Quantum neural ordinary and partial differential equations

• URL: http://arxiv.org/abs/2508.18326v1
• Date: Sun, 24 Aug 2025 18:43:44 GMT
• Title: Quantum neural ordinary and partial differential equations
• Authors: Yu Cao, Shi Jin, Nana Liu,
• Abstract summary: 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.
• Score: 38.77776626953413
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a unified framework called Quantum Neural Ordinary and Partial Differential Equations (QNODEs and QNPDEs) that brings the continuous-time formalism of classical neural ODEs/PDEs into quantum machine learning and quantum control. We define QNODEs as the evolution of finite-dimensional quantum systems, and QNPDEs as infinite-dimensional (continuous-variable) counterparts, governed by generalised Schrodinger-type Hamiltonian dynamics with unitary evolution, coupled with a corresponding loss function. Notably, this formalism permits gradient estimation using an adjoint-state method, facilitating efficient learning of quantum dynamics, and other dynamics that can be mapped (relatively easily) to quantum dynamics. Using this method, we present quantum algorithms for computing gradients with and without time discretisation, enabling efficient gradient computation that would otherwise be less efficient on classical devices. The formalism subsumes a wide array of applications, including quantum state preparation, Hamiltonian learning, learning dynamics in open systems, and the learning of both autonomous and non-autonomous classical ODEs and PDEs. In many of these applications, we consider scenarios where the Hamiltonian has relatively few tunable parameters, yet the corresponding classical simulation remains inefficient, making quantum approaches advantageous for gradient estimation. This continuous-time perspective can also serve as a blueprint for designing novel quantum neural network architectures, generalising discrete-layered models into continuous-depth models.

Related papers

• Quantum Filtering and Stabilization of Dissipative Quantum Systems via Augmented Neural Ordinary Differential Equations [5.819937157229223]
  AQNODE is a framework that learns quantum trajectories and dissipation parameters directly from measurement data.<n>Our approach integrates weak measurement data to reconstruct qubit states and time-dependent decoherence rates.<n> AQNODE is a scalable, differentiable, and experimentally compatible framework for real-time modeling and control of dissipative quantum systems.
  arXiv  Detail & Related papers  (2025-09-08T20:23:45Z)
• Numerical solution of quantum Landau-Lifshitz-Gilbert equation [0.0]
  Landau-Lifshitz-Gilbert (LLG) equation has long served as a cornerstone for modeling magnetization dynamics in magnetic systems.<n>Inspired by the need to incorporate quantum effects into spin dynamics, recently a quantum generalization of the LG equation is proposed.<n>We develop a robust numerical methodology tailored to this quantum LG framework.
  arXiv  Detail & Related papers  (2025-06-24T13:04:07Z)
• VQC-MLPNet: An Unconventional Hybrid Quantum-Classical Architecture for Scalable and Robust Quantum Machine Learning [50.95799256262098]
  Variational quantum circuits (VQCs) hold promise for quantum machine learning but face challenges in expressivity, trainability, and noise resilience.<n>We propose VQC-MLPNet, a hybrid architecture where a VQC generates the first-layer weights of a classical multilayer perceptron during training, while inference is performed entirely classically.
  arXiv  Detail & Related papers  (2025-06-12T01:38:15Z)
• State-Based Quantum Simulation: Releasing the Powers of Quantum States and Copies [0.0]
  We introduce a method for quantum simulation in which the Hamiltonian is decomposed in terms of states.<n>We show how classical nonlinear and time-delayed ordinary differential equations can be simulated with the state-based method.
  arXiv  Detail & Related papers  (2025-05-20T04:05:26Z)
• Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems [77.88054335119074]
  We use FNOs to model the evolution of random quantum spin systems. We apply FNOs to a compact set of Hamiltonian observables instead of the entire $2n$ quantum wavefunction.
  arXiv  Detail & Related papers  (2024-09-05T07:18:09Z)
• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits [63.733312560668274]
  Given a quantum circuit containing d tunable RZ gates and G-d Clifford gates, can a learner perform purely classical inference to efficiently predict its linear properties? We prove that the sample complexity scaling linearly in d is necessary and sufficient to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. We devise a kernel-based learning model capable of trading off prediction error and computational complexity, transitioning from exponential to scaling in many practical settings.
  arXiv  Detail & Related papers  (2024-08-22T08:21:28Z)
• Quantum data learning for quantum simulations in high-energy physics [55.41644538483948]
  We explore the applicability of quantum-data learning to practical problems in high-energy physics. We make use of ansatz based on quantum convolutional neural networks and numerically show that it is capable of recognizing quantum phases of ground states. The observation of non-trivial learning properties demonstrated in these benchmarks will motivate further exploration of the quantum-data learning architecture in high-energy physics.
  arXiv  Detail & Related papers  (2023-06-29T18:00:01Z)
• Deep learning-based quantum algorithms for solving nonlinear partial differential equations [3.312385039704987]
  Partial differential equations frequently appear in the natural sciences and related disciplines. We explore the potential for enhancing a classical deep learning-based method for solving high-dimensional nonlinear partial differential equations.
  arXiv  Detail & Related papers  (2023-05-03T10:17:51Z)
• The Quantum Path Kernel: a Generalized Quantum Neural Tangent Kernel for Deep Quantum Machine Learning [52.77024349608834]
  Building a quantum analog of classical deep neural networks represents a fundamental challenge in quantum computing. Key issue is how to address the inherent non-linearity of classical deep learning. We introduce the Quantum Path Kernel, a formulation of quantum machine learning capable of replicating those aspects of deep machine learning.
  arXiv  Detail & Related papers  (2022-12-22T16:06:24Z)
• Quantum algorithms for quantum dynamics: A performance study on the spin-boson model [68.8204255655161]
  Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter-approximation of the time-evolution operator. variational quantum algorithms have become an indispensable alternative, enabling small-scale simulations on present-day hardware. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage.
  arXiv  Detail & Related papers  (2021-08-09T18:00:05Z)
• Variational classical networks for dynamics in interacting quantum matter [0.0]
  We introduce a variational class of wavefunctions based on complex networks of classical spins akin to artificial neural networks. We show that our method can be applied to any quantum many-body system with a well-defined classical limit.
  arXiv  Detail & Related papers  (2020-07-31T14:03:37Z)

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.