Population annealing: Theory and application in spin glasses

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.Comment: 16 pages, 10 figures, 4 table

Population Annealing Monte Carlo Studies of Ising Spin Glasses

Spin glasses are spin-lattice models with quenched disorder and frustration. The mean field long-range Sherrington-Kirkpatrick (SK) model was solved by Parisi and displays replica symmetry breaking (RSB), but the more realistic short-range Edwards-Anderson (EA) model is still not solved. Whether the EA spin glass phase has many pairs of pure states as described by the RSB scenario or a single pair of pure states as described by two-state scenarios such as the droplet/scaling picture is not known yet. Rigorous analytical calculations of the EA model are not available at present and efficient numerical simulations of spin glasses are crucial in making progresses in the field. In addition to being a prototypical example of a classical disordered system with many interesting equilibrium as well as nonequilibrium properties, spin glasses are of great importance across multiple fields from neural networks, various combinatorial optimization problems to benchmark tests of quantum annealing machines. Therefore, it is important to gain a better understanding of the spin glass models. In an effort to do so, our work has two main parts, one is to develop an efficient algorithm called population annealing Monte Carlo and the other is to explore the physics of spin glasses using thermal boundary conditions. We present a full characterization of the population annealing algorithm focusing on its equilibration properties and make a systematic comparison of population annealing with two well established simulation methods, parallel tempering Monte Carlo and simulated annealing Monte Carlo. We show numerically that population annealing is similar in performance to parallel tempering, each has its own strengths and weaknesses and both algorithms outperform simulated annealing in combinatorial optimization problems. In thermal boundary conditions, all eight combinations of periodic vs antiperiodic boundary conditions in the three spatial directions appear in the ensemble with their respective Boltzmann weights, thus minimizing finite-size effects due to domain walls. With thermal boundary conditions and sample stiffness extrapolation, we show that our data is consistent with a two-state picture, not the RSB picture for the EA model. Thermal boundary conditions also provides an elegant way to study the phenomena of temperature chaos and bond chaos, and our results are again in agreement with the droplet/scaling scenario

Unconstrained Tree Tensor Network: An adaptive gauge picture for enhanced performance

We introduce a variational algorithm to simulate quantum many-body states based on a tree tensor network ansatz which releases the isometry constraint usually imposed by the real-space renormalization coarse-graining: This additional numerical freedom, combined with the loop-free topology of the tree network, allows one to maximally exploit the internal gauge invariance of tensor networks, ultimately leading to a computationally flexible and efficient algorithm able to treat open and periodic boundary conditions on the same footing. We benchmark the novel approach against the 1D Ising model in transverse field with periodic boundary conditions and discuss the strategy to cope with the broken translational invariance generated by the network structure. We then perform investigations on a state-of-the-art problem, namely the bilinear-biquadratic model in the transition between dimer and ferromagnetic phases. Our results clearly display an exponentially diverging correlation length and thus support the most recent guesses on the peculiarity of the transition.Comment: 11 pages, 13 figure

Reentrant Behavior of the Spinodal Curve in a Nonequilibrium Ferromagnet

The metastable behavior of a kinetic Ising--like ferromagnetic model system in which a generic type of microscopic disorder induces nonequilibrium steady states is studied by computer simulation and a mean--field approach. We pay attention, in particular, to the spinodal curve or intrinsic coercive field that separates the metastable region from the unstable one. We find that, under strong nonequilibrium conditions, this exhibits reentrant behavior as a function of temperature. That is, metastability does not happen in this regime for both low and high temperatures, but instead emerges for intermediate temperature, as a consequence of the non-linear interplay between thermal and nonequilibrium fluctuations. We argue that this behavior, which is in contrast with equilibrium phenomenology and could occur in actual impure specimens, might be related to the presence of an effective multiplicative noise in the system.Comment: 7 pages, 4 figures; Final version to appear in Phys. Rev. E; Section V has been revise

Scalable Emulation of Sign-Problem−-Free Hamiltonians with Room Temperature p-bits

The growing field of quantum computing is based on the concept of a q-bit which is a delicate superposition of 0 and 1, requiring cryogenic temperatures for its physical realization along with challenging coherent coupling techniques for entangling them. By contrast, a probabilistic bit or a p-bit is a robust classical entity that fluctuates between 0 and 1, and can be implemented at room temperature using present-day technology. Here, we show that a probabilistic coprocessor built out of room temperature p-bits can be used to accelerate simulations of a special class of quantum many-body systems that are sign-problem$-$free or stoquastic, leveraging the well-known Suzuki-Trotter decomposition that maps a $d$-dimensional quantum many body Hamiltonian to a $d$+1-dimensional classical Hamiltonian. This mapping allows an efficient emulation of a quantum system by classical computers and is commonly used in software to perform Quantum Monte Carlo (QMC) algorithms. By contrast, we show that a compact, embedded MTJ-based coprocessor can serve as a highly efficient hardware-accelerator for such QMC algorithms providing several orders of magnitude improvement in speed compared to optimized CPU implementations. Using realistic device-level SPICE simulations we demonstrate that the correct quantum correlations can be obtained using a classical p-circuit built with existing technology and operating at room temperature. The proposed coprocessor can serve as a tool to study stoquastic quantum many-body systems, overcoming challenges associated with physical quantum annealers.Comment: Fixed minor typos and expanded Appendi

Monte-Carlo simulation of supercooled liquids using a self-consistent local temperature

We combine Creutz energy conservation with Kawasaki spin exchange to simulate the microcanonical dynamics of a system of interacting particles. Relaxation occurs via Glauber spin-flip activation using a self-consistent temperature. Heterogeneity in the dynamics comes from finite-size constraints on the spin exchange that yield a distribution of correlated regions. The simulation produces a high-frequency response that can be identified with the boson peak, and a lower-frequency peak that contains non-Debye relaxation and non-Arrhenius activation, similar to the primary response of supercooled liquids.Comment: 16 pages, 4 figure

GPU accelerated population annealing algorithm

Population annealing is a promising recent approach for Monte Carlo simulations in statistical physics, in particular for the simulation of systems with complex free-energy landscapes. It is a hybrid method, combining importance sampling through Markov chains with elements of sequential Monte Carlo in the form of population control. While it appears to provide algorithmic capabilities for the simulation of such systems that are roughly comparable to those of more established approaches such as parallel tempering, it is intrinsically much more suitable for massively parallel computing. Here, we tap into this structural advantage and present a highly optimized implementation of the population annealing algorithm on GPUs that promises speed-ups of several orders of magnitude as compared to a serial implementation on CPUs. While the sample code is for simulations of the 2D ferromagnetic Ising model, it should be easily adapted for simulations of other spin models, including disordered systems. Our code includes implementations of some advanced algorithmic features that have only recently been suggested, namely the automatic adaptation of temperature steps and a multi-histogram analysis of the data at different temperatures.Comment: 12 pages, 3 figures and 5 tables, code at https://github.com/LevBarash/PAisin