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Journal ArticleDOI

An electromechanical Ising Hamiltonian

01 Jun 2016-Science Advances (American Association for the Advancement of Science)-Vol. 2, Iss: 6
TL;DR: These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling.
Abstract: Solving intractable mathematical problems in simulators composed of atoms, ions, photons, or electrons has recently emerged as a subject of intense interest. We extend this concept to phonons that are localized in spectrally pure resonances in an electromechanical system that enables their interactions to be exquisitely fashioned via electrical means. We harness this platform to emulate the Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from the parametric resonances of multiple modes. The coupling between the mechanical spins is created by generating two-mode squeezed states, which impart correlations between modes that can imitate a random, ferromagnetic state or an antiferromagnetic state on demand. These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling.
Citations
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Journal ArticleDOI
04 Nov 2016-Science
TL;DR: It is shown that an optical processing approach based on a network of coupled optical pulses in a ring fiber can be used to model and optimize large-scale Ising systems, and a coherent Ising machine outperformed simulated annealing in terms of accuracy and computation time for a 2000-node complete graph.
Abstract: The analysis and optimization of complex systems can be reduced to mathematical problems collectively known as combinatorial optimization. Many such problems can be mapped onto ground-state search problems of the Ising model, and various artificial spin systems are now emerging as promising approaches. However, physical Ising machines have suffered from limited numbers of spin-spin couplings because of implementations based on localized spins, resulting in severe scalability problems. We report a 2000-spin network with all-to-all spin-spin couplings. Using a measurement and feedback scheme, we coupled time-multiplexed degenerate optical parametric oscillators to implement maximum cut problems on arbitrary graph topologies with up to 2000 nodes. Our coherent Ising machine outperformed simulated annealing in terms of accuracy and computation time for a 2000-node complete graph.

555 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state.
Abstract: This paper draws attention to a hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state. The basic constituents are stochastic nanomagnets which switch randomly between the ±1 Ising states and can be monitored continuously with standard electronics. Their mutual interactions can be short or long range, and their strengths can be reconfigured as needed to solve specific problems and to anneal the system at room temperature. The natural laws of statistical mechanics guide the network of stochastic nanomagnets at GHz speeds through the collective states with an emphasis on the low energy states that represent optimal solutions. As proof-of-concept, we present simulation results for standard NP-complete examples including a 16-city traveling salesman problem using experimentally benchmarked models for spin-transfer torque driven stochastic nanomagnets.

173 citations

Journal ArticleDOI
TL;DR: In this article, a large-scale optical Ising machine with a spatial light modulator was designed and experimentally demonstrated, where the spin variables were encoded in a binary phase modulation of the field and the light propagation can be tailored to minimize an Ising Hamiltonian.
Abstract: Quantum and classical physics can be used for mathematical computations that are hard to tackle by conventional electronics. Very recently, optical Ising machines have been demonstrated for computing the minima of spin Hamiltonians, paving the way to new ultrafast hardware for machine learning. However, the proposed systems are either tricky to scale or involve a limited number of spins. We design and experimentally demonstrate a large-scale optical Ising machine based on a simple setup with a spatial light modulator. By encoding the spin variables in a binary phase modulation of the field, we show that light propagation can be tailored to minimize an Ising Hamiltonian with spin couplings set by input amplitude modulation and a feedback scheme. We realize configurations with thousands of spins that settle in the ground state in a low-temperature ferromagneticlike phase with all-to-all and tunable pairwise interactions. Our results open the route to classical and quantum photonic Ising machines that exploit light spatial degrees of freedom for parallel processing of a vast number of spins with programmable couplings.

166 citations

Journal ArticleDOI
TL;DR: A hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state is drawn attention.
Abstract: This paper draws attention to a hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state. The basic constituents are stochastic nanomagnets which switch randomly between the $\pm 1$ Ising states and can be monitored continuously with standard electronics. Their mutual interactions can be short or long range, and their strengths can be reconfigured as needed to solve specific problems and to anneal the system at room temperature. The natural laws of statistical mechanics guide the network of stochastic nanomagnets at GHz speeds through the collective states with an emphasis on the low energy states that represent optimal solutions. As proof-of-concept, we present simulation results for standard NP-complete examples including a 16-city traveling salesman problem using experimentally benchmarked models for spin-transfer torque driven stochastic nanomagnets.

138 citations


Cites background from "An electromechanical Ising Hamilton..."

  • ...Many diverse systems have been proposed to solve NP-hard optimization problems such as those based on simulated annealing [2], DNA[3, 4], quantum annealing [5, 6], Cellular Neural Networks [7–9], CMOS [10], trapped ions [11], electromechanics [12], optics [13–20], and magnets [21–23]....

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Book ChapterDOI
03 Jun 2019
TL;DR: In this paper, the phase dynamics of coupled self-sustaining nonlinear oscillators are shown to be governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph.
Abstract: We present a new way to make Ising machines, i.e., using networks of coupled self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of subharmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., adding noise, and turning subharmonic locking on and off smoothly) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We present working hardware prototypes using CMOS electronic oscillators.

110 citations

References
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Journal ArticleDOI
13 May 1983-Science
TL;DR: There is a deep and useful connection between statistical mechanics and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters), and a detailed analogy with annealing in solids provides a framework for optimization of very large and complex systems.
Abstract: There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.

41,772 citations

Journal ArticleDOI
TL;DR: The field of cavity optomechanics explores the interaction between electromagnetic radiation and nano-or micromechanical motion as mentioned in this paper, which explores the interactions between optical cavities and mechanical resonators.
Abstract: We review the field of cavity optomechanics, which explores the interaction between electromagnetic radiation and nano- or micromechanical motion This review covers the basics of optical cavities and mechanical resonators, their mutual optomechanical interaction mediated by the radiation pressure force, the large variety of experimental systems which exhibit this interaction, optical measurements of mechanical motion, dynamical backaction amplification and cooling, nonlinear dynamics, multimode optomechanics, and proposals for future cavity quantum optomechanics experiments In addition, we describe the perspectives for fundamental quantum physics and for possible applications of optomechanical devices

4,031 citations

Journal ArticleDOI
TL;DR: In this article, the consequences of back-action of light confined in whispering-gallery dielectric micro-cavities, and presents a unified treatment of its two manifestations: namely the parametric instability (mechanical amplification and oscillation) and radiation pressure backaction cooling.
Abstract: The coupling of mechanical and optical degrees of freedom via radiation pressure has been a subject of early research in the context of gravitational wave detection. Recent experimental advances have allowed studying for the first time the modifications of mechanical dynamics provided by radiation pressure. This paper reviews the consequences of back-action of light confined in whispering-gallery dielectric micro-cavities, and presents a unified treatment of its two manifestations: notably the parametric instability (mechanical amplification and oscillation) and radiation pressure back-action cooling. Parametric instability offers a novel “photonic clock” which is driven purely by the pressure of light. In contrast, radiation pressure cooling can surpass existing cryogenic technologies and offers cooling to phonon occupancies below unity and provides a route towards cavity Quantum Optomechanics

2,438 citations

Journal ArticleDOI
06 Oct 2011-Nature
TL;DR: In this article, a coupled, nanoscale optical and mechanical resonator formed in a silicon microchip is used to cool the mechanical motion down to its quantum ground state (reaching an average phonon occupancy number of 0.85±0.08).
Abstract: The simple mechanical oscillator, canonically consisting of a coupled mass–spring system, is used in a wide variety of sensitive measurements, including the detection of weak forces and small masses. On the one hand, a classical oscillator has a well-defined amplitude of motion; a quantum oscillator, on the other hand, has a lowest-energy state, or ground state, with a finite-amplitude uncertainty corresponding to zero-point motion. On the macroscopic scale of our everyday experience, owing to interactions with its highly fluctuating thermal environment a mechanical oscillator is filled with many energy quanta and its quantum nature is all but hidden. Recently, in experiments performed at temperatures of a few hundredths of a kelvin, engineered nanomechanical resonators coupled to electrical circuits have been measured to be oscillating in their quantum ground state. These experiments, in addition to providing a glimpse into the underlying quantum behaviour of mesoscopic systems consisting of billions of atoms, represent the initial steps towards the use of mechanical devices as tools for quantum metrology or as a means of coupling hybrid quantum systems. Here we report the development of a coupled, nanoscale optical and mechanical resonator formed in a silicon microchip, in which radiation pressure from a laser is used to cool the mechanical motion down to its quantum ground state (reaching an average phonon occupancy number of 0.85±0.08). This cooling is realized at an environmental temperature of 20 K, roughly one thousand times larger than in previous experiments and paves the way for optical control of mesoscale mechanical oscillators in the quantum regime.

2,073 citations

Journal ArticleDOI
TL;DR: The main theoretical and experimental aspects of quantum simulation have been discussed in this article, and some of the challenges and promises of this fast-growing field have also been highlighted in this review.
Abstract: Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, ie, quantum simulation Quantum simulation promises to have applications in the study of many problems in, eg, condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field

1,941 citations