Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search

• URL: http://arxiv.org/abs/2603.26039v2
• Date: Fri, 03 Apr 2026 03:09:29 GMT
• Title: Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search
• Abstract summary: We show that the RMN method numerically achieves a quadratic convergence rate with respect to the error $varepsilon$, implying a complexity of $O(sqrtN/M loglog (1/varepsilon)$.
• Score: 17.013842168748127
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Grover's algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems of size $N$. Recent studies have reformulated this task as a maximization problem on the unitary manifold and solved it via linearly convergent Riemannian gradient ascent (RGA) methods, resulting in a complexity of $O(\sqrt{N/M} \log (1/\varepsilon))$, where $M$ denotes the number of target items. In this work, we adopt the Riemannian modified Newton (RMN) method to solve the quantum search problem, under the assumption that the ratio $ M/N$ is known. We show that, in this setting, the Riemannian Newton direction is collinear with the Riemannian gradient in the sense that the Riemannian gradient is always an eigenvector of the corresponding Riemannian Hessian. As a result, without additional overhead, the proposed RMN method numerically achieves a quadratic convergence rate with respect to the error $\varepsilon$, implying a complexity of $O(\sqrt{N/M} \log\log (1/\varepsilon))$. Furthermore, our approach remains Grover-compatible, namely, it relies exclusively on the standard Grover diffusion and oracle operators to ensure algorithmic implementability, and its parameter update process can be efficiently precomputed on classical computers.

Related papers

• Provably Efficient Algorithms for S- and Non-Rectangular Robust MDPs with General Parameterization [85.91302339486673]
  We study robust Markov decision processes (RMDPs) with general policy parameterization under s-rectangular and non-rectangular uncertainty sets.<n>We prove novel Lipschitz and Lipschitz-smoothness properties for general policy parameterizations that extends to infinite state spaces.<n>We design a projected gradient descent algorithm for s-rectangular uncertainty and a Frank-Wolfe algorithm for non-rectangular uncertainty.
  arXiv  Detail & Related papers  (2026-02-11T21:44:20Z)
• A Grover-compatible manifold optimization algorithm for quantum search [17.013842168748127]
  Grover's algorithm is a fundamental quantum algorithm that offers a quadratic speedup for the unstructured search problem.<n>We show that Grover's algorithm matches the speedup of $O(qrstN)$ achieved by Grover's algorithm.
  arXiv  Detail & Related papers  (2025-12-09T10:01:55Z)
• A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval [56.67706781191521]
  In this work, we present a robust phase retrieval problem where the task is to recover an unknown signal. Our proposed oracle avoids the need for computationally spectral descent, using a simple gradient step and outliers.
  arXiv  Detail & Related papers  (2024-09-07T06:37:23Z)
• Riemannian Bilevel Optimization [35.42472057648458]
  We focus in particular on batch and gradient-based methods, with the explicit goal of avoiding second-order information. We propose and analyze $mathrmRF2SA$, a method that leverages first-order gradient information. We provide explicit convergence rates for reaching $epsilon$-stationary points under various setups.
  arXiv  Detail & Related papers  (2024-05-22T20:49:01Z)
• Convergence and Complexity Guarantee for Inexact First-order Riemannian Optimization Algorithms [18.425648833592312]
  We show that tBMM converges to an $ilon$-stationary point within $O(epsilon-2)$. Under a mild iteration, the results still hold when the subproblem is solved in iterations in each provided the total optimality gap is bounded.
  arXiv  Detail & Related papers  (2024-05-05T22:53:14Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Stochastic Optimization for Non-convex Problem with Inexact Hessian Matrix, Gradient, and Function [99.31457740916815]
  Trust-region (TR) and adaptive regularization using cubics have proven to have some very appealing theoretical properties. We show that TR and ARC methods can simultaneously provide inexact computations of the Hessian, gradient, and function values.
  arXiv  Detail & Related papers  (2023-10-18T10:29:58Z)
• Gauss-Newton Temporal Difference Learning with Nonlinear Function Approximation [11.925232472331494]
  A Gauss-Newton Temporal Difference (GNTD) learning method is proposed to solve the Q-learning problem with nonlinear function approximation. In each iteration, our method takes one Gauss-Newton (GN) step to optimize a variant of Mean-Squared Bellman Error (MSBE) We validate our method via extensive experiments in several RL benchmarks, where GNTD exhibits both higher rewards and faster convergence than TD-type methods.
  arXiv  Detail & Related papers  (2023-02-25T14:14:01Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• Explicit Second-Order Min-Max Optimization: Practical Algorithms and Complexity Analysis [71.05708939639537]
  We propose and analyze several inexact regularized Newton-type methods for finding a global saddle point of emphconcave unconstrained problems.<n>Our method improves the existing line-search-based min-max optimization by shaving off an $O(loglog(1/eps)$ factor in the required number of Schur decompositions.
  arXiv  Detail & Related papers  (2022-10-23T21:24:37Z)
• Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization [94.19177623349947]
  Non-smooth non optimization problems emerge in machine learning and business making. Two core challenges impede the development of efficient methods with finitetime convergence guarantee. Two-phase versions of GFM and SGFM are also proposed and proven to achieve improved large-deviation results.
  arXiv  Detail & Related papers  (2022-09-12T06:53:24Z)
• On Riemannian Gradient-Based Methods for Minimax Problems [24.199289678553896]
  We propose a class of Riemanian-based methods to solve minimax problems. We prove that our RGDA has a sample complexity of $tildeO(kappa4eps-3)$. We also show that our Acc-RSGDA achieves a lower sample complexity on $tildeO(kappa4eps-3)$.
  arXiv  Detail & Related papers  (2020-10-13T00:54:00Z)
• On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems [75.58134963501094]
  This paper analyzes the trajectories of gradient descent (SGD) We show that SGD avoids saddle points/manifolds with $1$ for strict step-size policies.
  arXiv  Detail & Related papers  (2020-06-19T14:11:26Z)
• Riemannian Stochastic Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifold [7.257751371276488]
  R-ProxSGD and R-ProxSPB are generalizations of proximal SGD and proximal SpiderBoost. R-ProxSPB algorithm finds an $epsilon$-stationary point with $O(epsilon-3)$ IFOs in the online case, and $O(n+sqrtnepsilon-3)$ IFOs in the finite-sum case.
  arXiv  Detail & Related papers  (2020-05-03T23:41:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.