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arXiv:1805.01371 · New Journal of Physics

Networks of non-equilibrium condensates for global optimization

Kirill P. Kalinin, Natalia G. Berloff

Published Oct 16, 2018• 51 citations
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Abstract

Recently several gain-dissipative platforms based on the networks of optical parametric oscillators, lasers and various non-equilibrium Bose–Einstein condensates have been proposed and realised as analogue Hamiltonian simulators for solving large-scale hard optimisation problems. However, in these realisations the parameters of the problem depend on the node occupancies that are not known a priori , which limits the applicability of the gain-dissipative simulators to the classes of problems easily solvable by classical computations. We show how to overcome this difficulty and formulate the principles of operation of such simulators for solving the NP-hard large-scale optimisation problems such as constant modulus continuous quadratic optimisation and quadratic binary optimisation for any general matrix. To solve such problems any gain-dissipative simulator has to implement a feedback mechanism for the dynamical adjustment of the gain and coupling strengths.

DOI
10.1088/1367-2630/aae8ae

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Recently several gain-dissipative platforms based on the networks of optical parametric oscillators, lasers and various non-equilibrium Bose–Einstein condensates have been proposed and realised as analogue Hamiltonian simulators for solving large-scale hard optimisation problems. However, in these realisations the parameters of the problem depend on the node occupancies that are not known a priori , which limits the applicability of the gain-dissipative simulators to the classes of problems easily solvable by classical computations. We show how to overcome this difficulty and formulate the principles of operation of such simulators for solving the NP-hard large-scale optimisation problems such as constant modulus continuous quadratic optimisation and quadratic binary optimisation for any general matrix. To solve such problems any gain-dissipative simulator has to implement a feedback mechanism for the dynamical adjustment of the gain and coupling strengths.
Recently several gain-dissipative platforms based on the network of optical parametric oscillators, lasers and various non-equilibrium Bose-Einstein condensates have been proposed as analog Hamiltonian simulators for solving large-scale hard optimisation problems. The parameters of such problems depend on the node occupancies that are not a priory known, which limits the applicability of the gain-dissipative simulators to the classes of problems easily solvable by classical computations. We show how to overcome this difficulty and formulate algorithms for solving the NP-hard large-scale optimisation problems such as constant modulus continuous quadratic optimisation for any general matrix. We show that to solve such problems any gain-dissipative simulator has to implement a feedback mechanism for the dynamical adjustment of the gain, so that occupancy of each node is the same. Based on the principle of operation of such simulators we propose a novel class of classical gain-dissipative algorithms and show its advantage in comparison with classical algorithms. The estimates of the time operation of the physical implementation of the gain-dissipative simulators for large matrices show the speed-up of the several orders of magnitude in comparison with classical computations.
Gain-dissipative platforms consisting of lasers, optical parametric oscillators and nonequilibrium condensates operating at the condensation or coherence threshold have been recently proposed as efficient analog simulators of the two-local spin Hamiltonians with continuous or discrete degrees of freedom. We show that nonequilibrium condensates above the threshold arranged in an interacting network may realize k-local Hamiltonians with k>2 and lead to nontrivial phase configurations. Similarly, many gain-dissipative systems that can be manipulated by optical means can bring about the ground state of the k-local Hamiltonians and solve higher-order binary optimization problems. We show how to facilitate the search for the global solution by invoking complex couplings in the system and demonstrate the efficiency of the method on the sets of complex problems. This approach offers a highly flexible new kind of computation based on gain-dissipative simulators with complex coupling switching.
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.
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.
Classical spin models with discrete or continuous degrees of freedom arise in many studies of complex physical systems. A wide class of hard real-life optimization problems can be formulated as a minimization of a spin Hamiltonian. Here we show how to simulate the discrete Ising and n-state planar Potts models with or without external fields using the physical gain-dissipative platforms with continuous phases, such as lasers and various nonequilibrium Bose-Einstein condensates. The underlying operational principle originates from a combination of resonant and nonresonant pumping. Our results lay grounds for the physical simulations of a broad range of Hamiltonians with complex interactions that can vary in time and space and with combined symmetries.
Photon Bose-Einstein condensates (pBECs) are a platform of interest for both the study of fundamental phenomena in many-body physics and the implementation of next-generation analog simulators for solving optimization problems. In our physical system, the network of coupled pBECs in a microcavity setup is created by individually laser-pumping trap sites that are formed by permanent nanostructuring and connected via waveguides to their nearest neighbours. Due to the roughness and non-flat landscape of the microcavity mirrors, trap sites are excited at different frequencies, making the relative phases undefined. This is crucial for finding the ground state of a network of pBECs - a problem that is NP-complete, which makes this platform a promising candidate for analog optimization problem solver. We propose an approach for in-situ coherence tuning using a permanent nanostructuring technique with active feedback on excited trap site frequencies and report our preliminary results on the creation of an uncoupled coherent network of pBECs using this approach.