Kirkpatrick Spin Glass Research Articles

A benchmarking study of quantum algorithms for combinatorial optimization

We study the performance scaling of three quantum algorithms for combinatorial optimization: measurement-feedback coherent Ising machines (MFB-CIM), discrete adiabatic quantum computation (DAQC), and the Dürr–Høyer algorithm for quantum minimum finding (DH-QMF) that is based on Grover’s search. We use MaxCut problems as a reference for comparison, and time-to-solution (TTS) as a practical measure of performance for these optimization algorithms. For each algorithm, we analyze its performance in solving two types of MaxCut problems: weighted graph instances with randomly generated edge weights attaining 21 equidistant values from −1 to 1; and randomly generated Sherrington–Kirkpatrick (SK) spin glass instances. We empirically find a significant performance advantage for the studied MFB-CIM in comparison to the other two algorithms. We empirically observe a sub-exponential scaling for the median TTS for the MFB-CIM, in comparison to the almost exponential scaling for DAQC and the proven Õ2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{{{{\\mathcal{O}}}}}\\left(\\sqrt{{2}^{n}}\\right)$$\\end{document} scaling for DH-QMF. We conclude that the MFB-CIM outperforms DAQC and DH-QMF in solving MaxCut problems.

Unlearning regularization for Boltzmann machines

Boltzmann machines (BMs) are graphical models with interconnected binary units, employed for the unsupervised modeling of data distributions. When trained on real data, BMs show the tendency to behave like critical systems, displaying a high susceptibility of the model under a small rescaling of the inferred parameters. This behavior is not convenient for the purpose of generating data, because it slows down the sampling process, and induces the model to overfit the training-data. In this study, we introduce a regularization method for BMs to improve the robustness of the model under rescaling of the parameters. The new technique shares formal similarities with the unlearning algorithm, an iterative procedure used to improve memory associativity in Hopfield-like neural networks. We test our unlearning regularization on synthetic data generated by two simple models, the Curie–Weiss ferromagnetic model and the Sherrington–Kirkpatrick spin glass model. We show that it outperforms Lp -norm schemes and discuss the role of parameter initialization. Eventually, the method is applied to learn the activity of real neuronal cells, confirming its efficacy at shifting the inferred model away from criticality and coming out as a powerful candidate for actual scientific implementations.

Role of quantum fluctuation ininducing ergodicity inthe spin glass phase and its effect inquantum annealing.

We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that aquantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

A transport equation approach for deep neural networks with quenched random weights

We consider a multi-layer Sherrington–Kirkpatrick spin-glass as a model for deep restricted Boltzmann machines with quenched random weights and solve for its free energy in the thermodynamic limit by means of Guerra’s interpolating techniques under the RS and 1RSB ansatz. In particular, we recover the expression already known for the replica-symmetric case. Further, we drop the restriction constraint by introducing intra-layer connections among spins and we show that the resulting system can be mapped into a modular Hopfield network, which is also addressed via the same techniques up to the first step of replica symmetry breaking.

Spherical Spin Glass Model with External Field

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

Study of longitudinal fluctuations of the Sherrington–Kirkpatrick model

We study finite-size corrections to the free energy of the Sherrington–Kirkpatrick spin glass in the low-temperature phase where replica symmetry is broken. We investigate the role of longitudinal fluctuations in these corrections, neglecting the transverse contribution. In particular, we are interested in the value of exponent that controls the finite volume corrections: is defined by the relation , N being the total number of spins. We perform both an analytical and numerical estimate of the analytical result for . From both the approaches, we get the result: .

The effects of the one-step replica symmetry breaking on the Sherrington–Kirkpatrick spin glass model in the presence of random field with a joint Gaussian probability density function for the exchange interactions and random fields

The Sherrington–Kirkpatrick Ising spin glass model, in the presence of a random magnetic field, is investigated within the framework of the one-step replica symmetry breaking. The two random variables (exchange integral interaction Jij and random magnetic field hi) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ, assuming positive and negative values. The thermodynamic properties, the three different phase diagrams and system’s parameters are computed with respect to the natural parameters of the joint Gaussian probability density function at non-zero and zero temperatures. The low temperature negative entropy controversy, a result of the replica symmetry approach, has been partly remedied in the current study, leading to a less negative result. In addition, the present system possesses two successive spin glass phase transitions with characteristic temperatures.

Rare events analysis of temperature chaos in the Sherrington–Kirkpatrick model

We investigate the question of temperature chaos in the Sherrington–Kirkpatrick spin glass model, applying a recently proposed rare events based data analysis method to existing Monte Carlo data. Thanks to this new method, temperature chaos is now observable for this model, even with the limited size systems that can currently be simulated.

The Sherrington–Kirkpatrick spin glass model in the presence of a random field with a joint Gaussian probability density function for the exchange interactions and random fields

The magnetic systems with disorder form an important class of systems, which are under intensive studies, since they reflect real systems. Such a class of systems is the spin glass one, which combines randomness and frustration. The Sherrington–Kirkpatrick Ising spin glass with random couplings in the presence of a random magnetic field is investigated in detail within the framework of the replica method. The two random variables (exchange integral interaction and random magnetic field) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ. The thermodynamic properties and phase diagrams are studied with respect to the natural parameters of both random components of the system contained in the probability density. The de Almeida–Thouless line is explored as a function of temperature, ρ and other system parameters. The entropy for zero temperature as well as for non zero temperatures is partly negative or positive, acquiring positive branches as h0 increases.

Ground states of the Sherrington–Kirkpatrick spin glass with Levy bonds

Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) ∝ 1/|J|1+α, |J| > 1, are investigated for a range of powers α. The results are compared with those for the Sherrington–Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground-state energy densities with α, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves at infinite system size attain universally the Parisi-energy of the SK as long as the second moment of P(J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω SK = 2/3 already well above α = 2, at which point it reaches a minimum. This result is justified with a speculative calculation of a random energy model with Levy bonds. The exponent ρ that describes the variations of the ground-state energy fluctuations with system size decays monotonically from its SK value for decreasing α and appears to vanish at α = 1.

Interpolating the Sherrington–Kirkpatrick replica trick

Interpolation techniques have become, in the past decades, a powerful approach to describe several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented in the cavity field technique, or its variants such as stochastic stability and random overlap structures. However the first and most famous approach to mean field statistical mechanics with quenched disorder is the replica trick. Among the models where these methods have been used (namely, dealing with frustration and complexity), probably the best known is the Sherrington–Kirkpatrick spin glass. In this paper we apply the interpolation scheme to the original replica trick framework and test it directly on the cited paradigmatic model. Although the problem, at a mathematical level, has been deeply investigated by Talagrand, it is still rich in information from a theoretical physics perspective; in fact, by treating the number of replicas n ∈ (0, 1] as an interpolating parameter (far from its original interpretation) the proof of the attendant commutativity of the zero replica and the infinite volume limits can be easily obtained. Further, within this perspective, we can naturally think of n as a quenching temperature close to that introduced in off-equilibrium approaches to gain some new insight into our understanding of the off-equilibrium features encountered in equilibrium statistical mechanics of spin glasses.

Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobitechnique

During the last few years, through the combined effort of the insight coming from physicalintuition and computer simulation, and the exploitation of rigorous mathematical methods,the main features of the mean-field Sherrington–Kirkpatrick spin glass model have beenfirmly established. In particular, it has been possible to prove the existence and uniquenessof the infinite-volume limit for the free energy, and its Parisi expression, in terms of avariational principle involving a functional order parameter. Even the expected property ofultrametricity, for the infinite-volume states, seems to be near to a complete proof.The main structural feature of this model, and related models, is the deep phenomenon ofspontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. Byexpanding on our previous work, the aim of this paper is to investigate a generalframework, where replica symmetry breaking is embedded in a kind of mechanical schemeof the Hamilton–Jacobi type. Here, the analog of the ‘time’ variable is a parametercharacterizing the strength of the interaction, while the ‘space’ variables rule outquantitatively the broken replica symmetry pattern. Starting from the simple cases, whereannealing is assumed, or replica symmetry, we build up a progression of dynamical systems,with an increasing number of space variables, which allow us to weaken the effect of thepotential in the Hamilton–Jacobi equation as the level of symmetry breaking is increased.This new machinery allows us to work out mechanically the generalK-step RSB solutions, in a different interpretation with respect to the replica trick, andeasily reveals their properties such as existence or uniqueness.

An eigenvalue method for computing the largest relaxation time of disorderedsystems

We consider the dynamics of finite size disordered systems as defined by a masterequation satisfying detailed balance. The master equation can be mapped ontoa Schrödinger equation in configuration space, where the quantum HamiltonianH has the generic form of an Anderson localization tight-binding model. The largest relaxation timeteq governingthe convergence towards Boltzmann equilibrium is determined by the lowest non-vanishing eigenvalueE1 = 1/teq ofH (the lowest eigenvaluebeing E0 = 0). So therelaxation time teq can be computed without simulating the dynamics by any eigenvalue method able to compute the first excitedenergy E1. Here we use the ‘conjugate gradient’ method to determineE1 for each disordered sample and present numerical results on the statistics of the relaxation timeteq over disordered samples of a given size for two models: (i) forthe random walk in a self-affine potential of Hurst exponentH on a two-dimensionalsquare of size L × L, we findthe activated scaling lnteq(L)∼Lψ with ψ = H as expected; (ii) for the dynamics of the Sherrington–Kirkpatrick spin glass model ofN spins, we findthe growth lnteq(N)∼Nψ with ψ = 1/3 inagreement with most previous Monte Carlo measures. In addition, we find that the rescaled distribution of(lnteq) decaysas e−uη forlarge u with a tail exponent of order . We give a rare-event interpretation of this value, that points towards a sample-to-samplefluctuation exponent of order for the barrier.

A mechanical approach to mean field spin models

Inspired by the bridge pioneered by Guerra among statistical mechanics on lattice and analytical mechanics on 1+1 continuous Euclidean space time, we built a self-consistent method to solve for the thermodynamics of mean field models defined on lattice, whose order parameters self-average. We show the whole procedure by analyzing in full detail the simplest test case, namely, the Curie–Weiss model. Further, we report some applications also to models whose order parameters do not self-average by using the Sherrington–Kirkpatrick spin glass as a guide.

The interaction between multioverlaps in the high temperature phase of the Sherrington–Kirkpatrick spin glass

In this paper we explore the joint behavior of a finite number of multioverlaps in the high temperature phase of the Sherrington–Kirkpatrick model. Extending the work by Talagrand [Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models, A series of Modern Surveys in Mathematics Vol. 46 (Springer-Verlag, Berlin, 2003)], we show that, when these objects are scaled to have nontrivial limiting distributions, the joint behavior is described by a Gaussian process with an explicit covariance structure.

An exact relation between free energy fluctuations and bond chaos in theSherrington–Kirkpatrick model

Using a variant of the interpolating Hamiltonian technique, we show that there exists, inthe Sherrington–Kirkpatrick spin glass, an exact connection between the sample-to-samplefluctuations of the free energy and bond chaos involving two- and four-replica overlapsbetween replicas with different but correlated bonds. This relation is used to derive anupper bound of the fluctuations.

Fluctuations of the free energy in the diluted SK-model

We derive limit theorems for the fluctuations of the free energy in a diluted version of the Sherrington–Kirkpatrick model. Our proofs are based on the classical approach of Aizenman et al. [M. Aizenman, J.L. Lebowitz, D. Ruelle, Some rigorous results on the Sherrington–Kirkpatrick spin glass model, Comm. Math. Phys. 112 (1987) 3–20].

Hysteretic optimization for the Sherrington–Kirkpatrick spin glass

Hysteretic optimization is a heuristic optimization method based on the observation that magnetic samples are driven into a low-energy state when demagnetized by an oscillating magnetic field of decreasing amplitude. We show that hysteretic optimization is very good for finding ground states of Sherrington–Kirkpatrick spin glass systems. With this method it is possible to get good statistics for ground state energies for large samples of systems consisting of up to about 2000 spins. The way we estimate error rates may be useful for some other optimization methods as well. Our results show that both the average and the width of the ground state energy distribution converges faster with increasing size than expected from earlier studies.

Universality in Sherrington–Kirkpatrick's spin glass model

It was already clear to Sherrington and Kirkpatrick [Phys. Rev. B 17 (1978) 4384–4403] that the limiting free energy in Sherrington–Kirkpatrick's Spin Glass Model does not depend on the particular distribution of environments. We give here a mathematical proof of this fact.

Central limit theorem for fluctuations in the high temperature region of the Sherrington–Kirkpatrick spin glass model

In a region above the Almeida–Thouless line, where we are able to control the thermodynamic limit of the Sherrington–Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale 1/N, for large N. The method we employ is based on the idea we recently developed of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the free energy.