Integrating Neural Networks with a Quantum Simulator for State Reconstruction.

We demonstrate quantum many-body state reconstruction from experimental data generated by a programmable quantum simulator by means of a neural-network model incorporating known experimental errors. Specifically, we extract restricted Boltzmann machine wave functions from data produced by a Rydberg quantum simulator with eight and nine atoms in a single measurement basis and apply a novel regularization technique to mitigate the effects of measurement errors in the training data. Reconstructions of modest complexity are able to capture one- and two-body observables not accessible to experimentalists, as well as more sophisticated observables such as the Rényi mutual information. Our results open the door to integration of machine learning architectures with intermediate-scale quantum hardware.

[1]  S. Ravets,et al.  Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models , 2016, Nature.

[2]  Christoph Becker,et al.  Identifying quantum phase transitions using artificial neural networks on experimental data , 2018, Nature Physics.

[3]  Norbert M. Linke,et al.  Machine learning assisted readout of trapped-ion qubits , 2018, Journal of Physics B: Atomic, Molecular and Optical Physics.

[4]  D. Gross,et al.  Efficient quantum state tomography. , 2010, Nature communications.

[5]  D. Deng,et al.  Quantum Entanglement in Neural Network States , 2017, 1701.04844.

[6]  I. Bloch,et al.  Crystallization in Ising quantum magnets , 2015, Science.

[7]  Michael Knap,et al.  Classifying snapshots of the doped Hubbard model with machine learning , 2018, Nature Physics.

[8]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  M. Bukov,et al.  QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains , 2016, 1610.03042.

[10]  Antoine Browaeys,et al.  Synthetic three-dimensional atomic structures assembled atom by atom , 2017, Nature.

[11]  O. Landon-Cardinal,et al.  Practical variational tomography for critical one-dimensional systems , 2014, 1412.0686.

[12]  R. van Bijnen,et al.  Adiabatic formation of Rydberg crystals with chirped laser pulses , 2011, 1103.2096.

[13]  M. Lukin,et al.  Probing many-body dynamics on a 51-atom quantum simulator , 2017, Nature.

[14]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

[15]  Markus Greiner,et al.  A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice , 2009, Nature.

[16]  Roger G. Melko,et al.  QuCumber: wavefunction reconstruction with neural networks , 2018, SciPost Physics.

[17]  M. Lukin,et al.  Dynamical crystallization in the dipole blockade of ultracold atoms. , 2009, Physical review letters.

[18]  Yee Whye Teh,et al.  A Fast Learning Algorithm for Deep Belief Nets , 2006, Neural Computation.

[19]  Roger G. Melko,et al.  Learnability scaling of quantum states: Restricted Boltzmann machines , 2019 .

[20]  Giacomo Torlai,et al.  Latent Space Purification via Neural Density Operators. , 2018, Physical review letters.

[21]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[22]  I. Bloch,et al.  Many-body interferometry of a Rydberg-dressed spin lattice , 2016, Nature Physics.

[23]  Roger G. Melko,et al.  Reconstructing quantum states with generative models , 2018, Nature Machine Intelligence.

[24]  D. Gross,et al.  Focus on quantum tomography , 2013 .

[25]  S. Sachdev.  Quantum Phase Transitions: A first course , 1999 .

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

[27]  O. Gühne,et al.  03 21 7 2 3 M ar 2 00 6 Scalable multi-particle entanglement of trapped ions , 2006 .

[28]  I. Bloch,et al.  Coherent many-body spin dynamics in a long-range interacting Ising chain , 2017, 1705.08372.

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

[30]  Tarun Grover,et al.  Entanglement and the sign structure of quantum states , 2014, 1412.3534.

[31]  Eric R. Anschuetz,et al.  Atom-by-atom assembly of defect-free one-dimensional cold atom arrays , 2016, Science.

[32]  Yi Zhang,et al.  Entanglement entropy of critical spin liquids. , 2011, Physical review letters.

[33]  Philipp Slusallek,et al.  Introduction to real-time ray tracing , 2005, SIGGRAPH Courses.

[35]  J. Cirac,et al.  Neural-Network Quantum States, String-Bond States, and Chiral Topological States , 2017, 1710.04045.

[36]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[37]  Sylvain Schwartz,et al.  High-Fidelity Control and Entanglement of Rydberg-Atom Qubits. , 2018, Physical review letters.

[38]  Matthew Rispoli,et al.  Quantum thermalization through entanglement in an isolated many-body system , 2016, Science.

[39]  J. Fiurášek,et al.  Quantum inference of states and processes , 2002, quant-ph/0210146.

[40]  Vinay V Ramasesh,et al.  Quantum-gas microscope for fermionic atoms. , 2015, Physical review letters.

[41]  David P. DiVincenzo,et al.  The complexity of stoquastic local Hamiltonian problems , 2006, Quantum Inf. Comput..

[42]  Roger G. Melko,et al.  Learning Thermodynamics with Boltzmann Machines , 2016, ArXiv.

[43]  Geoffrey E. Hinton,et al.  Reducing the Dimensionality of Data with Neural Networks , 2006, Science.

[44]  Franco Nori,et al.  QuTiP 2: A Python framework for the dynamics of open quantum systems , 2012, Comput. Phys. Commun..

[45]  King-Sun Fu,et al.  IEEE Transactions on Pattern Analysis and Machine Intelligence Publication Information , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[46]  M. Punk,et al.  Dislocation-mediated melting of one-dimensional Rydberg crystals , 2011, 1104.2603.

[47]  M. Lavagna.  Quantum Phase Transitions , 2001, cond-mat/0102119.

[48]  Immanuel Bloch,et al.  Single-spin addressing in an atomic Mott insulator , 2011, Nature.

[49]  Matthew B Hastings,et al.  Measuring Renyi entanglement entropy in quantum Monte Carlo simulations. , 2010, Physical review letters.

[50]  Nicolas Le Roux,et al.  Representational Power of Restricted Boltzmann Machines and Deep Belief Networks , 2008, Neural Computation.

[51]  Peter T. Brown,et al.  Probing the Quench Dynamics of Antiferromagnetic Correlations in a 2D Quantum Ising Spin System , 2018, Physical Review X.

[52]  C. F. Roos,et al.  Efficient tomography of a quantum many-body system , 2016, Nature Physics.

[53]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[54]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[55]  D. Gross,et al.  Experimental quantum compressed sensing for a seven-qubit system , 2016, Nature Communications.

[56]  Geoffrey E. Hinton,et al.  A Learning Algorithm for Boltzmann Machines , 1985, Cogn. Sci..

[57]  C. Schwemmer,et al.  Permutationally invariant quantum tomography. , 2010, Physical review letters.

[58]  M. Rispoli,et al.  Measuring entanglement entropy in a quantum many-body system , 2015, Nature.

[59]  J. van Leeuwen,et al.  Neural Networks: Tricks of the Trade , 2002, Lecture Notes in Computer Science.

[60]  Geoffrey E. Hinton.  Training Products of Experts by Minimizing Contrastive Divergence , 2002, Neural Computation.

[61]  Peter Zoller,et al.  Probing Rényi entanglement entropy via randomized measurements , 2018, Science.

[62]  Matthias Troyer,et al.  Neural-network quantum state tomography , 2018 .

[63]  M. Greiner,et al.  Site-resolved imaging of a fermionic Mott insulator , 2015, Science.

[64]  H. Weimer,et al.  Two-stage melting in systems of strongly interacting Rydberg atoms. , 2010, Physical review letters.