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:2608.23002

Exact and Optimal Recursive Quantum Search via Hilbert-Space Decomposition

John Burke, Ciaran McGoldrick

quant-phcs.DSSubmitted Aug 24, 2026• Announced Aug 25, 2026• 0 citations
arXivHTMLPDF

Abstract

Current approaches to quantum search fail to deeply exploit extant structure in the underlying Hilbert space. Decomposing the search by this structure empowers new strategies and formulations for quantum search and algorithm design. We present a new decomposition technique acting directly on this structure by recursively decomposing the Hilbert space and constructing the search operator from reflections over the resulting partition. When initial and target states factorise over this partition, dynamics reduce to a single rotation in a two-dimensional plane at each level, with angle given by a scalar recurrence. This recurrence avoids error accumulation from separately bounding success probabilities at each level, yielding an exact state description enabling treatment of the recursion as a whole. We obtain the target state deterministically and derive oracle and non-oracle costs independently of the search setting. For unstructured search, our approach attains the simultaneously optimal $Θ(\sqrt{N})$ oracle and non-oracle gate counts. For spatial search on $d$-dimension grids, it recovers the $O(\sqrt{N})$ time for $d\geq3$ and the $O\bigl(\sqrt{N}(\log N)^{3/2}\bigr)$ bound of Aaronson and Ambainis for $d=2$. The exact description of the recursion extends over our decomposition to new subdivision structures and provides a new approach for applying and analysing recursion in quantum algorithm design.

Comments
28 pages, 3 figures

Related papers

Quantum Search without Global Diffusion

John Burke, Ciaran McGoldrick

Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle. Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the $O(\sqrt{N})$ oracle complexity of Grover search when each partition contains at least $\log_2(\log_2 N)$ qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.

quant-pharXiv:2604.15435•Apr 20, 2026•0 citations

Exact and Fixed-Point Grover Search with Qudits

Tanay Roy

Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.

quant-pharXiv:2607.24658•Jul 28, 2026•0 citations

Deterministic Quantum Search via Recursive Oracle Expansion

Burke, John, Ciarán McGoldrick

We introduce a novel deterministic quantum search algorithm that provides a practical alternative to conventional probabilistic search approaches. Our scheme eliminates the inherent uncertainty of quantum search without relying on arbitrary phase rotations, a key limitation of other deterministic methods. The algorithm achieves certainty by recursively expanding the base oracle so that it marks all states prefixed by the same two bits as the target, encompassing exactly one-quarter of the search space. This enables a step-by-step reduction of the superposition until the target state can be measured with certainty. The algorithm achieves deterministic success with a query complexity of $O(N^{\log_2(3)/2}) \approx O(N^{0.7925})$, falling between Grover's $O(\sqrt{N})$ scaling and the classical $O(N)$. Our approach relies exclusively on two-qubit nearest-neighbour diffusion operators, avoiding global diffusion entirely. We show that, despite the increased query complexity, this design reduces the total number of two-qubit gates required for diffusion by more than an order of magnitude for search spaces up to at least 18 qubits, with even greater advantages on hardware with limited qubit connectivity. The scheme's inherent determinism, reliance on simple nearest-neighbour, low-depth operations, and scalable recursive structure make it well-suited for hardware implementation. Additionally, we show that the algorithm naturally supports partial database search, enabling deterministic identification of selected target bits without requiring a full search, further broadening its applicability.

arXiv:2507.15797•Jul 21, 2025•0 citations

Universal construction for the unsorted quantum search algorithms

Xijia Miao

The multiple-quantum operator algebra formalism has been exploited to construct generally an unsorted quantum search algorithm. The exponential propagator and its corresponding effective Hamiltonian are constructed explicitly that describe in quantum mechanics the time evolution of a multi- particle two-state quantum system from the initial state to the output of the unsorted quantum search problem. The exponential propagator usually may not be compatible with the mathematical structure and principle of the search problem and hence is not a real quantum search network, but it can be further decomposed into a product of a series of the oracle unitary operations such as the selective phase-shift operations and the nonselective unitary operations which can be expressed further as a sequence of elementary building blocks such as one-qubit quantum gates and the two-qubit diagonal phase gates, resulting in that the decomposed propagator is compatible with the mathematical structure and principle of the search problem and thus, becomes a real quantum search network. The decomposition for the propagator can be achieved with the help of the operator algebra structure and symmetry of the effective Hamiltonian, and the properties of the multiple-quantum operator algebra spaces, especially the characteristic transformation behavior of the multiple-quantum operators under the z-axis rotations. It has been shown that the computational complexity of the search algorithm is dependent upon that of the numerical multidimensional integration and hence it is believed that the search algorithm could solve efficiently the unsorted search problem. An NMR device is also proposed to solve efficiently the unsorted search problem in polynomial time.

arXiv:quant-ph/0101126•Jan 26, 2001•4 citations

Quantum Search in Hilbert Space

Michail A. Zak

A proposed quantum-computing algorithm would perform a search for an item of information in a database stored in a Hilbert-space memory structure. The algorithm is intended to make it possible to search relatively quickly through a large database under conditions in which available computing resources would otherwise be considered inadequate to perform such a task. The algorithm would apply, more specifically, to a relational database in which information would be stored in a set of N complex orthonormal vectors, each of N dimensions (where N can be exponentially large). Each vector would constitute one row of a unitary matrix, from which one would derive the Hamiltonian operator (and hence the evolutionary operator) of a quantum system. In other words, all the stored information would be mapped onto a unitary operator acting on a quantum state that would represent the item of information to be retrieved. Then one could exploit quantum parallelism: one could pose all search queries simultaneously by performing a quantum measurement on the system. In so doing, one would effectively solve the search problem in one computational step. One could exploit the direct- and inner-product decomposability of the unitary matrix to make the dimensionality of the memory space exponentially large by use of only linear resources. However, inasmuch as the necessary preprocessing (the mapping of the stored information into a Hilbert space) could be exponentially expensive, the proposed algorithm would likely be most beneficial in applications in which the resources available for preprocessing were much greater than those available for searching.

•Jan 1, 2003•0 citations

Quantum search algorithms on the hypercube

Birgit Hein, Gregor Tanner

We investigate a set of discrete-time quantum search algorithms on the n -dimensional hypercube following a proposal by Shenvi et al (2003 Phys. Rev. A 67 052307). We show that there exists a whole class of quantum search algorithms in the symmetry-reduced space which perform a search of a marked vertex in time of order where N = 2 n , the number of vertices. In analogy to Grover's algorithm, the spatial search is effectively facilitated through a rotation in a two-level subspace of the full Hilbert space. In the hypercube, these two-level systems are introduced through avoided crossings. We give estimates on the quantum states forming the two-level subspaces at the avoided crossings and derive improved estimates on the search times.

arXiv:0906.3094•Journal of Physics A Mathematical and Theoretical•Jan 30, 2009•26 citations
Current approaches to quantum search fail to deeply exploit extant structure in the underlying Hilbert space. Decomposing the search by this structure empowers new strategies and formulations for quantum search and algorithm design. We present a new decomposition technique acting directly on this structure by recursively decomposing the Hilbert space and constructing the search operator from reflections over the resulting partition. When initial and target states factorise over this partition, dynamics reduce to a single rotation in a two-dimensional plane at each level, with angle given by a scalar recurrence. This recurrence avoids error accumulation from separately bounding success probabilities at each level, yielding an exact state description enabling treatment of the recursion as a whole. We obtain the target state deterministically and derive oracle and non-oracle costs independently of the search setting. For unstructured search, our approach attains the simultaneously optimal Θ(N)Θ(\sqrt{N})Θ(N​) oracle and non-oracle gate counts. For spatial search on ddd-dimension grids, it recovers the O(N)O(\sqrt{N})O(N​) time for d≥3d\geq3d≥3 and the O(N(log⁡N)3/2)O\bigl(\sqrt{N}(\log N)^{3/2}\bigr)O(N​(logN)3/2) bound of Aaronson and Ambainis for d=2d=2d=2. The exact description of the recursion extends over our decomposition to new subdivision structures and provides a new approach for applying and analysing recursion in quantum algorithm design.
Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle. Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the O(N)O(\sqrt{N})O(N​) oracle complexity of Grover search when each partition contains at least log⁡2(log⁡2N)\log_2(\log_2 N)log2​(log2​N) qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.
Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.
We introduce a novel deterministic quantum search algorithm that provides a practical alternative to conventional probabilistic search approaches. Our scheme eliminates the inherent uncertainty of quantum search without relying on arbitrary phase rotations, a key limitation of other deterministic methods. The algorithm achieves certainty by recursively expanding the base oracle so that it marks all states prefixed by the same two bits as the target, encompassing exactly one-quarter of the search space. This enables a step-by-step reduction of the superposition until the target state can be measured with certainty. The algorithm achieves deterministic success with a query complexity of O(Nlog⁡2(3)/2)≈O(N0.7925)O(N^{\log_2(3)/2}) \approx O(N^{0.7925})O(Nlog2​(3)/2)≈O(N0.7925), falling between Grover's O(N)O(\sqrt{N})O(N​) scaling and the classical O(N)O(N)O(N). Our approach relies exclusively on two-qubit nearest-neighbour diffusion operators, avoiding global diffusion entirely. We show that, despite the increased query complexity, this design reduces the total number of two-qubit gates required for diffusion by more than an order of magnitude for search spaces up to at least 18 qubits, with even greater advantages on hardware with limited qubit connectivity. The scheme's inherent determinism, reliance on simple nearest-neighbour, low-depth operations, and scalable recursive structure make it well-suited for hardware implementation. Additionally, we show that the algorithm naturally supports partial database search, enabling deterministic identification of selected target bits without requiring a full search, further broadening its applicability.
The multiple-quantum operator algebra formalism has been exploited to construct generally an unsorted quantum search algorithm. The exponential propagator and its corresponding effective Hamiltonian are constructed explicitly that describe in quantum mechanics the time evolution of a multi- particle two-state quantum system from the initial state to the output of the unsorted quantum search problem. The exponential propagator usually may not be compatible with the mathematical structure and principle of the search problem and hence is not a real quantum search network, but it can be further decomposed into a product of a series of the oracle unitary operations such as the selective phase-shift operations and the nonselective unitary operations which can be expressed further as a sequence of elementary building blocks such as one-qubit quantum gates and the two-qubit diagonal phase gates, resulting in that the decomposed propagator is compatible with the mathematical structure and principle of the search problem and thus, becomes a real quantum search network. The decomposition for the propagator can be achieved with the help of the operator algebra structure and symmetry of the effective Hamiltonian, and the properties of the multiple-quantum operator algebra spaces, especially the characteristic transformation behavior of the multiple-quantum operators under the z-axis rotations. It has been shown that the computational complexity of the search algorithm is dependent upon that of the numerical multidimensional integration and hence it is believed that the search algorithm could solve efficiently the unsorted search problem. An NMR device is also proposed to solve efficiently the unsorted search problem in polynomial time.
A proposed quantum-computing algorithm would perform a search for an item of information in a database stored in a Hilbert-space memory structure. The algorithm is intended to make it possible to search relatively quickly through a large database under conditions in which available computing resources would otherwise be considered inadequate to perform such a task. The algorithm would apply, more specifically, to a relational database in which information would be stored in a set of N complex orthonormal vectors, each of N dimensions (where N can be exponentially large). Each vector would constitute one row of a unitary matrix, from which one would derive the Hamiltonian operator (and hence the evolutionary operator) of a quantum system. In other words, all the stored information would be mapped onto a unitary operator acting on a quantum state that would represent the item of information to be retrieved. Then one could exploit quantum parallelism: one could pose all search queries simultaneously by performing a quantum measurement on the system. In so doing, one would effectively solve the search problem in one computational step. One could exploit the direct- and inner-product decomposability of the unitary matrix to make the dimensionality of the memory space exponentially large by use of only linear resources. However, inasmuch as the necessary preprocessing (the mapping of the stored information into a Hilbert space) could be exponentially expensive, the proposed algorithm would likely be most beneficial in applications in which the resources available for preprocessing were much greater than those available for searching.
We investigate a set of discrete-time quantum search algorithms on the n -dimensional hypercube following a proposal by Shenvi et al (2003 Phys. Rev. A 67 052307). We show that there exists a whole class of quantum search algorithms in the symmetry-reduced space which perform a search of a marked vertex in time of order where N = 2 n , the number of vertices. In analogy to Grover's algorithm, the spatial search is effectively facilitated through a rotation in a two-level subspace of the full Hilbert space. In the hypercube, these two-level systems are introduced through avoided crossings. We give estimates on the quantum states forming the two-level subspaces at the avoided crossings and derive improved estimates on the search times.