Learning ground states of quantum Hamiltonians with graph networks

• URL: http://arxiv.org/abs/2110.06390v1
• Date: Tue, 12 Oct 2021 22:56:16 GMT
• Title: Learning ground states of quantum Hamiltonians with graph networks
• Authors: Dmitrii Kochkov and Tobias Pfaff and Alvaro Sanchez-Gonzalez and Peter Battaglia and Bryan K. Clark
• Abstract summary: Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. We use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations.
• Score: 6.024776891570197
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects.

Related papers

• Variational quantum algorithms for scanning the complex spectrum of non-Hermitian systems [0.0]
  We propose a variational method for solving the non-Hermitian Hamiltonian on a quantum computer. The energy is set as a parameter in the cost function and can be tuned to obtain the whole spectrum. Our work suggests an avenue for solving non-Hermitian quantum many-body systems with variational quantum algorithms on near-term noisy quantum computers.
  arXiv  Detail & Related papers  (2023-05-31T12:50:22Z)
• Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states [0.0]
  We consider energy problems for quantum many-body systems with extensive Hamiltonians and finite-range interactions. We prove that variational optimization problems for matrix product states, tree tensor networks, and the multiscale entanglement renormalization ansatz are free of barren plateaus.
  arXiv  Detail & Related papers  (2023-03-31T22:49:49Z)
• Physics-Informed Neural Networks for Quantum Eigenvalue Problems [1.2891210250935146]
  Eigenvalue problems are critical to several fields of science and engineering. We use unsupervised neural networks for discovering eigenfunctions and eigenvalues for differential eigenvalue problems. The network optimization is data-free and depends solely on the predictions of the neural network.
  arXiv  Detail & Related papers  (2022-02-24T18:29:39Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Variational Adiabatic Gauge Transformation on real quantum hardware for effective low-energy Hamiltonians and accurate diagonalization [68.8204255655161]
  We introduce the Variational Adiabatic Gauge Transformation (VAGT) VAGT is a non-perturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit. The accuracy of VAGT is tested trough numerical simulations, as well as simulations on Rigetti and IonQ quantum computers.
  arXiv  Detail & Related papers  (2021-11-16T20:50:08Z)
• 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)
• Physics informed neural networks for continuum micromechanics [68.8204255655161]
  Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong non-linear solutions by optimization. It is shown, that the domain decomposition approach is able to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world $mu$CT-scans.
  arXiv  Detail & Related papers  (2021-10-14T14:05:19Z)
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition [59.20417091220753]
  We present a constructive algorithm for generating quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.
  arXiv  Detail & Related papers  (2021-04-01T19:06:00Z)
• Variational Monte Carlo calculations of $\mathbf{A\leq 4}$ nuclei with an artificial neural-network correlator ansatz [62.997667081978825]
  We introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei. We compute the binding energies and point-nucleon densities of $Aleq 4$ nuclei as emerging from a leading-order pionless effective field theory Hamiltonian.
  arXiv  Detail & Related papers  (2020-07-28T14:52:28Z)
• Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies [0.0]
  We show that a new approach to optimization with constraints can be applied to complex quantum systems. The developed approach together with the provided open source software can be applied to the optimal control of noisy quantum systems.
  arXiv  Detail & Related papers  (2020-07-02T17:53:01Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.