Bulletin of the American Physical Society
APS March Meeting 2020
Volume 65, Number 1
Monday–Friday, March 2–6, 2020; Denver, Colorado
Session R39: Machine Learning for Quantum Matter IIIFocus

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Sponsoring Units: DCOMP GDS DMP Chair: Annabelle Bohrdt, Tech Univ Muenchen Room: 703 
Thursday, March 5, 2020 8:00AM  8:36AM 
R39.00001: Frustrated magnets and fermions with Neural Network Quantum States Invited Speaker: Kenny Jing Choo NeuralNetwork 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 J1J2 model on the square lattice. While early representations of manybody 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. 
Thursday, March 5, 2020 8:36AM  8:48AM 
R39.00002: Learning the Ground State Wavefunction of Periodic Systems Using Recurrent Neural Networks Christopher Roth, Allan MacDonald Modeling quantum manybody 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 J1J2 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. 
Thursday, March 5, 2020 8:48AM  9:00AM 
R39.00003: Calculating Renyi Entropies with Neural Autoregressive Quantum States Zhaoyou Wang, Emily J Davis Entanglement entropy is an essential metric for characterizing quantum manybody 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 highorder 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 manybody systems. 
Thursday, March 5, 2020 9:00AM  9:12AM 
R39.00004: Probabilistic Simulation of Quantum Circuits with the Transformer Juan Carrasquilla, Di Luo, Felipe Perez, Bryan Clark, Ashley Milsted, Maksims Volkovs, Mario Aolita In this work, we present an exact probabilistic formulation of quantum dynamics through positive valueoperator measurements (POVM). In this formulation, unitary dynamics and quantum channels are represented by quasistochastic 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 stateoftheart 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. 
Thursday, March 5, 2020 9:12AM  9:48AM 
R39.00005: Variational optimization in the AI era Invited Speaker: Bryan Clark The variational method has been a cornerstone approach to tackling the quantum manybody problem since the beginnings of quantum mechanics. Throughout this history, wavefunctions 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 wavefunction 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. 
Thursday, March 5, 2020 9:48AM  10:00AM 
R39.00006: Deep neural network solution of the electronic Schrödinger equation Jan Hermann, Zeno Schätzle, Frank Noe 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 highquality approximations, because its accuracy is limited in principle only by the flexibility of the used wavefunction 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 Jastrowfactor and backflow constructions as entry points for graphconvolutional neural networks, which ensure the exact permutational antisymmetry. We demonstrate that PauliNet outperforms comparable stateoftheart 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. 
Thursday, March 5, 2020 10:00AM  10:12AM 
R39.00007: AbInitio Solution of the ManyElectron Schrödinger Equation with Deep Neural Networks James Spencer, David Pfau, Alex Matthews, W Matthew C Foulkes 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 approximators^{1} and have been recently used in bosonic^{2} and lattice systems^{3}. 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 learning^{4}. 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 SlaterJastrow wavefunctions. 
Thursday, March 5, 2020 10:12AM  10:24AM 
R39.00008: Towards neural network quantum states with nonabelian symmetries Tom Vieijra, Corneel Casert, Jannes Nys, Wesley De Neve, Jutho Haegeman, Jan Ryckebusch, Frank Verstraete Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum manybody 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. 
Thursday, March 5, 2020 10:24AM  10:36AM 
R39.00009: Designing neural networks for stationary states in open quantum manybody systems Nobuyuki Yoshioka, Ryusuke Hamazaki We propose a new variational scheme based on the neuralnetwork quantum states to simulate the stationary states of open quantum manybody 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 stationarystate search problem into finding a zeroenergy 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 transversefield Ising models in both one and two dimensions and the XYZ model in one dimension. 
Thursday, March 5, 2020 10:36AM  10:48AM 
R39.00010: Deep LearningEnhanced Variational Monte Carlo Method for Quantum ManyBody Physics Li Yang, Zhaoqi Leng, Li Li, Ankit Patel, Wenjun Hu, Han Pu Artificial neural networks have been successfully incorporated into variational Monte Carlo method (VMC) to study quantum manybody systems. However, there have been few systematic studies of exploring quantum manybody 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 onedimensional (1D) SU(N) spin chain. Our numerical results of the groundstate energies with up to 16 layers of DNN show excellent agreement with the BetheAnsatz 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 manybody calculations. 
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