We apply a generalized Kibble-Zurek out-of-equilibrium scaling ansatz to simulated annealing when approaching the spin-glass transition at temperature $T=0$ of the two-dimensional Ising model with random $J= \pm 1$ couplings. Analyzing the spin-glass order parameter and the excess energy as functions of the system size and the annealing velocity in Monte Carlo simulations with Metropolis dynamics, we find scaling where the energy relaxes slower than the spin-glass order parameter, i.e., there are two different dynamic exponents. The values of the exponents relating the relaxation time scales to the system length, $τ\sim L^z$, are $z=8.28 \pm 0.03$ for the relaxation of the order parameter and $z=10.31 \pm 0.04$ for the energy relaxation. We argue that the behavior with dual time scales arises as a consequence of the entropy-driven ordering mechanism within droplet theory. We point out that the dynamic exponents found here for $T \to 0$ simulated annealing are different from the temperature-dependent equilibrium dynamic exponent $z_{\rm eq}(T)$, for which previous studies have found a divergent behavior; $z_{\rm eq}(T\to 0) \to \infty$. Thus, our study shows that, within Metropolis dynamics, it is easier to relax the system to one of its degenerate ground states than to migrate at low temperatures between regions of the configuration space surrounding different ground states. In a more general context of optimization, our study provides an example of robust dense-region solutions for which the excess energy (the conventional cost function) may not be the best measure of success.
Probing the lowest energy configuration of a complex system by quantum annealing was recently found to be more effective than its classical, thermal counterpart. Comparing classical and quantum Monte Carlo annealing protocols on the random two-dimensional Ising model we confirm the superiority of quantum annealing relative to classical annealing. We also propose a theory of quantum annealing, based on a cascade of Landau-Zener tunneling events. For both classical and quantum annealing, the residual energy after annealing is inversely proportional to a power of the logarithm of the annealing ti…
In this paper we show quantum fluctuation effect of fully frustrated Ising spin systems. Quantum annealing has been expected to be an efficient method to find ground state of optimization problems. However it is not clear when to use the quantum annealing. In order to clarify when the quantum annealing works well, we have to study microscopic properties of quantum annealing. In fully frustrated Ising spin systems, there are macroscopically degenerated ground states. When we apply quantum annealing to fully frustrated systems, we cannot obtain each ground state with the same probability. This n…
Quantum annealing method has been widely attracted attention in statistical physics and information science since it is expected to be a powerful method to obtain the best solution of optimization problem as well as simulated annealing. The quantum annealing method was incubated in quantum statistical physics. This is an alternative method of the simulated annealing which is well-adopted for many optimization problems. In the simulated annealing, we obtain a solution of optimization problem by decreasing temperature (thermal fluctuation) gradually. In the quantum annealing, in contrast, we dec…
Recently, Heim, Ronnow, Isakov and Troyer [Science 348 (2015) 215] have reported that Monte Carlo simulations for the Ising spin glass model on the square lattice in the physically relevant continuous-imaginary-time limit do not show superiority of quantum annealing (QA) using transverse field against classical annealing (CA). Although the QA schedule that they had used has been using conventionally, however the QA schedule mathematically has no guarantee that the used schedule is the best QA schedule for performance of optimization. We propose a new QA schedule for studying transverse-field-b…
We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to quantum annealing of the random-field Ising model to accelerate convergence toward the target ground state. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instant…
We study quantum annealing of pure and random transverse Ising chains that are coupled to boson baths, using a novel numerical method based on the combination of the quasi-adiabatic propagation path integral (QUAPI) and the matrix product state (MPS) formalisms. We present numerical results on systems up to 64 spins and conclude that the baths disturb quantum annealing of pure and random Ising chains. The numerical method to compute the reduced density matrix is described in detail.
We study first order quantum phase transitions in mean-field spin glasses. We solve the quantum Random Energy Model using elementary methods and show that at the transition the eigenstate suddenly projects onto the unperturbed ground state and that the gap between the lowest states is exponentially small in the system size. We argue that this is a generic feature of all `Random First Order' models, which includes benchmarks such as random satisfiability. We introduce a two-time instanton to calculate this gap in general, and discuss the consequences for quantum annealing.
We propose a nonadiabatic approach to quantum annealing, in which we repeat quantum annealing in nonadiabatic time scales, and collect the final states of many realizations to find the ground state among them. In this way, we replace the diffculty of long annealing time in adiabatic quantum annealing by another problem of the number of nonsidabatic (short-time) trials. The one-dimensional transverse-field Ising model is used to test this idea, and it is shown that nonadiabatic quantum annealing has the same computational complexity to find the ground state as the conventional adiabatic anneali…
We discuss an Ising spin glass where each $S=1/2$ spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the qu…
We show clear evidence of a quadratic speedup of a quantum annealing (QA) Schroedinger dynamics over a Glauber master-equation simulated annealing (SA) for a random Ising model in one dimension, via an equal-footing exact deterministic dynamics of the Jordan-Wigner fermionized problems. This is remarkable, in view of the arguments of Katzgraber et al., PRX 4, 021008 (2014), since SA does not encounter any phase transition, while QA does. We also find a second remarkable result: that a "quantum-inspired" imaginary-time Schroedinger QA provides a further exponential speedup, i.e., an asymptotic …
Population annealing is an efficient sequential Monte Carlo algorithm for simulating equilibrium states of systems with rough free energy landscapes. The theory of population annealing is presented, and systematic and statistical errors are discussed. The behavior of the algorithm is studied in the context of large-scale simulations of the three-dimensional Ising spin glass and the performance of the algorithm is compared to parallel tempering. It is found that the two algorithms are similar in efficiency though with different strengths and weaknesses.
Noisy mean field annealing (NMFA) is an algorithm that mimics a coherent Ising machine (CIM), which is an optical system for solving Ising problems. The NMFA has reproduced the solver performance of the CIM for systems of limited size even though it simplifies the interaction between spins with a mean-field approximation. However, recent experiments observing various thermodynamic quantities have revealed that the CIM can capture the phase transitions of the two-dimensional Ising models that the mean field cannot capture. This situation leads to a fundamental question as to how well the NMFA c…
We propose a checking parameter utilizing the breaking of the Jarzynski equality in the simulated annealing method using the Monte Carlo method. This parameter is based on the Jarzynski equality. By using this parameter, to detect that the system is in global minima of the free energy under gradual temperature reduction is possible. Thus, by using this parameter, one is able to investigate the efficiency of annealing schedules. We apply this parameter to the +-J Ising spin glass model. The application to the Gaussian Ising spin glass model is also mentioned. We discuss that the breaking of the…
Simulated Quantum Annealing (SQA), that is emulating a Quantum Annealing (QA) dynamics on a classical computer by a Quantum Monte Carlo whose parameters are changed during the simulation, is a well established computational strategy to cope with the exponentially large Hilbert space. It has enjoyed some early successes but has also raised more recent criticisms. Here we investigate, on the paradigmatic case of a one-dimensional transverse field Ising chain, two issues related to SQA in its Path-Integral implementation: the question of Monte Carlo vs physical (Schrödinger) dynamics and the issu…
Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, $H_I$. Here, we point to the existence of an analytical solution for such a problem for an arbitrary $H_I$ beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotoni…
We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the…
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the transverse Ising model, in which the transverse field is a function of time similar to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We have solved the time-dependent Schrö…
A partially annealed mean-field spin-glass model with a locally embedded pattern is studied. The model consists of two dynamical variables, spins and interactions, that are in contact with thermal baths at temperatures T_S and T_J, respectively. Unlike the quenched system, characteristic correlations among the interactions are induced by the partial annealing. The model exhibits three phases, which are paramagnetic, ferromagnetic and spin-glass phases. In the ferromagnetic phase, the embedded pattern is stably realized. The phase diagram depends significantly on the ratio of two temperatures n…
We combine population annealing Monte Carlo and several thermodynamic integration techniques to measure equilibrium vibrational and configurational entropies in the metastable fluid regime beyond the dynamic glass transition. We obtain results for a three-dimensional binary mixture hard sphere system. Our results suggest that the configurational entropy vanishes before the equilibrium pressure diverges, which implies that an underlying thermodynamic glass transition exists for this system. The computational methods are general and can be applied to a variety of glass forming systems but are re…
Here we first discuss briefly the quantum annealing technique. We then study the quantum annealing of Sherrington-Kirkpatrick spin glass model with the tuning of both transverse and longitudinal fields. Both the fields are time-dependent and vanish adiabatically at the same time, starting from high values. We solve, for rather small systems, the time-dependent Schrodinger equation of the total Hamiltonian by employing a numerical technique. At the end of annealing we obtain the final state having high overlap with the exact ground state(s) of classical spin glass system (obtained independently…
We compare the performance of quantum annealing (QA, through Schrödinger dynamics) and simulated annealing (SA, through a classical master equation) on the $p$-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second-order ($p=2$, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition ($p \ge 3$, i.e., with multi-spin Ising interactions), in contrast, the classical annealing dynamics appears to remain stu…
By applying a recently proposed mapping, we derive exactly the upper phase boundary of several Ising spin glass models defined over static graphs and random graphs, generalizing some known results and providing new ones.
Glasses prepared by physical vapour deposition have been shown to be remarkably more stable than those prepared by standard cooling protocols, with properties that appear to be similar to systems aged for extremely long times. When subjected to a rapid rise in temperature, ultrastable glasses anneal towards the liquid in a qualitatively different manner than ordinary glasses, with the seeming competition of different timescales and lengthscales. We numerically reproduce the phenomenology of ultrastable glass annealing with a kinetically constrained model, a three dimensional East model with so…
We propose a novel type of minor-embedding (ME) in simulated-annealing-based Ising machines. The Ising machines can solve combinatorial optimization problems. Many combinatorial optimization problems are mapped to find the ground (lowest-energy) state of the logical Ising model. When connectivity is restricted on Ising machines, ME is required for mapping from the logical Ising model to a physical Ising model, which corresponds to a specific Ising machine. Herein we discuss the guiding principle of ME design to achieve a high performance in Ising machines. We derive the proposed ME based on a …
We show that the effective action of the quantum spherical spin glass is invariant under a generalized form of the Becchi-Rouet-Stora-Tyutin(BRST) supersymmetry. The Ward identities associated to this invariance indicate that the spin glass order parameter must vanish, and as a result the annealed average is exact in this model. We present new results for for the free energy, entropy and specific heat. Due to quantum effects the entropy remains finite and the specific het vanishes at zero temperature. results for the phase diagram coincide with those obtained by different formalisms. At zero t…
We show that, above the critical temperature, if the dimension D of a given Ising spin glass model is sufficiently high, its free energy can be effectively expressed through the free energy of a related Ising model. When, in a large sense, D is infinite, in the paramagnetic phase and on its boundary the mapping is exact. In this limit the method provides a general and simple rule to obtain exactly the upper phase boundaries. We provide even simple effective rules to find crossover surfaces and correlation functions. We apply the mapping to several spin glass models.
The field-cooled magnetization (FCM) processes of Ising spin glasses under relatively small fields are investigated by experiment on Fe_{0.55}Mn_{0.45}TiO_3 and by numerical simulation on the three-dimensional Edwards-Anderson model. Both results are explained in a unified manner by means of the droplet picture. In particular, the cusp-like behavior of the FCM is interpreted as evidence, not for an equilibrium phase transition under a finite magnetic field, but for a dynamical (`blocking') transition frequently observed in glassy systems.
We determine accurate values of ordering temperatures and critical exponents for Ising Spin Glass transitions in dimension 4, using a combination of finite size scaling and non-equilibrium scaling techniques. We find that the exponents $η$ and $z$ vary with the form of the interaction distribution, indicating non-universality at Ising spin glass transitions. These results confirm conclusions drawn from numerical data for dimension 3.
Here is the first part of the summary of my work on random Ising model using real-space renormalization group (RSRG), also known as a Migdal-Kadanoff one. This approximate renormalization scheme was applied to the analysis thermodynamic properties of the model, and of probabilistic properties of a pair correlator, which is a fluctuating object in disordered systems. PACS numbers: 02.50.-r, 05.20.-y, 05.70.Fh, 64.60.ae, 75.10.-b, 75.10.Hk, 87.10.+e Keywords: statistical physics, magnetism, spin glass, neural networks
Frustration represents an essential feature in the behavior of magnetic materials when constraints on the microscopic Hamiltonian cannot be satisfied simultaneously. This gives rise to exotic phases of matter including spin liquids, spin ices, and stripe phases. Here we demonstrate an approach to understanding the microscopic effects of frustration by computing the phases of a 468-spin Shastry-Sutherland Ising Hamiltonian using a quantum annealer. Our approach uses mean-field boundary conditions to mitigate effects of finite size and defects alongside an iterative quantum annealing protocol to…
We study the role of Hamiltonian complexity in the performance of quantum annealers. We consider two general classes of annealing Hamiltonians: stoquastic ones, which can be simulated efficiently using the quantum Monte Carlo algorithm, and nonstoquastic ones, which cannot be treated efficiently. We implement the latter by adding antiferromagnetically coupled two-spin driver terms to the traditionally studied transverse-field Ising model, and compare their performance to that of similar stoquastic Hamiltonians with ferromagnetically coupled additional terms. We focus on a model of long-range I…
We consider the random field Ising model and show rigorously that the spin glass susceptibility at equilibrium is always bounded by the ferromagnetic susceptibility, and therefore that no spin glass phase can be present at equilibrium out of the ferromagnet critical line. When the magnetization is, however, fixed to values smaller than the equilibrium one, a glassy phase can exist, as we show explicitly on the Bethe lattice.
Dynamic nonlinear (cubic) susceptibility in quantum d-dimensional Ising spin glass with short-range interactions is investigated on the basis of quantum droplet model and quantum-mechanical nonlinear response theory. Nonlinear response depends on the tunneling rate for a droplet which regulates the strength of quantum fluctuations. It shows a strong dependence on the distribution of droplet free energies and on the droplet length scale average. Comparison with recent experiments on quantum spin glasses like disordered dipolar quantum Ising magnet is discussed.
We investigate the distributions of the link overlap, P(Q), in 3-dimensional Ising spin glasses. We use clustering methodology to identify a set of pairs of states from different Gibbs states, and calculate its contribution to P(Q). We find that the distribution over this set does not become trivial as the system size increases.
We argue that the chaotic temperature effect predicted in Ising spin glasses should be stronger when one considers continuous (XY, Heisenberg) kind of spins, due to bigger entropic fluctuations. We then discuss the behavior of 3d spin glasses using the Migdal-Kadanoff Renormalization Group for Ising and XY spins, where we show explicitly that the chaotic length scale, the length beyond which equilibrium configurations are completly reshuffled when one changes temperature, could be far smaller for XY than for Ising spins. These results could thus explain why experiments are seeing a stronger re…
We conjecture the existence of a relationship between frustration and the transition point at zero temperature of Ising spin glasses. The relation reveals that, in several Ising spin glass models, the concentration of ferromagnetic bonds is close to the critical concentration at zero temperature when the output of a function about frustration is equal to unity. The function is the derivative of the average number of frustrated plaquettes with respect to the average number of antiferromagnetic bonds. This relation is conjectured in Ising spin glasses with binary couplings on two-dimensional lat…
The field-cooled (FC) process on an Ising spin-glass model is investigated by a standard Monte Calro (MC) method on one hand, and the equilibrium magnetization of the same system is evaluated by the exchang MC method on the other hand. The two types of simulation reveal intriguing glassy (nonequilibrium) dynamic properties of the FC magnetization (FCM) of the system. Particularly, the FCM decrease is observed when the FC process is halted at a low temperater, although its value is smaller than the corresponding equilibrium value. We present a comprehensive interpretation of such peculiar pheno…
Nearest-neighbor-interaction Ising spin glasses are studied on three different hierarchical lattices, all of them belonging to the Wheatstone-Bridge family. It is shown that the spin-glass lower critical dimension in these lattices should be greater than 2.32. Finite-temperature spin-glass phases are found for a lattice of fractal dimension $D \approx 3.58$ (whose unit cell is obtained from a simple construction of a part of the cubic lattice), as well as for a lattice of fractal dimension close to five.
We consider the spatial correlation function of the two-dimensional Ising spin glass under out-equilibrium conditions. We pay special attention to the scaling limit reached upon approaching zero temperature. The field-theory of a non-interacting field makes a surprisingly good job at describing the spatial shape of the correlation function of the out-equilibrium Edwards-Anderson Ising model in two dimensions.
The sample-to-sample fluctuations of the free energy in finite-dimensional Ising spin glasses are calculated, using the replica method, from higher order terms in the replica number $n$. It is shown that the Parisi symmetry breaking scheme does not give the correct answers for these higher order terms. A modified symmetry breaking scheme with the same stability is shown to resolve the problem.
Identities and inequalities are proved for the order parameters, correlation functions and their derivatives of the Ising spin glass. The results serve as additional evidence that the ferromagnetic phase is composed of two regions, one with strong ferromagnetic ordering and the other with the effects of disorder dominant. The Nishimori line marks a crossover between these two regions.
We investigate the ferromagnetic-glassy transitions which separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice +-J (Edwards-Anderson) Ising model with bond distribution $P(J) = p δ(J - 1) + (1-p) δ(J + 1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly re…
We analyze the limits inherent to the inverse reconstruction of a pairwise Ising spin glass based on susceptibility propagation. We establish the conditions under which the susceptibility propagation algorithm is able to reconstruct the characteristics of the network given first- and second-order local observables, evaluate eventual errors due to various types of noise in the originally observed data, and discuss the scaling of the problem with the number of degrees of freedom.
Equilibrium numerical data on the three dimensional bimodal interaction Ising spin glass up to size L=48 show that corrections to scaling, which are known to be strong, behave in a non-monotonic manner with size. Extrapolation to the infinite size thermodynamic limit is difficult; however the large L data indicate that the ordering temperature Tc lies significantly higher than the values which have been estimated from previous numerical work limited to smaller sizes. In view of the present results it is at the least premature to conclude that the three dimensional bimodal and Gaussian Ising sp…
Extensive simulations are made of the spin glass susceptibility and correlation length in five dimension Ising Spin Glasses (ISGs) with Gaussian and bimodal interaction distributions. Once the transition temperature is accurately established using a standard criterion, critical exponents and correction terms can be readily estimated by extrapolating measurements made in the thermodynamic limit regime. The data show that the critical exponents of the susceptibility $γ$ and of the correlation length $ν$ depend on the form of the interaction distribution. This observation implies that quite gener…
The one-dimensional long-range Ising spin glass provides useful insights into the properties of finite dimensional spin glasses with short-range interactions. The defect energy renormalization group equations derived for it by Kotliar, Anderson and Stein have been re-examined and a new fixed point has been found. The fixed points which have previously been studied are found to be inappropriate. It is shown that the renormalization group equations themselves directly imply that the spin glass phase is replica symmetric, in agreement with droplet model expectations, when the range exponent $σ$ l…
By means of extensive computer simulations we analyze in detail the two dimensional $\pm J$ Ising spin glass with ferromagnetic next-nearest-neighbor interactions. We found a crossover from ferromagnetic to ``spin glass'' like order both from numerical simulations and analytical arguments. We also present evidences of a second crossover from the ``spin glass'' behavior to a paramagnetic phase for the largest volume studied.
