Rapid fair sampling of the XY spin Hamiltonian with a laser simulator

Vishwa Pal,Nir Davidson,Simon Mahler,Chene Tradonsky,Asher A Friesem

doi:10.1103/physrevresearch.2.033008

Vishwa Pal, Nir Davidson + Show 3 more

Open Access

https://doi.org/10.1103/physrevresearch.2.033008

Copy DOI

Abstract

Coupled oscillators such as lasers, OPO's and BEC polaritons can rapidly and efficiently dissipate into a stable phase locked state that can be mapped onto the minimal energy (ground state) of classical spin Hamiltonians. However, for degenerate or near degenerate ground state manifolds, statistical fair sampling is required to obtain a complete knowledge of the minimal energy state, which needs many repetitions of simulations under identical conditions. We show that with dissipatively coupled lasers such fair sampling can be achieved rapidly and accurately by exploiting the many longitudinal modes of each laser to form an ensemble of identical but independent simulators, acting in parallel. We fairly sampled the ground state manifold of square, triangular and Kagome lattices by measuring their coherence function identifying manifolds composed of a single, doubly degenerate, and highly degenerate ground states, respectively.

Highlights

Various combinatorial optimization problems that occur, for example, in social networks, neural networks, management of large data sets, artificial intelligence, spin glass, drug discovery, protein folding, and traveling salesmen, are considered to be computationally hard problems [1,2]
2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel (Received 26 December 2019; accepted 11 June 2020; published 2 July 2020). Coupled oscillators such as lasers, optical parametric oscillators, and Bose-Einstein-condensate polaritons can rapidly and efficiently dissipate into a stable phase-locked state that can be mapped onto the minimal energy of classical spin Hamiltonians
Such optimization problems can be mapped into classical spin systems (Ising or XY Hamiltonian), where they are reduced to finding the global minimum of the spin Hamiltonian [3,4,5,6]

Summary

INTRODUCTION

Various combinatorial optimization problems that occur, for example, in social networks, neural networks, management of large data sets, artificial intelligence, spin glass, drug discovery, protein folding, and traveling salesmen, are considered to be computationally hard problems [1,2]. Each laser has approximately 250 longitudinal modes that form an ensemble of approximately 250 identical but independent simulators of the XY spin Hamiltonian This provides a massive parallelism that enables rapid and accurate fair sampling of the ground-state manifold. For a triangular lattice with negative coupling, we measure an oscillatory coherence function, indicating a doubly degenerate ground state, which could not be directly inferred from the far-field diffraction pattern. When the internal phases of each laser are varied, the coherence function is modified into a directional coherence function, corresponding to new stripe phase ordering All these experimental results agree with a theoretical model based on statistical fair sampling of the ground-state manifold. We observe and explain an intriguing ensemble average difference of π between the phases of nearest-neighbor lasers that are negatively coupled, even in triangular lattices

EXPERIMENTAL ARRANGEMENT AND PROCEDURES

FAIR SAMPLING OF THE GROUND-STATE MANIFOLD

CONCLUDING REMARKS

Numerical simulations

Findings

Analytical results