Boltzmann sampling for an XY model using a non-degenerate optical parametric oscillator network

We present an experimental scheme of implementing multiple spins in a classical XY model using a non-degenerate optical parametric oscillator (NOPO) network. We built an NOPO network to simulate a one-dimensional XY Hamiltonian with 5000 spins and externally controllable effective temperatures. The XY spin variables in our scheme are mapped onto the phases of multiple NOPO pulses in a single ring cavity and interactions between XY spins are implemented by mutual injections between NOPOs. We show the steady-state distribution of optical phases of such NOPO pulses is equivalent to the Boltzmann distribution of the corresponding XY model. Estimated effective temperatures converged to the setting values, and the estimated temperatures and the mean energy exhibited good agreement with the numerical simulations of the Langevin dynamics of NOPO phases.

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

[2]  Ken-ichi Kawarabayashi,et al.  A coherent Ising machine for 2000-node optimization problems , 2016, Science.

[3]  M. W. Johnson,et al.  Quantum annealing with manufactured spins , 2011, Nature.

[4]  Yoshihisa Yamamoto,et al.  Mapping of Ising models onto injection-locked laser systems. , 2011, Optics express.

[5]  Kanti V. Mardia,et al.  Bayesian Methods in Structural Bioinformatics , 2012 .

[6]  K. Hukushima,et al.  Exchange Monte Carlo Method and Application to Spin Glass Simulations , 1995, cond-mat/9512035.

[7]  Kazuyuki Aihara,et al.  A fully programmable 100-spin coherent Ising machine with all-to-all connections , 2016, Science.

[8]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[9]  K. Mardia,et al.  Protein Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data , 2007, Biometrics.

[10]  C. Langmead,et al.  The von Mises Graphical Model: Structure Learning (CMU-CS-11-108 / CMU-CB-11-100) , 2011 .

[11]  Nir Davidson,et al.  Observing geometric frustration with thousands of coupled lasers. , 2013, Physical review letters.

[12]  R. Spigler,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[13]  Pavlos G. Lagoudakis,et al.  Realizing the classical XY Hamiltonian in polariton simulators. , 2016, Nature materials.

[14]  Yukito Iba.  EXTENDED ENSEMBLE MONTE CARLO , 2001 .

[15]  Kanti V. Mardia,et al.  A multivariate von mises distribution with applications to bioinformatics , 2008 .

[16]  Yoshua Bengio,et al.  On the Challenges of Physical Implementations of RBMs , 2013, AAAI.

[17]  Geoffrey E. Hinton,et al.  Reinforcement Learning with Factored States and Actions , 2004, J. Mach. Learn. Res..

[18]  Arecchi,et al.  Theory of phase locking of globally coupled laser arrays. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[19]  Anna Levit,et al.  Free Energy–Based Reinforcement Learning Using Quantum Monte Carlo and Quantum Annealing , 2018 .

[20]  Yoshihisa Yamamoto,et al.  Large-scale Ising spin network based on degenerate optical parametric oscillators , 2016, Nature Photonics.

[21]  Pierre Baldi,et al.  Computing with Arrays of Coupled Oscillators: An Application to Preattentive Texture Discrimination , 1990, Neural Computation.

[22]  D. Mattis.  Transfer matrix in plane-rotator model , 1984 .

[23]  Yoshihisa Yamamoto,et al.  Boltzmann Sampling by Degenerate Optical Parametric Oscillator Network for Structure-Based Virtual Screening , 2016, Entropy.

[24]  H. Mabuchi,et al.  Topological defect formation in 1D and 2D spin chains realized by network of optical parametric oscillators , 2016, 1605.08121.

[25]  Thomas Erneux,et al.  Stability conditions for coupled lasers: series coupling versus parallel coupling , 1993 .

[26]  Shuhei Tamate,et al.  Simulating the classical XY model with a laser network , 2016, 1608.00358.

[27]  Mandel,et al.  Global coupling with time delay in an array of semiconductor lasers , 2000, Physical review letters.

[28]  Richard S. Zemel,et al.  Lending direction to neural networks , 1995, Neural Networks.

[29]  Ryan Hamerly,et al.  Large-scale artificial spin network based on time-multiplexed degenerate optical parametric oscillators for coherent Ising machine , 2017, LASE.