In this paper, we introduce a novel approach to solve the many-body Schrodinger equation by the tensor neural network. Based on the tensor product structure, we can do the direct numerical integration by using fixed quadrature points for the functions constructed by the tensor neural network within tolerable computational complexity. Especially, we design several types of efficient numerical methods to treat the variable-coupled Coulomb potentials with high accuracy. The corresponding machine learning method is built for solving many-body Schrodinger equation. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.In this paper, we introduce a novel approach to solve the many-body Schrodinger equation by the tensor neural network. Based on the tensor product structure, we can do the direct numerical integration by using fixed quadrature points for the functions constructed by the tensor neural network within tolerable computational complexity. Especially, we design several types of efficient numerical methods to treat the variable-coupled Coulomb potentials with high accuracy. The corresponding machine learning method is built for solving many-body Schrodinger equation. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.

arXiv:2209.12572Sep 26, 20220 citations

Verifiably exact solution of the electronic Schrödinger equation on quantum devices

Scott E. Smart, David A. Mazziotti

Quantum computers have the potential for a significant speedup of molecular computations. However, existing algorithms have limitations; quantum phase estimation (QPE) is intractable on current hardware while variational quantum eigensolvers (VQE) are dependent upon approximate wave functions without guaranteed convergence. In this paper we present an algorithm that yields verifiably exact solutions of the many-electron Schr\"odinger equation. Rather than solve the Schr\"odinger equation directly, we solve its contraction over all electrons except two, known as the contracted Schr\"odinger equation (CSE). The CSE generates a wave-function Ansatz, constructed from an iterative product of nonunitary two-body transformations, whose energy gradient with respect to the two-body operator of the current iteration vanishes if and only if the CSE is satisfied. Because the CSE implies the Schr\"odinger equation, the two-electron Ansatz provides a verifiably exact Ansatz for solving the many-electron Schr\"odinger equation. The exactness property contrasts with that of Ans\"atze built from the product of unitary two-body transformations where the gradient---the residual of the anti-Hermitian part of the CSE (ACSE)---can vanish without implying a solution of the Schr\"odinger equation. We demonstrate the algorithm on both simulators and noisy quantum computers with ${\mathrm{H}}_{2}$ dissociation and the rectangle-to-square transition in ${\mathrm{H}}_{4}$ .Quantum computers have the potential for a significant speedup of molecular computations. However, existing algorithms have limitations; quantum phase estimation (QPE) is intractable on current hardware while variational quantum eigensolvers (VQE) are dependent upon approximate wave functions without guaranteed convergence. In this paper we present an algorithm that yields verifiably exact solutions of the many-electron Schr\"odinger equation. Rather than solve the Schr\"odinger equation directly, we solve its contraction over all electrons except two, known as the contracted Schr\"odinger equation (CSE). The CSE generates a wave-function Ansatz, constructed from an iterative product of nonunitary two-body transformations, whose energy gradient with respect to the two-body operator of the current iteration vanishes if and only if the CSE is satisfied. Because the CSE implies the Schr\"odinger equation, the two-electron Ansatz provides a verifiably exact Ansatz for solving the many-electron Schr\"odinger equation. The exactness property contrasts with that of Ans\"atze built from the product of unitary two-body transformations where the gradient---the residual of the anti-Hermitian part of the CSE (ACSE)---can vanish without implying a solution of the Schr\"odinger equation. We demonstrate the algorithm on both simulators and noisy quantum computers with ${\mathrm{H}}_{2}$ dissociation and the rectangle-to-square transition in ${\mathrm{H}}_{4}$.

arXiv:2303.00758Physical Review AFeb 5, 202418 citations

Many-Body Schrödinger Equation

Hiroshi Isozaki

Mathematical physics studiesJan 1, 20236 citations

Monte-Carlo solution to the many-body schrödinger equation

John G. Zabolitzky

Progress in Particle and Nuclear PhysicsJan 1, 19869 citations

{\mathrm{H}}_{2}

dissociation and the rectangle-to-square transition in

{\mathrm{H}}_{4}