Information geometry of nonmonotonic quantum natural gradient

• URL: http://arxiv.org/abs/2510.18286v1
• Date: Tue, 21 Oct 2025 04:27:10 GMT
• Title: Information geometry of nonmonotonic quantum natural gradient
• Authors: Hideyuki Miyahara,
• Abstract summary: We investigate the properties of nonmonotonic quantum natural gradient (QNG)<n>Nonmonotonic QNG was shown to achieve faster convergence compared to conventional QNG.<n>We show that non-monotone quantum Fisher metrics can lead to faster convergence in QNG.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Natural gradient is an advanced optimization method based on information geometry, where the Fisher metric plays a crucial role. Its quantum counterpart, known as quantum natural gradient (QNG), employs the symmetric logarithmic derivative (SLD) metric, one of the quantum Fisher metrics. While quantization in physics is typically well-defined via the canonical commutation relations, the quantization of information-theoretic quantities introduces inherent arbitrariness. To resolve this ambiguity, monotonicity has been used as a guiding principle for constructing geometries in physics, as it aligns with physical intuition. Recently, a variant of QNG, which we refer to as nonmonotonic QNG in this paper, was proposed by relaxing the monotonicity condition. It was shown to achieve faster convergence compared to conventional QNG. In this paper, we investigate the properties of nonmonotonic QNG. To ensure the paper is self-contained, we first demonstrate that the SLD metric is locally optimal under the monotonicity condition and that non-monotone quantum Fisher metrics can lead to faster convergence in QNG. Previous studies primarily relied on a specific type of quantum divergence and assumed that density operators are full-rank. Here, we explicitly consider an alternative quantum divergence and extend the analysis to non-full-rank cases. Additionally, we explore how geometries can be designed using Petz functions, given that quantum Fisher metrics are characterized through them. Finally, we present numerical simulations comparing different quantum Fisher metrics in the context of parameter estimation problems in quantum circuit learning.

Related papers

• Stochastic Quantum Information Geometry and Speed Limits at the Trajectory Level [35.18016233072556]
  We bridge the gap between quantum information geometry and thermodynamics by introducing the Conditional Quantum Fisher Information (CQFI)<n>We show that the CQFI admits a decomposition into incoherent (population) and coherent (basis rotation) contributions, augmented by a transient interference cross-term absent at the ensemble level.
  arXiv  Detail & Related papers  (2026-01-18T16:23:26Z)
• Quantum Fisher information matrix via its classical counterpart from random measurements [5.726854405157353]
  Preconditioning with the quantum Fisher information matrix (QFIM) is a popular approach in quantum variational algorithms.<n>We show that averaging the classical Fisher information matrix over Haar-random measurement bases yields $mathbbE_Usimmu_H[FU(boldsymboltheta)] = frac12Q(boldsymboltheta)$ for pure states in $mathbbCN$.
  arXiv  Detail & Related papers  (2025-09-10T00:00:02Z)
• Quantum neural ordinary and partial differential equations [38.77776626953413]
  We present a unified framework that brings the continuous-time formalism of classical neural ODEs/PDEs into quantum machine learning and quantum control.<n>We define QNODEs as the evolution of finite-dimensional quantum systems, and QNPDEs as infinite-dimensional (continuous-variable) counterparts.
  arXiv  Detail & Related papers  (2025-08-24T18:43:44Z)
• Weighted Approximate Quantum Natural Gradient for Variational Quantum Eigensolver [5.873113584103881]
  Variational quantum eigensolver (VQE) is one of the most prominent algorithms using near-term quantum devices.<n>We propose a weighted Approximate Quantum Natural Gradient (WA-QNG) method tailored for $k$ of local Hamiltonians.
  arXiv  Detail & Related papers  (2025-04-07T11:18:09Z)
• Variational Quantum Subspace Construction via Symmetry-Preserving Cost Functions [36.94429692322632]
  We propose a variational strategy based on symmetry-preserving cost functions to iteratively construct a reduced subspace for extraction of low-lying energy states.<n>As a proof of concept, we test the proposed algorithms on H4 chain and ring, targeting both the ground-state energy and the charge gap.
  arXiv  Detail & Related papers  (2024-11-25T20:33:47Z)
• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• Quantum Natural Stochastic Pairwise Coordinate Descent [13.986982036653632]
  Variational quantum algorithms, optimized using gradient-based methods, often exhibit sub-optimal convergence performance.<n>Quantum natural gradient descent (QNGD) is a more efficient method that incorporates the geometry of the state space via a quantum information metric.<n>We formulate a novel quantum information metric and construct an unbiased estimator for this metric using single-shot measurements.
  arXiv  Detail & Related papers  (2024-07-18T18:57:29Z)
• Quantum natural gradient without monotonicity [0.0]
  The quantum natural gradient (QNG) was introduced and utilized for noisy intermediate-scale devices. In this paper, we propose generalized QNG by removing the condition of monotonicity. We provide analytical and numerical evidence showing that non-monotone QNG based on the SLD metric in terms of convergence speed.
  arXiv  Detail & Related papers  (2024-01-24T05:54:02Z)
• Wasserstein Quantum Monte Carlo: A Novel Approach for Solving the Quantum Many-Body Schr\"odinger Equation [56.9919517199927]
  "Wasserstein Quantum Monte Carlo" (WQMC) uses the gradient flow induced by the Wasserstein metric, rather than Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.
  arXiv  Detail & Related papers  (2023-07-06T17:54:08Z)
• Quantum Neural Estimation of Entropies [20.12693323453867]
  entropy measures quantify the amount of information and correlation present in a quantum system. We propose a variational quantum algorithm for estimating the von Neumann and R'enyi entropies, as well as the measured relative entropy and measured R'enyi relative entropy.
  arXiv  Detail & Related papers  (2023-07-03T17:30:09Z)
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory [44.99833362998488]
  We introduce quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple quantum numbers. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories.
  arXiv  Detail & Related papers  (2022-12-28T18:56:25Z)
• Direct measurement of quantum Fisher information [5.067521928161945]
  In the adiabatic perturbation theory, Berry curvature is related to the generalized force, and the quantum metric tensor is linked with energy fluctuation. In this article, we first adopt an alternative way to derive the link of energy fluctuation to the quantum Fisher information. We numerically testify the direct extraction of the quantum Fisher information based on adiabatic perturbation in two-level systems.
  arXiv  Detail & Related papers  (2022-08-05T13:07:31Z)
• Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry [3.5097082077065003]
  We introduce a generalization of quantum natural gradient descent to parameterized mixed states. We also provide a robust first-order approximating algorithm, Quantum-Probabilistic Mirror Descent. Our approaches extend previously sample-efficient techniques to allow for flexibility in model choice.
  arXiv  Detail & Related papers  (2022-06-09T17:58:15Z)

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.