Session R39: Machine Learning for Quantum Matter III

Frustrated magnets and fermions with Neural Network Quantum States

Neural-Network quantum states (NQS) have been recently proposed as a method to solve challenging interacting quantum problems. During this talk, I will discuss the application of NQS to two problems. First, we use a deep convolutional network to find the ground state of the frustrated J1-J2 model on the square lattice. While early representations of many-body quantum states in terms of neural networks are usually based on shallow architectures such as the restricted Boltzmann machine, the benefits of deeper architectures are emerging in the latest research. Here, we show that these deep convolutional NQS can achieve results that are competitive with other state of the art variational methods developed in the past decade. Second, we present an extension of NQS to model interacting fermionic problems. Borrowing techniques from quantum simulation, we directly map fermionic degrees of freedom to spin ones, and then use NQS to perform electronic structure calculations. On small test molecules, we achieve energies below chemical accuracy, and frequently improves upon coupled cluster methods.   

Learning the Ground State Wavefunction of Periodic Systems Using Recurrent Neural Networks

Modeling quantum many-body systems is enormously challenging due to the exponential scaling of Hilbert dimension with system size. Finding an Ansatz that efficiently compresses the wavefunction is key to simulating large systems. Here, we present an approach for simulating periodic quantum systems using long short term memory networks (LSTMs), whose recurrent structure is able to efficiently capture invariance to discrete translations in the bulk. We perform Variational Monte Carlo using an autoregressive Ansatz, where the probability amplitude associated with an electron having a particular quantum number is conditioned on the quantum numbers of the electrons behind it. These amplitudes and phases are iteratively generated by an LSTM, which is trained to minimize the energy using stochastic gradient descent. We show that such a formulation can be used to find the ground state of the 1D Hubbard and J1-J2 Heisenberg models for several hundred electrons. Furthermore, we can learn about the bulk by "growing" the sample; iteratively training the solution on progressively larger systems until the edge effects become negligible. We argue that such a scheme can be generalized more naturally to higher dimensions than Density Matrix Renormalization Group.   

Calculating Renyi Entropies with Neural Autoregressive Quantum States

Entanglement entropy is an essential metric for characterizing quantum many-body systems, but its numerical evaluation for neural network representations of quantum states has so far been inefficient and only demonstrated for the restricted Boltzmann machine architecture. We estimate generalized Renyi entropies S_n of an autoregressive neural quantum state using quantum Monte Carlo methods. A naive “direct sampling” approach performs well for small Renyi order n but fails for larger orders when benchmarked on a 1D Heisenberg model. We therefore propose an improved “conditional sampling” method exploiting the autoregressive structure of the network ansatz, which outperforms direct sampling in both 1D and 2D Heisenberg models. Conditional sampling facilities calculations of high-order Renyi entropies up to at least n > 30, which allows for a polynomial approximation of the von Neumann entropy as well as extraction of the largest eigenvalue of the reduced density matrix, and thus the single copy entanglement. By demonstrating good convergence even up to high Renyi order, our methods elucidate the potential of neural network quantum states in quantum Monte Carlo studies of entanglement entropy for many-body systems.   

Probabilistic Simulation of Quantum Circuits with the Transformer

In this work, we present an exact probabilistic formulation of quantum dynamics through positive value-operator measurements (POVM). In this formulation, unitary dynamics and quantum channels are represented by quasi-stochastic matrices acting on true probability distributions which specify the quantum state univocally. The probability distribution representation of the quantum state opens up the possibility of bridging the state-of-the-art techniques from machine learning into the simulation of quantum mechanics. Using the POVM formalism, we have developed a practical algorithm for the probabilistic simulation of quantum circuits with the Transformer, a powerful ansatz responsible for the most recent breakthroughs in the natural language processing research. The method is applied to state preparation of GHZ state and Linear Graph state up to 60 qubits, as well as variational quantum circuit preparation of the ground state of the Transverse Field Ising Model.   

Variational optimization in the AI era

The variational method has been a cornerstone approach to tackling the quantum many-body problem since the beginnings of quantum mechanics. Throughout this history, wave-functions have grown in number of parameters and generality. The eventual conclusion to this arc is to consider the variational space of all computer programs. Using tools and inspiration from AI, we have developed an approach to represent this class (computational graph states); a novel way to optimize tens of thousands of parameters within this space (supervised wave-function optimization); and multiple novel variational ansatz (neural net backflow, etc). We will describe these advancements and our effort to push forward, in the age of AI, the variational approach to the quantum many body problem.
  

Deep neural network solution of the electronic Schrödinger equation

The electronic Schrödinger equation describes fundamental properties of molecules and materials, but cannot be solved exactly for larger systems than a hydrogen atom. Quantum Monte Carlo is an apt approach for high-quality approximations, because its accuracy is limited in principle only by the flexibility of the used wave-function ansatz, but traditional trial wave functions are too rigid to take full advantage of this potential. Here, we greatly increase the flexibility of existing ansatzes by incorporating deep neural networks, which are known as superb universal function approximators. Our architecture, dubbed PauliNet, is built around the Hartree–Fock solution as a baseline, includes analytical cusp conditions, and uses the Jastrow-factor and backflow constructions as entry points for graph-convolutional neural networks, which ensure the exact permutational antisymmetry. We demonstrate that PauliNet outperforms comparable state-of-the-art trial wave functions on atoms, small molecules, and a strongly correlated model system. Our approach opens a new path towards highly accurate and systematically improvable electronic structure methods with explicit access to the corresponding wave function and hence a variety of electronic properties.   

Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks

Calculating analytic solutions to the Schrödinger equation is impossible except in a small number of special cases. Approximate solutions typically impose a fixed functional form on the wavefunction. Neural networks have shown impressive power as accurate practical function approximators¹ and have been recently used in bosonic² and lattice systems³. We show that deep neural networks can learn the ground state wavefunction of chemical systems given only the positions and charges of the nuclei using a combination of variational Monte Carlo (VMC) and optimisation methods from machine learning⁴. The neural network Ansatz, FermiNet, is compact yet flexible and gives more accurate energies than conventional Ansätze. We obtain ground state energies, ionisation potentials and electron affinities to within chemical accuracy on a variety of atoms and small molecules and outperform VMC using conventional Slater-Jastrow wavefunctions.

1. A. Krizhevsky, I. Sutskever, G.E. Hinton, NIPS'12, 1097-1105 (2012).
2. G. Carleo, M. Troyer, Science 356, 602-606 (2017).
3. D. Luo, B.K. Clark Phys. Rev. Lett. 122, 226401 (2019).
4. D. Pfau, J. S. Spencer, A. G. de G. Matthews, W. M. C. Foulkes, arXiv:1909.02487 (2019).

DP and JS contributed equally to this work.   

Towards neural network quantum states with nonabelian symmetries

Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum many-body wave functions, the parameterization of a quantum wavefunction with nonabelian symmetries in terms of a Boltzmann machine inherently leads to biased results due to the basis dependence. We demonstrate that this problem can be overcome by sampling in the basis of irreducible representations instead of spins, for which the corresponding ansatz respects the nonabelian symmetries of the system. We will show that this representation is connected to symmetric tensor network states.

We apply our methodology to find the ground states of the one-dimensional antiferromagnetic Heisenberg (AFH) model with spin-half and spin-1 degrees of freedom. The proposed ansatz can target excited states, which is illustrated by calculating the energy gap of the AFH model.

Implementing non-abelian symmetries in variational quantum states is an important step towards solving frustrated quantum systems. We demonstrate recent results using our ansatz on the frustrated J1-J2 model.   

Designing neural networks for stationary states in open quantum many-body systems

We propose a new variational scheme based on the neural-network quantum states to simulate the stationary states of open quantum many-body systems [1]. Using the high expressive power of the variational ansatz described by the restricted Boltzmann machines [2], which we dub as the neural stationary state ansatz, we compute the stationary states of quantum dynamics obeying the homogeneous Markovian quantum master equations. The mapping of the stationary-state search problem into finding a zero-energy ground state of an appropriate Hermitian operator allows us to apply the conventional variational Monte Carlo method for the optimization. Our method is shown to simulate various spin systems efficiently, i.e., the transverse-field Ising models in both one and two dimensions and the XYZ model in one dimension.

[1] N. Yoshioka and R. Hamazaki, Phys. Rev. B 99, 214306 (2019).
[2] G. Carleo and M. Troyer, Science 355, 602 (2017).   

Deep Learning-Enhanced Variational Monte Carlo Method for Quantum Many-Body Physics

Artificial neural networks have been successfully incorporated into variational Monte Carlo method (VMC) to study quantum many-body systems. However, there have been few systematic studies of exploring quantum many-body physics using deep neural networks (DNNs), despite of the tremendous success enjoyed by DNNs in many other areas in recent years. One main challenge of implementing DNN in VMC is the inefficiency of optimizing such networks with large number of parameters. We introduce an importance sampling gradient optimization (ISGO) algorithm (arXiv:1905.10730), which significantly improves the computational speed of training DNN in VMC. We design an efficient convolutional DNN architecture to compute the ground state of a one-dimensional (1D) SU(N) spin chain. Our numerical results of the ground-state energies with up to 16 layers of DNN show excellent agreement with the Bethe-Ansatz exact solution. Furthermore, we also calculate the loop correlation function using the wave function obtained. Our work demonstrates the feasibility and advantages of applying DNNs to numerical quantum many-body calculations.