Ising Model Research Articles

Error estimation in current noisy quantum computers

One of the main important features of the noisy intermediate-scale quantum (NISQ) era is the correct evaluation and consideration of errors. In this paper, we analyse the main sources of errors in current (IBM) quantum computers and we present a useful tool (TED-qc) designed to facilitate the total error probability expected for any quantum circuit. We propose this total error probability as the best way to estimate a lower bound for the fidelity in the NISQ era, avoiding the necessity of comparing the quantum calculations with any classical one. In order to contrast the robustness of our tool we compute the total error probability that may occur in three different quantum models: 1) the Ising model, 2) the Quantum-Phase Estimation (QPE), and 3) the Grover’s algorithm. For each model, the main quantities of interest are computed and benchmarked against the reference simulator’s results as a function of the error probability for a representative and statistically significant sample size. The analysis is satisfactory in more than the 99%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$99\\%$$\\end{document} of the cases. In addition, we study how error mitigation techniques are able to eliminate the noise induced during the measurement. These results have been calculated for the IBM quantum computers, but both the tool and the analysis can be easily extended to any other quantum computer.

Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field

Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field

The two upper critical dimensions of the Ising and Potts models

We derive the exact actions of the Q-state Potts model valid on any graph, first for the spin degrees of freedom, and second for the Fortuin-Kasteleyn clusters. In both cases the field is a traceless Q-component scalar field Φα. For the Ising model (Q = 2), the field theory for the spins has upper critical dimension \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{\ ext{c}}}^{{\ ext{spin}}}$$\\end{document} = 4, whereas for the clusters it has \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{\ ext{c}}}^{{\ ext{cluster}}}$$\\end{document} = 6. As a consequence, the probability for three points to be in the same cluster is not given by mean-field theory for d within 4 < d < 6. We estimate the associated universal structure constant as \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=\\sqrt{6-d}+\\mathcal{O}{\\left(6-d\\right)}^{3/2}$$\\end{document}. This shows that some observables in the Ising model have an upper critical dimension of 4, while others have an upper critical dimension of 6. Combining perturbative results from the ϵ = 6 – d expansion with a non-perturbative treatment close to dimension d = 4 allows us to locate the shape of the critical domain of the Potts model in the whole (Q, d) plane.

Dually Weighted Multi-matrix Models as a Path to Causal Gravity-Matter Systems

AbstractWe introduce a dually-weighted multi-matrix model that for a suitable choice of weights reproduce two-dimensional Causal Dynamical Triangulations (CDT) coupled to the Ising model. When Ising degrees of freedom are removed, this model corresponds to the CDT-matrix model introduced by Benedetti and Henson (Phys Lett B 678:222, 2009). We present exact as well as approximate results for the Gaussian averages of characters of a Hermitian matrix A and $$A^2$$ A 2 for a given representation and establish the present limitations that prevent us to solve the model analytically. This sets the stage for the formulation of more sophisticated matter models coupled to two-dimensional CDT as dually weighted multi-matrix models providing a complementary view to the standard simplicial formulation of CDT-matter models.

Nonperturbative thermodynamic extrinsic curvature of the anyon gas

Thermodynamic extrinsic curvature is a new mathematical tool in thermodynamic geometry. By using the thermodynamic extrinsic curvature, one may obtain a more complete geometric representation of the critical phenomena and thermodynamics. We introduce nonperturbative thermodynamic extrinsic curvature of an ideal two-dimensional gas of anyons. Using extrinsic curvature, we find new fixed points in nonperturbative thermodynamics of the anyon gas that particles behave as semions. Here, we investigate the critical behavior of thermodynamic extrinsic curvature of two-dimensional Kagome Ising model near the critical point [Formula: see text] in a constant magnetic field and show that it behaves as [Formula: see text] with [Formula: see text], where [Formula: see text] denotes the critical exponent of the specific heat. Then, we consider the three-dimensional spherical model and show that the scaling behavior is [Formula: see text], where [Formula: see text]. Finally, using a general argument, we show that extrinsic curvature [Formula: see text] has two different scaling behaviors for positive and negative [Formula: see text]. For [Formula: see text], our results indicate that [Formula: see text]. However, for [Formula: see text], we found a different scaling behavior, where [Formula: see text].

A bootstrap study of minimal model deformations

For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document} deformation in two dimensions.

Frustration—no frustration crossover and phase transitions in 2D spin models with zig-zag structures

Three 2D spin models made of frustrated zig-zag chains with competing interactions which, by exact summation with respect to some degrees of freedom, can be replaced by an effective temperature-dependent interaction, were considered. The first model, exactly solvable Ising chains coupled by only four-spin interactions, does not exhibit any finite temperature phase transition; nevertheless, temperature can trigger a frustration–no frustration crossover accompanied by gigantic specific heat. A similar effect was observed in several two-leg ladder models (Weiguo 2020 arXiv:2006.08921v2; 2020 2006.15087v1). The anisotropic Ising chains coupled by a direct interchain interaction and, competing with it, indirect interaction via spins located between chains, are analyzed using the exact Onsager’s equation and linear perturbation renormalization group (LPRG). Depending on the parameter set, such a model exhibits one antiferromagnetic (AF) or ferromagnetic (FM) phase transition or three phase transitions with a re-entrant disordered phase between AF and FM ones. The LPRG method was also used to study coupled uniaxial XXZ chains which, for example, can be a minimal model to describe the magnetic properties of compounds in which uranium and rare earth atoms form zig-zag chains. As with the Ising model, for a certain set of parameters, the model can undergo three phase transitions. However, both intrachain and interchain plain interactions si,jxsk,lx+si,jysk,ly can eliminate the re-entrant disordered phase, and then only one transition takes place. Additionally, the XXZ model can undergo temperature-induced metamagnetic transition.

A differentiable programming framework for spin models

We introduce a novel framework for simulating spin models using differentiable programming, an approach that leverages the advancements in machine learning and computational efficiency. We focus on three distinct spin systems: the Ising model, the Potts model, and the Cellular Potts model, demonstrating the practicality and scalability of our framework in modeling these complex systems. Additionally, this framework allows for the optimization of spin models, which can adjust the parameters of a system by a defined objective function. In order to simulate these models, we adapt the Metropolis-Hastings algorithm to a differentiable programming paradigm, employing batched tensors for simulating spin lattices. This adaptation not only facilitates the integration with existing deep learning tools but also significantly enhances computational speed through parallel processing capabilities, as it can be implemented on different hardware architectures, including GPUs and TPUs.

Compensation temperature and hysteresis behaviors of the ferrimagnetic bilayer zigzag graphyne nanoribbon with RKKY interaction

In this study, a bilayer zigzag graphyne nanoribbon with RKKY interaction is described using a ferrimagnetic mixed-spin (2, 5/2) Ising model. The compensation behaviors and hysteresis characteristics of the system generated by various physical parameters, such as crystal fields of sublattices, interlayer exchange coupling and the intralayer exchange coupling are explored using Monte Carlo simulation. In the research results, we found that among the 30 possible spin configurations that may occur in the ground state of the system, only 21 spin configurations are stable. At the same time, under certain parameters, the system exhibit interesting behaviors such as compensation phenomena and multi-loop hysteresis. This can provide a theoretical reference for multi-state magnetic recording.

Spin textures of coherent photons with SU(4) symmetry

We show coherent photons with spin and orbital angular momentum have symmetry of special unitary group of degree four, SU(4), both theoretically and experimentally. We show the degree of polarisation could be a measure to characterise the quantum coherency of a spin state, which was significantly affected by the partial trace out of orbital degree of freedom. By changing the amplitudes of the classically entangled state, we observed a crossover from quantum Heisenberg spin to classical Ising spin. We also demonstrated projections of coherent states, which recover the coherency of the spin state, while the SU(4) state is projected to be a simple product state of SU(2)×SU(2). Based on the developed platform, we have explored spin texture of coherent photons, and found a unique spin profile reminiscent of the Bengal cat upon wrapping to a three-dimensional sphere. We have controlled the phase between singlet and triplet states and found the spin texture rotated upon increasing the phase.

Exploring mean field spin models with multiple populations: A theoretical framework

This paper thoroughly investigates the analysis of a modified generalized mean-field Ising model. The thermodynamic limit of free energy is investigated, and both the upper and lower bounds are derived. The lower bound is derived using a standard entropic variational principle, whereas a decoupling method is employed to derive the upper bound. Although the two bounds differ in form, they exhibit the same local maxima. Finally, the impact of a slight external field on the magnetic order is analyzed.

Training an Ising machine with equilibrium propagation

Ising machines, which are hardware implementations of the Ising model of coupled spins, have been influential in the development of unsupervised learning algorithms at the origins of Artificial Intelligence (AI). However, their application to AI has been limited due to the complexities in matching supervised training methods with Ising machine physics, even though these methods are essential for achieving high accuracy. In this study, we demonstrate an efficient approach to train Ising machines in a supervised way through the Equilibrium Propagation algorithm, achieving comparable results to software-based implementations. We employ the quantum annealing procedure of the D-Wave Ising machine to train a fully-connected neural network on the MNIST dataset. Furthermore, we demonstrate that the machine’s connectivity supports convolution operations, enabling the training of a compact convolutional network with minimal spins per neuron. Our findings establish Ising machines as a promising trainable hardware platform for AI, with the potential to enhance machine learning applications.

Enhanced network inference from sparse incomplete time series through automatically adapted L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} regularization

Reconstructing dynamics of complex systems from sparse, incomplete time series data is a challenging problem with applications in various domains. Here, we develop an iterative heuristic method to infer the underlying network structure and parameters governed by Ising dynamics from incomplete spin configurations based on sparse and small-sized samples. Our method iterates between imputing missing spin states given current coupling strengths and re-estimating couplings from completed spin state data. Central to our approach is the novel application of adaptive l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_1$$\\end{document} regularization on updating coupling strengths, which features an automatic adjustment of the regularization strength throughout the iterative inference process. By doing so, we aim at preventing over-fitting and enforcing the sparsity of couplings without access to ground truth parameters. We demonstrate that this approach accurately recovers parameters and imputes missing spins even with substantial missing data and short time series, providing improvements in the inference of Ising model parameters even for relatively small sample sizes.

Behavior of quantum coherence in quantum phase transitions of two-dimensional XY and ising models

We investigate the quantum coherence behavior of the ground states of 2D Heisenberg XY model and 2D Ising model with a transverse field on square lattices using the Quantum Renormalization Group (QRG) method. Our analysis focused on the ground state density matrix and its marginal states, revealing non-analytic behavior of quantum coherence (especially two-site coherence) near the critical point. This behavior allowed us to detect quantum phase transitions (QPT) in these models. By examining the scaling behavior of the maximum derivative of quantum coherence with system size, we determined the critical exponent of coherence for both models and the length exponent of the Ising model. Additionally, we investigated the time evolution of coherence in both models. Our results closely align with those obtained from entanglement analysis, that is while quantum coherence requires fewer computational calculations compared to discord and entanglement approaches.

Quantum critical fluctuations in a transverse-field Ising magnet.

CoNb2O6is a model system for a spin-1/2 one-dimensional (1D) transverse-field Ising magnet (TFIM) with a rather low 3D Neel ordering temperature atTN= 2.95 K. We studied CoNb2O6using ultrasound measurements down to 0.3 K in transverse magnetic fields applied along thebdirection. Upon entering the 3D ordered state, we observe pronounced anomalies in the transverse acoustic modec66. In particular, from 1.3 to 1.5 K and around 4.7 T, this mode reveals an almost diverging softening, which is considerably reduced at lower and higher magnetic fields. &#xD;We interpret this as an influence of quantum critical fluctuations emerging from the quantum critical point (QCP) of the 1D Ising spin chains at about 4.75 T, which lies below the QCP of the 3D ordering at about 5.4 T. This is clear experimental evidence of the predicted generic phase diagram for a TFIM with superimposed 3D ordering.

Separability Transitions in Topological States Induced by Local Decoherence.

We study states with intrinsic topological order subjected to local decoherence from the perspective of separability, i.e., whether a decohered mixed state can be expressed as an ensemble of short-range entangled pure states. We focus on toric codes and the X-cube fracton state and provide evidence for the existence of decoherence-induced separability transitions that precisely coincide with the threshold for the feasibility of active error correction. A key insight is that local decoherence acting on the "parent" cluster states of these models results in a Gibbs state. As an example, for the 2D (3D) toric code subjected to bit-flip errors, we show that the decohered density matrix can be written as a convex sum of short-range entangled states for p>p_{c}, where p_{c} is related to the paramagnetic-ferromagnetic transition in the 2D (3D) random bond Ising model along the Nishimori line.

Primary and secondary order parameters in the fully frustrated transverse-field Ising model on the square lattice

Primary and secondary order parameters in the fully frustrated transverse-field Ising model on the square lattice

Quantum annealing of a frustrated magnet

Quantum annealing, which involves quantum tunnelling among possible solutions, has state-of-the-art applications not only in quickly finding the lowest-energy configuration of a complex system, but also in quantum computing. Here we report a single-crystal study of the frustrated magnet α-CoV2O6, consisting of a triangular arrangement of ferromagnetic Ising spin chains without evident structural disorder. We observe quantum annealing phenomena resulting from time-reversal symmetry breaking in a tiny transverse field. Below ~ 1 K, the system exhibits no indication of approaching the lowest-energy state for at least 15 hours in zero transverse field, but quickly converges towards that configuration with a nearly temperature-independent relaxation time of ~ 10 seconds in a transverse field of ~ 3.5 mK. Our many-body simulations show qualitative agreement with the experimental results, and suggest that a tiny transverse field can profoundly enhance quantum spin fluctuations, triggering rapid quantum annealing process from topological metastable Kosterlitz-Thouless phases, at low temperatures.

Bayesian hidden mark interaction model for detecting spatially variable genes in imaging-based spatially resolved transcriptomics data.

Recent technology breakthroughs in spatially resolved transcriptomics (SRT) have enabled the comprehensive molecular characterization of cells whilst preserving their spatial and gene expression contexts. One of the fundamental questions in analyzing SRT data is the identification of spatially variable genes whose expressions display spatially correlated patterns. Existing approaches are built upon either the Gaussian process-based model, which relies on ad hoc kernels, or the energy-based Ising model, which requires gene expression to be measured on a lattice grid. To overcome these potential limitations, we developed a generalized energy-based framework to model gene expression measured from imaging-based SRT platforms, accommodating the irregular spatial distribution of measured cells. Our Bayesian model applies a zero-inflated negative binomial mixture model to dichotomize the raw count data, reducing noise. Additionally, we incorporate a geostatistical mark interaction model with a generalized energy function, where the interaction parameter is used to identify the spatial pattern. Auxiliary variable MCMC algorithms were employed to sample from the posterior distribution with an intractable normalizing constant. We demonstrated the strength of our method on both simulated and real data. Our simulation study showed that our method captured various spatial patterns with high accuracy; moreover, analysis of a seqFISH dataset and a STARmap dataset established that our proposed method is able to identify genes with novel and strong spatial patterns.

A discrete time crystal order in the driven Ising model: thermal effects, jamming analogy and quench dynamics

We explore the emergence of a discrete time crystalline (DTC) order and its stability against thermal fluctuations in a driven kinetic Ising model on a two-dimensional square lattice using the drive protocol invented in a recent work (Gambetta et al 2019 Phys. Rev. E 100 060105(R)). The DTC order is found to be quite robust in the presence of thermal fluctuations. We construct the resulting three-dimensional phase diagram for the DTC order, which manifests a striking resemblance to the jamming phase diagram proposed by Liu and Nagel. This finding may suggest a new way to view the DTC order as a new type of nonequilibrium soft matter. The quench dynamics exhibits a unique feature due to the nature of the employed drive protocol, namely, breakdown of the inverse relationship between the domain growth and defect relaxation, which holds in the usual quench dynamics of the kinetic Ising model.