We show that quantum information geometry can be used to characterize Grover's searching algorithm. Specifically, quantifying the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric, we uncover that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuos approximation of the parametric quantum out-put state in Grover's algorithm.We show that quantum information geometry can be used to characterize Grover's searching algorithm. Specifically, quantifying the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric, we uncover that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuos approximation of the parametric quantum out-put state in Grover's algorithm.

AIP conference proceedingsJan 1, 20129 citations

Generalization of Grover’s algorithm to multiobject search in quantum computing, Part I

Goong Chen, S. A. Fulling, Jeesen Chen

Computational matematics series/Computational mathematics seriesFeb 14, 20021 citation

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.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:2512.08432Dec 10, 20250 citations

The geometry of entanglement and Grover's algorithm

Toshihiro Iwai, Naoki Hayashi, Kimitake Mizobe

A measure of entanglement with respect to a bipartite partition of n -qubit has been defined and studied from the viewpoint of Riemannian geometry (Iwai 2007 J. Phys. A: Math. Theor. 40 12161 ). This paper has two aims. One is to study further the geometry of entanglement, and the other is to investigate Grover's search algorithms, both the original and the fixed-point ones, in reference with entanglement. As the distance between the maximally entangled states and the separable states is known already in the previous paper, this paper determines the set of maximally entangled states nearest to a typical separable state which is used as an initial state in Grover's search algorithms, and to find geodesic segments which realize the above-mentioned distance. As for Grover's algorithms, it is already known that while the initial and the target states are separable, the algorithms generate sequences of entangled states. This fact is confirmed also in the entanglement measure proposed in the previous paper, and then a split Grover algorithm is proposed which generates sequences of separable states only with respect to the bipartite partition.A measure of entanglement with respect to a bipartite partition of n -qubit has been defined and studied from the viewpoint of Riemannian geometry (Iwai 2007 J. Phys. A: Math. Theor. 40 12161 ). This paper has two aims. One is to study further the geometry of entanglement, and the other is to investigate Grover's search algorithms, both the original and the fixed-point ones, in reference with entanglement. As the distance between the maximally entangled states and the separable states is known already in the previous paper, this paper determines the set of maximally entangled states nearest to a typical separable state which is used as an initial state in Grover's search algorithms, and to find geodesic segments which realize the above-mentioned distance. As for Grover's algorithms, it is already known that while the initial and the target states are separable, the algorithms generate sequences of entangled states. This fact is confirmed also in the entanglement measure proposed in the previous paper, and then a split Grover algorithm is proposed which generates sequences of separable states only with respect to the bipartite partition.

Journal of Physics A Mathematical and TheoreticalFeb 26, 20084 citations

Problem-Solution Symmetry in Grover’s Quantum Search Algorithm

Fumiaki Morikoshi

International Journal of Theoretical PhysicsFeb 7, 20116 citations

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.