Quantum physics informed neural networks for multi-variable partial differential equations

• URL: http://arxiv.org/abs/2503.12244v1
• Date: Sat, 15 Mar 2025 19:55:33 GMT
• Title: Quantum physics informed neural networks for multi-variable partial differential equations
• Authors: Giorgio Panichi, Sebastiano Corli, Enrico Prati,
• Abstract summary: We introduce an architecture specifically designed to compute second-order (and higher-order) derivatives without relying on nested automatic differentiation methods.<n>This approach mitigates the unwanted side effects associated with nested gradients in simulations, paving the way for more efficient and accurate implementations.<n>As a proof-of-concept, we solve a one-dimensional instance of the heat equation, demonstrating its effectiveness in handling PDEs.
• Score: 1.024113475677323
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum Physics-Informed Neural Networks (QPINNs) integrate quantum computing and machine learning to impose physical biases on the output of a quantum neural network, aiming to either solve or discover differential equations. The approach has recently been implemented on both the gate model and continuous variable quantum computing architecture, where it has been demonstrated capable of solving ordinary differential equations. Here, we aim to extend the method to effectively address a wider range of equations, such as the Poisson equation and the heat equation. To achieve this goal, we introduce an architecture specifically designed to compute second-order (and higher-order) derivatives without relying on nested automatic differentiation methods. This approach mitigates the unwanted side effects associated with nested gradients in simulations, paving the way for more efficient and accurate implementations. By leveraging such an approach, the quantum circuit addresses partial differential equations -- a challenge not yet tackled using this approach on continuous-variable quantum computers. As a proof-of-concept, we solve a one-dimensional instance of the heat equation, demonstrating its effectiveness in handling PDEs. Such a framework paves the way for further developments in continuous-variable quantum computing and underscores its potential contributions to advancing quantum machine learning.

Related papers

• 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)
• Evaluation of phase shifts for non-relativistic elastic scattering using quantum computers [39.58317527488534]
  This work reports the development of an algorithm that makes it possible to obtain phase shifts for generic non-relativistic elastic scattering processes on a quantum computer.
  arXiv  Detail & Related papers  (2024-07-04T21:11:05Z)
• Parallel Quantum Computing Simulations via Quantum Accelerator Platform Virtualization [44.99833362998488]
  We present a model for parallelizing simulation of quantum circuit executions. The model can take advantage of its backend-agnostic features, enabling parallel quantum circuit execution over any target backend.
  arXiv  Detail & Related papers  (2024-06-05T17:16:07Z)
• A multiple-circuit approach to quantum resource reduction with application to the quantum lattice Boltzmann method [39.671915199737846]
  We introduce a multiple-circuit algorithm for a quantum lattice Boltzmann method (QLBM) solve of the incompressible Navier--Stokes equations.<n>The presented method is validated and demonstrated for 2D lid-driven cavity flow.
  arXiv  Detail & Related papers  (2024-01-20T15:32:01Z)
• Self-Adaptive Physics-Informed Quantum Machine Learning for Solving Differential Equations [0.0]
  Chebyshevs have shown significant promise as an efficient tool for both classical and quantum neural networks to solve differential equations.<n>We adapt and generalize this framework in a quantum machine learning setting for a variety of problems.<n>The results indicate a promising approach to the near-term evaluation of differential equations on quantum devices.
  arXiv  Detail & Related papers  (2023-12-14T18:46:35Z)
• Efficient estimation of trainability for variational quantum circuits [43.028111013960206]
  We find an efficient method to compute the cost function and its variance for a wide class of variational quantum circuits. This method can be used to certify trainability for variational quantum circuits and explore design strategies that can overcome the barren plateau problem.
  arXiv  Detail & Related papers  (2023-02-09T14:05:18Z)
• Quantum Kernel Methods for Solving Differential Equations [21.24186888129542]
  We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented.
  arXiv  Detail & Related papers  (2022-03-16T18:56:35Z)
• Quantum Model-Discovery [19.90246111091863]
  Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks. Our results show a promising path to Quantum Model Discovery (QMoD) on the interface between classical and quantum machine learning approaches.
  arXiv  Detail & Related papers  (2021-11-11T18:45:52Z)
• Representation Learning via Quantum Neural Tangent Kernels [10.168123455922249]
  Variational quantum circuits are used in quantum machine learning and variational quantum simulation tasks. Here we discuss these problems, analyzing variational quantum circuits using the theory of neural tangent kernels. We analytically solve the dynamics in the frozen limit, or lazy training regime, where variational angles change slowly and a linear perturbation is good enough.
  arXiv  Detail & Related papers  (2021-11-08T01:30:34Z)
• 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)
• Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems [40.20472268839781]
  We generalize the reaction-diffusion equation in statistical physics, Schr"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics. We take finite difference method to discretize NPDE for finding numerical solution. Basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated.
  arXiv  Detail & Related papers  (2021-03-10T00:05:46Z)
• Solving nonlinear differential equations with differentiable quantum circuits [21.24186888129542]
  We propose a quantum algorithm to solve systems of nonlinear differential equations. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits. We show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space.
  arXiv  Detail & Related papers  (2020-11-20T13:21:11Z)

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.