Ising Energy Research Articles

Efficient sampling of ground and low-energy Ising spin configurations with a coherent Ising machine

We show that the nonlinear stochastic dynamics of a measurement-feedback-based coherent Ising machine (MFB-CIM) in the presence of quantum noise can be exploited to sample degenerate ground and low-energy spin configurations of the Ising model. We formulate a general discrete-time Gaussian-state model of the MFB-CIM which faithfully captures the nonlinear dynamics present at and above system threshold. This model overcomes the limitations of both mean-field models, which neglect quantum noise, and continuous-time models, which assume long photon lifetimes. Numerical simulations of our model show that when the MFB-CIM is operated in a quantum-noise-dominated regime with short photon lifetimes (i.e., low cavity finesse), homodyne monitoring of the system can efficiently produce samples of low-energy Ising spin configurations, requiring many fewer roundtrips to sample than suggested by established high-finesse, continuous-time models. We find that sampling performance is robust to, or even improved by, turning off or altogether reversing the sign of the parametric drive, but performance is critically reduced in the absence of optical nonlinearity. For the class of MAX-CUT problems with binary-signed edge weights, the number of roundtrips sufficient to fully sample all spin configurations up to the first-excited Ising energy, including all degeneracies, scales as $1.08^N$. At a problem size of $N = 100$ with a few dozen (median of 20) such desired configurations per instance, we have found median sufficient sampling times of $6\times10^6$ roundtrips; in an experimental implementation of an MFB-CIM with a 10 GHz repetition rate, this corresponds to a wall-clock sampling time of 60 ms.

Reinforcement learning enhanced quantum-inspired algorithm for combinatorial optimization

Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables a stable single-player version of self-play training and helps the agent escape local optima. The training on any problem instance can be accelerated by applying transfer learning from an agent trained on randomly generated problems. Our approach allows sampling high quality solutions to the Ising problem with high probability and outperforms both baseline heuristics and a black-box hyperparameter optimization approach.

Machine Learning as a Flaring Storm Warning Machine: Was a Warning Machine for the 2017 September Solar Flaring Storm Possible?

Abstract Machine learning is now one of the methodologies of choice for flare forecasting, and supervised techniques, in both their traditional and deep versions, are becoming more frequently used for prediction in this area of space weather. Most studies assess the prediction effectiveness of machine-learning methods by computing confusion matrices, which are typically highly non-diagonal, particularly in applications concerning the forecasting of X-class flares. The present study suggests that the reliability of the outcomes of a supervised machine-learning method could be better assessed by using it as a warning machine, sounding binary alerts unrolled over time, and by comparing the number of alerts sounded by the machine in specific time windows with the number of events actually observed in those time windows. Indeed, when applied to the prediction of the events associated with the 2017 September solar storm, a hybrid LASSO algorithm was able to sound alerts every day a flare actually occurred; it also identified the corresponding flare class. In addition, the machine was able to predict with some accuracy a reliable proxy of the energy budget daily released by magnetic reconnection during the entire course of the storm. Finally, the analysis shows that the combination of sparsity-enhancing machine learning and feature ranking could allow the identification of the prominent role that the Ising energy played as an active region property in the forecasting process.

Dynamics of strongly coupled disordered dissipative spin-boson systems

Spin-boson Hamiltonians are an effective description for numerous quantum many-body systems such as atoms coupled to cavity modes, quantum electrodynamics in circuits and trapped ion systems. While reaching the limit of strong coupling is possible in current experiments, the understanding of the physics in this parameter regime remains a challenge, especially when disorder and dissipation are taken into account. Here we investigate a regime where the many-body spin dynamics can be related to a Ising energy function defined in terms of the spin-boson couplings. While in the coherent weak coupling regime it is known that an effective description in terms of spin Hamiltonian is possible, we show that a similar viewpoint can be adopted in the presence of dissipation and strong couplings. The resulting many-body dynamics features approximately thermal regimes, separated by out-of-equilibrium ones in which detailed balance is broken. Moreover, we show that under appropriately chosen conditions one can even achieve cooling of the spin degrees of freedom. This points towards the possibility of using strongly coupled dissipative spin-boson systems for engineering complex energy landscapes together with an appropriate cooling dynamics.

A single shot coherent Ising machine based on a network of injection-locked multicore fiber lasers

Combinatorial optimization problems over large and complex systems have many applications in social networks, image processing, artificial intelligence, computational biology and a variety of other areas. Finding the optimized solution for such problems in general are usually in non-deterministic polynomial time (NP)-hard complexity class. Some NP-hard problems can be easily mapped to minimizing an Ising energy function. Here, we present an analog all-optical implementation of a coherent Ising machine (CIM) based on a network of injection-locked multicore fiber (MCF) lasers. The Zeeman terms and the mutual couplings appearing in the Ising Hamiltonians are implemented using spatial light modulators (SLMs). As a proof-of-principle, we demonstrate the use of optics to solve several Ising Hamiltonians for up to thirteen nodes. Overall, the average accuracy of the CIM to find the ground state energy was ~90% for 120 trials. The fundamental bottlenecks for the scalability and programmability of the presented CIM are discussed as well.

Testing and Improving a Set of Morphological Predictors of Flaring Activity

Efficient prediction of solar flares relies on parameters that quantify the eruptive capability of solar active regions. Several such quantitative predictors have been proposed in the literature, inferred mostly from photospheric magnetograms and/or white-light observations. Two of them are the Ising energy and the sum of the total horizontal magnetic field gradient. The former has been developed from line-of-sight magnetograms, while the latter uses sunspot detections and characteristics, based on continuum images. Aiming to include these parameters in an automated prediction scheme, we test their applicability on regular photospheric magnetic field observations provided by the Helioseismic and Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory (SDO). We test their efficiency as predictors of flaring activity on a representative sample of active regions and investigate possible modifications of these quantities. The Ising energy appears to be an efficient predictor, and the efficiency is even improved if it is modified to describe interacting magnetic partitions or sunspot umbrae. The sum of the horizontal magnetic field gradient appears to be slightly more promising than the three variations of the Ising energy we implement in this article. The new predictors are also compared with two very promising predictors: the effective connected magnetic field strength and the total unsigned non-neutralized current. Our analysis shows that the efficiency of morphological predictors depends on projection effects in a nontrivial way. All four new predictors are found useful for inclusion in an automated flare forecasting facility, such as the Flare Likelihood and Region Eruption Forecasting (FLARECAST), but their utility, among others, will ultimately be determined by the validation effort underway in the framework of the FLARECAST project.

Partially disordered antiferromagnetism and multiferroic behavior in a frustrated Ising systemCoCl2−2SC(NH2)2

We investigate partially disordered antiferromagnetism in CoCl$_2$-2SC(NH$_2$)$_2$, in which a-b plane hexagonal layers are staggered along the c-axis, rather than stacked. A robust 1/3 state forms in applied magnetic fields which the spins are locked, varying neither as a function of temperature nor field. By contrast, in zero field, partial antiferromagnetic order occurs, in which free spins are available to create a Curie-like magnetic susceptibility. We report measurements of the crystallographic structure, and the specific heat, magnetization, and electric polarization down to $T$ = 50 mK and up to $\mu_{0}H$ = 60 T. The Co$^{2+}$ $S = 3/2$ spins are Ising-like and form distorted hexagonal layers. The Ising energy scale is well separated from the magnetic exchange, and both energy scales are accessible to the measurements allowing us to cleanly parameterize them. In transverse fields, a quantum Ising phase transition can be observed at 2 T. Finally we find that magnetic exchange striction induces changes in the electric polarization up to 3 $\mu$C/m$^2$ and single-ion magnetic anisotropy effects induce a much a larger electric polarization change of 300 $\mu$C/m$^2$.

On separation efficiency

AbstractRising energy costs have renewed interest in energy efficient separations. Such efficiencies can be defined as the free energy of unmixing divided by the work done on and the heat added to the process. The greatest efficiencies occur when the energy cost far exceeds any capital cost. In this limit, rate processes are not important, and efficiency is governed by thermodynamics. Efficiencies for gas absorption and stripping are then typically around 50%. The efficiencies of liquid–liquid extractions are lower, especially when one solute is wanted at higher concentration. The efficiencies of membrane separations are higher, but these require high pressure, normally obtained mechanically. As a result, membrane efficiencies should be reduced by any Carnot efficiency of generating the mechanical energy. These results, which are consistent with earlier estimates for distillation, have implications for carbon dioxide capture as a route to mitigating global warming. © 2012 American Institute of Chemical Engineers AIChE J, 2012

Physical consideration of an image in image restoration using Bayes’ formula

We consider image restoration by Bayes’ formula and investigate the relationship between an image and a prior probability from the following two viewpoints: hyperparameter estimation and the accuracy of a restored image. The Q -Ising model is adopted as a prior probability in Bayes’ formula. Not the Q -Ising energy, but the Potts energy plays an important role in the hyperparameter estimation. From the viewpoint of the hyperparameter estimation, the relationship between a natural image and a prior probability is characterized through the Potts energy and magnetization of an image. The Potts energy and magnetization of an image are defined by a set of pixels’ state of an image. The closer to the average Potts energy and magnetization over a prior probability the Potts energy and magnetization of a natural image is, the closer to the true value of a hyperparameter the estimated value of a hyperparameter from a degraded image is. For the accuracy of a restored image, the image which has a smaller Q -Ising energy is better restored by Bayes’ formula composed of the Q -Ising prior. The consideration for the relationship between an image and a prior probability is expected to be valid for a more complicated prior probability.

Random Walk over Basins of Attraction to Construct Ising Energy Landscapes

An efficient algorithm is developed to construct disconnectivity graphs by a random walk over basins of attraction. This algorithm can detect a large number of local minima, find energy barriers between them, and estimate local thermal averages over each basin of attraction. It is applied to the Sherrington-Kirkpatrick (SK) spin glass Hamiltonian where existing methods have difficulties even for a moderate number of spins. Finite-size results are used to make predictions in the thermodynamic limit that match theoretical approximations and recent findings on the free energy landscapes of SK spin glasses.

Glauber model in a quantum representation

The Glauber model is reconsidered based on a quantum formulation of the master equation. Unlike the conventional approach the temperature and the Ising energy are included from the beginning by introducing a Heisenberg-like picture of the second quantized operators. This method enables us to get an exact expression for the transition rate of a single flip process w(i)(sigma(i)) which is in accordance with the principle of detailed balance. The transition rate differs significantly from the conventional one due to Glauber in the low-temperature regime. Here the behavior is controlled by the Ising energy and not by the microscopic time scale.

Respiratory correlated cone-beam computed tomography on an isocentric C-arm

A methodology for 3D image reconstruction from retrospectively gated cone-beam CT projection data has been developed. A mobile x-ray cone-beam device consisting of an isocentric C-arm equipped with a flat panel detector was used to image a moving phantom. Frames for reconstruction were retrospectively selected from complete datasets based on the known rotation of the C-arm and a signal from a respiratory monitor. Different sizes of gating windows were tested. A numerical criterion for blur on the reconstructed image was suggested. The criterion is based on minimization of an Ising energy function, similar to approaches used in image segmentation or restoration. It is shown that this criterion can be used for the determination of the optimal gating window size. Images reconstructed from the retrospectively gated projection sequences using the optimal gating window data showed a significant improvement compared to images reconstructed from the complete projection datasets.

, , , and from 3D Ising energy and specific heat

We analyse Monte Carlo data for the energy and specific heat at and close to the critical point of the 3D cubic Ising model. From the finite-size scaling of the energy E and the specific heat C at criticality we obtain the estimate . Furthermore, one obtains precise estimates for the `backgrounds' (nonsingular parts) and . Fitting solely off-critical energy estimates to a scaling law, we find, depending on the choice of the reduced temperature, either and , or and . Including information from the data at , we obtain the estimate . We also determine the universal combination in both phases.

Computation of alloy phase diagrams at low temperatures

Standard statistical-mechanics techniques for alloy-Ising models such as Monte Carlo simulations or the cluster variation method usually present numerical problems at low temperatures or for highly stoichiometric compounds. Under these conditions, their application to complex alloy Hamiltonians, with extended pair and multi-site interactions, is non trivial and can be very computer-time demanding. In this work, we investigate the application of a low-temperature expansion of the thermodynamic potentials for Hamiltonians with many pair and multi-site interactions. In this way, analytic expressions can be obtained for the free energies from which temperature-composition phase diagrams for any alloy can easily be computed regardless of the complexity of the Ising energy expression. It is demonstrated that with only a few terms in the expansion, the low-temperature expansion is accurate up to temperatures where Monte Carlo simulations or cluster variation calculations are practical. Consequently, these three methods can be used as complimentary techniques to compute a single phase diagram. Furthermore, we also show that the coefficients of the low-temperature expansion can be computed from the same information used to build the cluster variational free energy, thereby making the low-temperature expansion very simple to use. We illustrate the application of this new approach by computing the fcc Pd-rich phase diagram of the Pd-V alloy.

Scaling fields and universality of the liquid-gas critical point.

We report an extensive Monte Carlo study of the density and energy fluctuations in a 2D Lennard-Jones fluid, within the grand-canonical ensemble. The results are analyzed within a finite-size-scaling theory incorporating the mixed character of the scaling fields that manifests particle-hole asymmetry. The limiting critical distribution of the density matches that of the magnetization in the 2D Ising model, illuminating the configurational basis of universality; the corrections to the limiting behavior contain an antisymmetric component which reflects the critical Ising energy operator, corroborating the anticipated structure of the scaling fields.