Deep learning-based quantum algorithms for solving nonlinear partial differential equations

• URL: http://arxiv.org/abs/2305.02019v2
• Date: Sat, 27 May 2023 19:17:07 GMT
• Title: Deep learning-based quantum algorithms for solving nonlinear partial differential equations
• Authors: Lukas Mouton, Florentin Reiter, Ying Chen, Patrick Rebentrost
• Abstract summary: 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.
• Score: 3.312385039704987
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the potential for enhancing a classical deep learning-based method for solving high-dimensional nonlinear partial differential equations with suitable quantum subroutines. First, with near-term noisy intermediate-scale quantum computers in mind, we construct architectures employing variational quantum circuits and classical neural networks in conjunction. While the hybrid architectures show equal or worse performance than their fully classical counterparts in simulations, they may still be of use in very high-dimensional cases or if the problem is of a quantum mechanical nature. Next, we identify the bottlenecks imposed by Monte Carlo sampling and the training of the neural networks. We find that quantum-accelerated Monte Carlo methods offer the potential to speed up the estimation of the loss function. In addition, we identify and analyse the trade-offs when using quantum-accelerated Monte Carlo methods to estimate the gradients with different methods, including a recently developed backpropagation-free forward gradient method. Finally, we discuss the usage of a suitable quantum algorithm for accelerating the training of feed-forward neural networks. Hence, this work provides different avenues with the potential for polynomial speedups for deep learning-based methods for nonlinear partial differential equations.

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)
• 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)
• Towards Neural Variational Monte Carlo That Scales Linearly with System Size [67.09349921751341]
  Quantum many-body problems are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, and the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm.
  arXiv  Detail & Related papers  (2022-12-21T19:00:04Z)
• Analytic theory for the dynamics of wide quantum neural networks [7.636414695095235]
  We study the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. For random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system.
  arXiv  Detail & Related papers  (2022-03-30T23:24:06Z)
• Simulating the Mott transition on a noisy digital quantum computer via Cartan-based fast-forwarding circuits [62.73367618671969]
  Dynamical mean-field theory (DMFT) maps the local Green's function of the Hubbard model to that of the Anderson impurity model. Quantum and hybrid quantum-classical algorithms have been proposed to efficiently solve impurity models. This work presents the first computation of the Mott phase transition using noisy digital quantum hardware.
  arXiv  Detail & Related papers  (2021-12-10T17:32:15Z)
• 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)
• Variational Quantum Optimization with Multi-Basis Encodings [62.72309460291971]
  We introduce a new variational quantum algorithm that benefits from two innovations: multi-basis graph complexity and nonlinear activation functions. Our results in increased optimization performance, two increase in effective landscapes and a reduction in measurement progress.
  arXiv  Detail & Related papers  (2021-06-24T20:16:02Z)
• Autoregressive Transformer Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation [5.668795025564699]
  We present an approach for tackling open quantum system dynamics. We compactly represent quantum states with autoregressive transformer neural networks. Efficient algorithms have been developed to simulate the dynamics of the Liouvillian superoperator.
  arXiv  Detail & Related papers  (2020-09-11T18:00:00Z)
• Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices [0.0]
  We introduce a quantum solver of contracted eigenvalue equations, the quantum analogue of classical methods for the energies. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.
  arXiv  Detail & Related papers  (2020-04-23T18:35:26Z)

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.