A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Schrodinger Equation

• URL: http://arxiv.org/abs/2406.00047v1
• Date: Tue, 28 May 2024 15:42:15 GMT
• Title: A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Schrodinger Equation
• Authors: Daniel Freedman, Eyal Rozenberg, Alex Bronstein,
• Abstract summary: A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. We propose a solution via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties.
• Score: 8.648660469053342
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature ("cusps") of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrodinger Equation.

Related papers

• Quantum simulation of the Fokker-Planck equation via Schrodingerization [33.76659022113328]
  This paper studies a quantum simulation technique for solving the Fokker-Planck equation. We employ the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations.
  arXiv  Detail & Related papers  (2024-04-21T08:53:27Z)
• Quantum tomography of helicity states for general scattering processes [65.268245109828]
  Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics. We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
  arXiv  Detail & Related papers  (2023-10-16T21:23:42Z)
• 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)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• A quantum-classical decomposition of Gaussian quantum environments: a stochastic pseudomode model [0.8258451067861933]
  We show that the effect of a Bosonic environment linearly coupled to a quantum system can be simulated by a Gaussian Lindblad master equation. For a subset of rational spectral densities, all parameters are explicitly specified without the need of any fitting procedure.
  arXiv  Detail & Related papers  (2023-01-18T14:17:17Z)
• Fermionic approach to variational quantum simulation of Kitaev spin models [50.92854230325576]
  Kitaev spin models are well known for being exactly solvable in a certain parameter regime via a mapping to free fermions. We use classical simulations to explore a novel variational ansatz that takes advantage of this fermionic representation. We also comment on the implications of our results for simulating non-Abelian anyons on quantum computers.
  arXiv  Detail & Related papers  (2022-04-11T18:00:01Z)
• 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)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Simulating Many-Body Systems with a Projective Quantum Eigensolver [0.0]
  We present a new hybrid quantum-classical algorithm for optimizing unitary coupled-cluster wave functions. We also introduce a selected variant of PQE (SPQE) that uses an adaptive ansatz built from arbitrary-order particle-hole operators.
  arXiv  Detail & Related papers  (2021-01-31T00:31:12Z)
• Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization [106.70006655990176]
  A distributional optimization problem arises widely in machine learning and statistics. We propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent. We prove that when the objective function satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly.
  arXiv  Detail & Related papers  (2020-12-21T18:33:13Z)
• Quantum annealing approach to Ionic Diffusion in Solid [0.0]
  Existing methods can only calculate the correlation factor analytically in the case of physically unrealistic models. We have mapped the problem into a quantum spin system described by the Ising Hamiltonian. We have calculated the correlation factor in a simple case with a known exact result by a variety of computational methods.
  arXiv  Detail & Related papers  (2019-12-31T10:06:40Z)

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.