Identifying variational wave functions that efficiently parametrize the
physically relevant states in the exponentially large Hilbert space is one of
the key tasks towards solving the quantum many-body problem. Powerful tools in
this context such as tensor network states have recently been complemented by
states derived from artificial neural networks (ANNs). Here, we propose and
investigate a new family of quantum states, coined generalized transfer matrix
states (GTMS), which bridges between the two mentioned approaches in the
framework of deep ANNs. In particular, we show by means of a constructive
embedding that the class of GTMS contains generic matrix product states while
at the same time being capable of capturing more long-ranged quantum
correlations that go beyond the area-law entanglement properties of tensor
networks. While the state amplitude of generic deep ANNs cannot be exactly
evaluated, that of a GTMS network can be analytically computed using transfer
matrix methods. With numerical simulations, we demonstrate how the GTMS network
learns random matrix product states in a supervised learning scheme, and how
augmenting the network by long-ranged couplings leads to the onset of
volume-law entanglement scaling. By means of an explicit example using
variational Monte Carlo, we also show that GTMS can parametrize critical
quantum many-body ground states to a good accuracy. Our findings suggest that
GTMS are a promising candidate for the study of critical and dynamical quantum
many-body systems.

提出并研究了一种名为广义转移矩阵状态（GTMS）的新型量子态，桥接了深度人工神经网络的两种方法，GTMS 包含了通用矩阵积态，并能够捕捉超越张量网络区域定律纠缠性质的更多长程量子关联。通过数值模拟和显式例子，表明 GTMS 是研究临界和动力学量子多体系统的有前途的候选方案。

从人工神经网络中推出广义传输矩阵状态

Generalized Transfer Matrix States from Artificial Neural Networks

Obtaining an accurate ground state wave function is one of the great
challenges in the quantum many-body problem. In this paper, we propose a new
class of wave functions, neural network backflow (NNB). The backflow approach,
pioneered originally by Feynman, adds correlation to a mean-field ground state
by transforming the single-particle orbitals in a configuration-dependent way.
NNB uses a feed-forward neural network to find the optimal transformation. NNB
directly dresses a mean-field state, can be systematically improved and
directly alters the sign structure of the wave-function. It generalizes the
standard backflow which we show how to explicitly represent as a NNB. We
benchmark the NNB on a Hubbard model at intermediate doping finding that it
significantly decreases the relative error, restores the symmetry of both
observables and single-particle orbitals, and decreases the double-occupancy
density. Finally, we illustrate interesting patterns in the weights and bias of
the optimized neural network.

本文提出了一种基于神经网络的波函数 —— 神经网络反流 (NNB)，并对 Hubbard 模型进行了测试，结果表明 NNB 显著降低了相对误差、恢复了观测值和单粒子轨道的对称性，降低了双占据密度。