Functional Tensor Network Solving Many-body Schr\"odinger Equation

TL;DR

This paper introduces the functional tensor network (FTN) method for solving the many-body Schrödinger equation in continuous space, achieving high accuracy and efficiency, and extending tensor network applications beyond lattice models.

Contribution

The paper proposes a novel FTN approach using tensor networks with functional bases to solve many-body quantum problems in continuous space, including strong correlations.

Findings

Accurate ground state solutions for coupled harmonic oscillators.

Effective handling of three-body interactions with high precision.

Linear complexity scaling with system size for MPS-based FTN.

Abstract

Schr\"odinger equation belongs to the most fundamental differential equations in quantum physics. However, the exact solutions are extremely rare, and many analytical methods are applicable only to the cases with small perturbations or weak correlations. Solving the many-body Schr\"odinger equation in the continuous spaces with the presence of strong correlations is an extremely important and challenging issue. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schr\"odinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave-function as tensor network. The observables, such as energy, can be calculated simply by tensor contractions. Simulating the ground state becomes solving a minimization problem defined by the tensor network. An efficient gradient-decent algorithm based on the automatically differentiable tensors is proposed. We here take matrix product state (MPS) as an example, whose complexity scales only linearly with the system size. We apply our approach to solve the ground state of coupled harmonic oscillators, and achieve high accuracy by comparing with the exact solutions. Reliable results are also given with the presence of three-body interactions, where the system cannot be decoupled to isolated oscillators. Our approach is simple and with well-controlled error, superior to the highly-nonlinear neural-network solvers. Our work extends the applications of tensor network from quantum lattice models to the systems in the continuous space. FTN can be used as a general solver of the differential equations with many variables. The MPS exemplified here can be generalized to, e.g., the fermionic tensor networks, to solve the electronic Schr\"odinger equation.