Global optimization of spin Hamiltonians with gain-dissipative systems

Recently, several platforms were proposed and demonstrated a proof-of-principle for finding the global minimum of the spin Hamiltonians such as the Ising and XY models using gain-dissipative quantum and classical systems. The implementation of dynamical adjustment of the gain and coupling strengths has been established as a vital feedback mechanism for analog Hamiltonian physical systems that aim to simulate spin Hamiltonians. Based on the principle of operation of such simulators we develop a novel class of gain-dissipative algorithms for global optimisation of NP-hard problems and show its performance in comparison with the classical global optimisation algorithms. These systems can be used to study the ground state and statistical properties of spin systems and as a direct benchmark for the performance testing of the gain-dissipative physical simulators. Our theoretical and numerical estimations suggest that for large problem sizes the analog simulator when built might outperform the classical computer computations by several orders of magnitude under certain assumptions about the simulator operation.

[1]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2011, SIAM Rev..

[2]  A. F. Adams,et al.  The Survey , 2021, Dyslexia in Higher Education.

[3]  Robert W. Harrison,et al.  Phase problem in crystallography , 1993 .

[4]  Yonina C. Eldar,et al.  Efficient phase retrieval of sparse signals , 2012, 2012 IEEE 27th Convention of Electrical and Electronics Engineers in Israel.

[5]  O. Bunk,et al.  Ptychographic X-ray computed tomography at the nanoscale , 2010, Nature.

[6]  J. Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[7]  Frank Vewinger,et al.  Variable potentials for thermalized light and coupled condensates , 2017, Nature Photonics.

[8]  N. Berloff,et al.  Blockchain platform with proof-of-work based on analog Hamiltonian optimisers , 2018, 1802.10091.

[9]  Shuzhong Zhang,et al.  Complex Quadratic Optimization and Semidefinite Programming , 2006, SIAM J. Optim..

[10]  Natalia G. Berloff,et al.  Networks of non-equilibrium condensates for global optimization , 2018, New Journal of Physics.

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

[12]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[13]  Andrew Lucas,et al.  Ising formulations of many NP problems , 2013, Front. Physics.

[14]  H. Nishimori.  Statistical Physics of Spin Glasses and Information Processing , 2001 .

[15]  A. Walther.  The Question of Phase Retrieval in Optics , 1963 .

[16]  R. Byer,et al.  Network of time-multiplexed optical parametric oscillators as a coherent Ising machine , 2014, Nature Photonics.

[17]  N. Berloff,et al.  Scenario of strongly nonequilibrated Bose-Einstein condensation , 2002 .

[18]  Scott Aaronson,et al.  NP-complete Problems and Physical Reality , 2005, Electron. Colloquium Comput. Complex..

[19]  Iain Dunning,et al.  What Works Best When? A Systematic Evaluation of Heuristics for Max-Cut and QUBO , 2018, INFORMS J. Comput..

[20]  Michael Chertkov,et al.  Optimal structure and parameter learning of Ising models , 2016, Science Advances.

[21]  J. Miao,et al.  Extending X-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes. , 2008, Annual review of physical chemistry.

[22]  西森 秀稔,et al.  Elements of Phase Transitions and Critical Phenomena , 2011 .

[23]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[24]  O. Bunk,et al.  Diffractive imaging for periodic samples: retrieving one-dimensional concentration profiles across microfluidic channels. , 2007, Acta crystallographica. Section A, Foundations of crystallography.

[25]  Hiroki Takesue,et al.  Boltzmann sampling for an XY model using a non-degenerate optical parametric oscillator network , 2018 .

[26]  Fred W. Glover,et al.  The unconstrained binary quadratic programming problem: a survey , 2014, Journal of Combinatorial Optimization.

[27]  Juliane Jung,et al.  The Traveling Salesman Problem: A Computational Study , 2007 .

[28]  An electrically pumped polaritonic lattice simulator , 2018 .

[29]  Terence Tao,et al.  Random matrices have simple spectrum , 2014, Comb..

[30]  Scott Aaronson,et al.  Guest Column: NP-complete problems and physical reality , 2005, SIGA.

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

[32]  Yonina C. Eldar,et al.  GESPAR: Efficient Phase Retrieval of Sparse Signals , 2013, IEEE Transactions on Signal Processing.

[33]  Mingjie Sun,et al.  Adaptive foveated single-pixel imaging with dynamic supersampling , 2016, Science Advances.

[34]  Fred W. Glover,et al.  Probabilistic GRASP-Tabu Search algorithms for the UBQP problem , 2013, Comput. Oper. Res..

[35]  van Aernout Enter.  Statistical Mechanics, A Short Treatise , 2000 .

[36]  西森 秀稔.  Statistical physics of spin glasses and information processing : an introduction , 2001 .

[37]  Raúl Rojas,et al.  Neural Networks - A Systematic Introduction , 1996 .

[38]  Jin-Kao Hao,et al.  Breakout Local Search for the Max-Cutproblem , 2013, Eng. Appl. Artif. Intell..

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

[40]  J. Doye,et al.  Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms , 1997, cond-mat/9803344.

[41]  J. Ambjorn,et al.  Shaken, but not stirred—Potts model coupled to quantum gravity , 2008, 0806.3506.

[42]  M. Wadati,et al.  Exactly Solvable Models in Statistical Mechanics , 1988 .

[43]  Boulevard Lavoisier,et al.  Breakout Local Search for the Max-Cut Problem , 2014 .

[44]  P. Wolynes,et al.  Spin glasses and the statistical mechanics of protein folding. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[45]  Jorge Nocedal,et al.  Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization , 1997, TOMS.