Continuous-variable neural-network quantum states and the quantum rotor model

We initiate the study of neural-network quantum state algorithms for analyzing continuousvariable lattice quantum systems in first quantization. A simple family of continuous-variable trial wavefunctons is introduced which naturally generalizes the restricted Boltzmann machine (RBM) wavefunction introduced for analyzing quantum spin systems. By virtue of its simplicity, the same variational Monte Carlo training algorithms that have been developed for ground state determination and time evolution of spin systems have natural analogues in the continuum. We offer a proof of principle demonstration in the context of ground state determination of a stoquastic quantum rotor Hamiltonian. Results are compared against those obtained from partial differential equation (PDE) based scalable eigensolvers. This study serves as a benchmark against which future investigation of continuous-variable neural quantum states can be compared, and points to the need to consider deep network architectures and more sophisticated training algorithms.

[1]  L. Reatto,et al.  The Ground State of Liquid He(4) , 1969 .

[2]  J. Glimm,et al.  Quantum Physics: A Functional Integral Point of View , 1981 .

[3]  M. Gutzwiller.  The Geometry of Quantum Chaos , 1985 .

[4]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[5]  White,et al.  Density-matrix algorithms for quantum renormalization groups. , 1993, Physical review. B, Condensed matter.

[6]  Martin J. Mohlenkamp,et al.  Numerical operator calculus in higher dimensions , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  Martin J. Mohlenkamp,et al.  Algorithms for Numerical Analysis in High Dimensions , 2005, SIAM J. Sci. Comput..

[9]  Tamara G. Kolda,et al.  An overview of the Trilinos project , 2005, TOMS.

[10]  J. Latorre,et al.  Matrix product states algorithms and continuous systems , 2006, cond-mat/0610530.

[11]  Ivan Oseledets,et al.  Tensor-Train Decomposition , 2011, SIAM J. Sci. Comput..

[12]  Nonlinear sigma models with compact hyperbolic target spaces , 2015, 1510.02129.

[13]  I. Oseledets,et al.  Calculating vibrational spectra of molecules using tensor train decomposition. , 2016, The Journal of chemical physics.

[14]  Pierre Vandergheynst,et al.  Geometric Deep Learning: Going beyond Euclidean data , 2016, IEEE Signal Process. Mag..

[15]  Dong-Ling Deng,et al.  Machine Learning Topological States , 2016, 1609.09060.

[16]  Alexander Veit,et al.  Using the Tensor-Train Approach to Solve the Ground-State Eigenproblem for Hydrogen Molecules , 2017, SIAM J. Sci. Comput..

[17]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[18]  Yusuke Nomura,et al.  Constructing exact representations of quantum many-body systems with deep neural networks , 2018, Nature Communications.

[19]  J. Chen,et al.  Equivalence of restricted Boltzmann machines and tensor network states , 2017, 1701.04831.

[20]  B. Swingle,et al.  Chaos in a quantum rotor model , 2019, 1901.10446.

[21]  Alicia J. Kollár,et al.  Hyperbolic lattices in circuit quantum electrodynamics , 2018, Nature.

[22]  Alicia J. Kollár,et al.  Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry. , 2019, Physical review. A.

[23]  Joan Bruna,et al.  Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges , 2021, ArXiv.

[24]  Yichen Huang,et al.  Neural Network Representation of Tensor Network and Chiral States. , 2017, Physical review letters.

[25]  Stephen R Clark,et al.  Compact neural-network quantum state representations of Jastrow and stabilizer states , 2021, 2103.09146.

[26]  A. Roggero,et al.  Exact representations of many-body interactions with restricted-Boltzmann-machine neural networks. , 2020, Physical review. E.

[27]  N. P. Breuckmann,et al.  Quantum phase transitions of interacting bosons on hyperbolic lattices , 2021, Journal of physics. Condensed matter : an Institute of Physics journal.