success probability in all examined cases, supporting it as the optimal ansatz among admissible compositions of global and local Grover operators. Guided by this numerically supported GRK ansatz, we derive an asymptotically tight upper bound on the maximal success probability within the GRK family and establish the corresponding lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols yields a strict, though subleading, improvement over the outer parallel Grover scheme. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.

quant-pharXiv:2603.01462•Physical Review A•Mar 3, 2026•1 citation

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method*

Tan Li, Shuo Zhang, Xiang-Qun Fu +4 more

For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O ( 1 / λ 0 ) rather than O ( 1 / λ ) , where λ 0 is a known lower bound of λ. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.

arXiv:1912.01773•Chinese Physics B•Nov 1, 2019•8 citations

Introducing structure to expedite quantum searching

Marcin Briański, Jan Gwinner, Vladyslav Hlembotskyi +3 more

We present a quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on noisy intermediate-scale quantum devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are, in fact, not unstructured, we also describe the partial uncompute technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use an asymptotically smaller number of elementary gates than Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example, Unique $k$-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.

arXiv:2006.05828•Physical Review A•Jun 28, 2021•13 citations

Quantum mechanics can find a needle in a haystack every time

F. Mohit, Joshua J. Guanzon, Jaden McKinlay +5 more

Grover's algorithm is one of the pioneering demonstrations of the advantages of quantum computing over its classical counterpart, providing - at most - a quadratic speed-up over the classical solution for unstructured database search. The original formulation of Grover's algorithm is non-deterministic, finding the answer with a probability that varies with the size of the search space and the number of marked elements. A recent reformulation introduced a deterministic form of Grover's algorithm that - in principle - finds the answer with certainty. Here we realise the deterministic Grover's algorithm on a programmable photonic integrated circuit, finding that it not only outperforms the original Grover's algorithm as predicted, but is also markedly more robust against technological imperfections. We explore databases of 4 to 10 elements, with every choice of a single marked element, achieving an average success probability of $99.77 \pm 0.05\%$.

arXiv:2506.06435•Jun 6, 2025•0 citations

Scaling Grover’s Search for Large Solution Spaces

Simone Faro, Francesco Pio Marino

Grover's algorithm offers a celebrated quadratic speedup for unstructured search problems, positioning it among the foundational tools of quantum computing. However, its effectiveness critically deteriorates when the number of valid solutions exceeds half of the total search space. In this work, we address this limitation by introducing a novel and efficient technique based on a virtual expansion of the input domain. Our approach extends the range of applicability of Grover's algorithm to high-density solution spaces, including those where the number of solutions is greater than N/2, without increasing the actual circuit depth or memory footprint. The core idea consists in appending one auxiliary qubit to the input register and modifying the oracle to mark only those solutions where the auxiliary qubit is in a fixed state. This effectively reduces the solution density in the extended space, restoring the geometric conditions required for successful amplitude amplification. We further analyze an extension of this strategy involving log n auxiliary qubits, which allows for tunable robustness at a moderate increase in computational cost. The proposed method is simple to implement on existing quantum hardware and maintains Grover's optimal asymptotic complexity in the expanded regime. Both theoretical analysis and empirical simulations confirm that our approach reliably outperforms the standard algorithm in scenarios with high solution density, thus broadening the practical utility of quantum search.

ACM International Symposium on High-Performance Parallel and Distributed Computing (HPDC)•Jul 20, 2025•1 citation

Unstructured database search is a fundamental problem that requires finding one or more elements satisfying a specific property within a set of candidates. While Grover's algorithm provides a quadratic speedup for finding a marked element when the number of solutions is known, finding all solutions when their number is unknown presents additional challenges. In this paper, we thoroughly analyze this problem by considering all the different regimes of combinations of classical, Grover and generalized fixed-point quantum search instances. We unify these into an adaptive Bayesian approach that maximizes the expected success probability per unit execution time at each searching instance. For a problem with a known number of solutions, much lower than the database size, we prove that our proposed approach is asymptotically optimal within the considered search model. Furthermore, we prove that for any probability distribution characterizing the likely number of solutions, in regimes where the classical verification time is at most 25% of the quantum amplitude amplification time, hybrid classical and Grover approaches are preferred to generalized fixed-point. Outside of this regime, we derive an upper bound on the maximal improvement achievable by fixed-point search compared to the best Grover-based score. Additionally, we determine the optimal transition point between classical sampling and quantum searching using an integral approximation of the probability distribution of the number of solutions. The analytical results are also supported by numerical simulations of the proposed algorithm and by comparisons between the generalized fixed-point and Grover-based search policies.

Search-base algorithms have widespread applications in different scenarios. Grover's quantum search algorithms and its generalization, amplitude amplification, provide a quadratic speedup over classical search algorithms for unstructured search. We consider the problem of searching with prior knowledge. More preciously, search for the solution among N items with a prior probability distribution. This letter proposes a new generalization of Grover's search algorithm which performs better than the standard Grover algorithm in average under this setting. We prove that our new algorithm achieves the optimal expected success probability of finding the solution if the number of queries is fixed.

Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is the standard and most extensively studied protocol for this task. In this work, we provide systematic numerical evidence that the GRK operator sequence gives the highest success probability in all examined cases, supporting it as the optimal ansatz among admissible compositions of global and local Grover operators. Guided by this numerically supported GRK ansatz, we derive an asymptotically tight upper bound on the maximal success probability within the GRK family and establish the corresponding lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols yields a strict, though subleading, improvement over the outer parallel Grover scheme. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.

For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O ( 1 / λ 0 ) rather than O ( 1 / λ ) , where λ 0 is a known lower bound of λ. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.

We present a quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on noisy intermediate-scale quantum devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are, in fact, not unstructured, we also describe the partial uncompute technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use an asymptotically smaller number of elementary gates than Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example, Unique

k

-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.

Grover's algorithm is one of the pioneering demonstrations of the advantages of quantum computing over its classical counterpart, providing - at most - a quadratic speed-up over the classical solution for unstructured database search. The original formulation of Grover's algorithm is non-deterministic, finding the answer with a probability that varies with the size of the search space and the number of marked elements. A recent reformulation introduced a deterministic form of Grover's algorithm that - in principle - finds the answer with certainty. Here we realise the deterministic Grover's algorithm on a programmable photonic integrated circuit, finding that it not only outperforms the original Grover's algorithm as predicted, but is also markedly more robust against technological imperfections. We explore databases of 4 to 10 elements, with every choice of a single marked element, achieving an average success probability of

99.77 \pm 0.05\%

.

Grover's algorithm offers a celebrated quadratic speedup for unstructured search problems, positioning it among the foundational tools of quantum computing. However, its effectiveness critically deteriorates when the number of valid solutions exceeds half of the total search space. In this work, we address this limitation by introducing a novel and efficient technique based on a virtual expansion of the input domain. Our approach extends the range of applicability of Grover's algorithm to high-density solution spaces, including those where the number of solutions is greater than N/2, without increasing the actual circuit depth or memory footprint. The core idea consists in appending one auxiliary qubit to the input register and modifying the oracle to mark only those solutions where the auxiliary qubit is in a fixed state. This effectively reduces the solution density in the extended space, restoring the geometric conditions required for successful amplitude amplification. We further analyze an extension of this strategy involving log n auxiliary qubits, which allows for tunable robustness at a moderate increase in computational cost. The proposed method is simple to implement on existing quantum hardware and maintains Grover's optimal asymptotic complexity in the expanded regime. Both theoretical analysis and empirical simulations confirm that our approach reliably outperforms the standard algorithm in scenarios with high solution density, thus broadening the practical utility of quantum search.