Spin systems with frustration and disorder are notoriously difficult to study, both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here, we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine.
Spin systems with frustration and disorder are notoriously difficult to study, both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here, we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine.
Spin glasses are experiencing a revival due to applications in quantum information\ntheory. In particular, they are the archetypal native benchmark problem for quantum annealing\nmachines. Furthermore, they find applications in fields as diverse as satisfiability,\nneural networks, and general combinatorial optimization problems. As such, developing\nand improving algorithms and methods to study these computationally complex systems\nis of paramount importance to many disciplines. This body of work attempts to attack\nthe problem of solving combinatorial optimization problems by simulating spin glasses\nfrom three sides: classical algorithm development, suggestions for quantum annealing\ndevice design, and improving measurements in realistic physical systems with inherent\nnoise. I begin with the introduction of a cluster algorithm based on Houdayer’s cluster algorithm\nfor two-dimensional Ising spin-glasses that is applicable to any space dimension\nand speeds up thermalization by several orders of magnitude at low temperatures where\nprevious algorithms have difficulty. I show improvement for the D-Wave chimera topology\nand the three-dimensional cubic lattice that increases with the size of the problem.\nOne consequence of adding cluster moves is that for problems with degenerate solutions,\nground-state sampling is improved. I demonstrate an ergodic algorithm to sample ground\nstates through the use of simple Monte Carlo with parallel tempering and cluster moves.\nIn addition, I present a non-ergodic algorithm to generate new solutions from a bank of\nknown solutions. I compare these results against results from quantum annealing utilizing\nthe D-Wave Inc. quantum annealing device. Finally, I present an algorithm for improving\nthe recovery of ground-state solutions from problems with noise by using thermal fluctuations\nto infer the correct solution at the Nishimori temperature. While this method has been\ndemonstrated analytically and numerically for trivial ferromagnetic and Gaussian distributions, a useful metric for more complex Gaussian distributions with added Gaussian noise\nis unavailable. We show improved recovery of numerical solutions on the chimera graph\nwith a ferromagnetic distribution and added Gaussian noise. Next, I direct my focus to the\ndesign of future generations of quantum annealers. The first design is the two-dimensional\nsquare-lattice bimodal spin glass with next-nearest ferrromagnetic interactions proposed\nby Lemke and Campbell claimed to exhibit a finite-temperature spin-glass state for a particular\nrelative strength of the next-nearest to nearest neighbor interactions. Our results\nfrom finite-temperature simulations show the system is in a paramagnetic state in the thermodynamic\nlimit, thus not useful for quantum annealing device designs that would benefit\nfrom a spin-glass phase transition. The second design is the diluted next-nearest neighbor\nIsing spin-glass with Gaussian interactions in an attempt to improve the estimation\nof critical parameter with smaller system sizes by implementing averaging of observables\nover different graph dilutions. To date, this model has shown no improvement. Finally,\nI make suggestions for the choice of distributions of interactions that are robust to noise\nand present a method for using previously unaccessible continuous distributions. I begin\nwith showing the best-case performance of quantum annealing devices. I show results for\nthe resilience, the probability that the ground-state solution has changed due to inherent\nanalog noise in the device, and present strategies for developing robust instance classes.\nThe analog noise is also detrimental to interactions chosen from continuous distributions.\nUsing Gaussian quadratures, I present a method for discretizing continuous distributions\nto reduce noise effects. Simulations on the D-Wave show that the average residual of\nthe ground-state energy with the true ground-state energy is calculated and shown to be\nsmaller in the case of the discrete distribution.
Spin glasses are disordered magnetic systems that have frustration. The most interesting feature of the physical properties of spin glasses is the complex structure of phase space that is caused by the competition between the quenched ferromagnetic and antiferromagnetic interactions of spin pairs. The popularity of spin glasses is two-fold. First of all, the extremely complicated energy landscapes built from the frustration among spin interactions gives spin glasses new properties that have never been seen in other ordered systems. One of these properties is a new transition between the paramagnetic phase and a disordered phase, the spin-glass phase. This new phenomenon triggers scientists' enthusiasm for deeply understanding the physical foundations and mathematical description for disordered systems such as spin glasses. Secondly, because analytical methods can no longer offer enough support for studying disordered systems, scientists use numerical tools like Markov-chain Monte Carlo to simulate the thermodynamics of the systems of interest as well as statistical methods to process big data generated from the simulations. Therefore, the research on such complex systems introduces a new paradigm to statistical physics and also opens a window into an overlapping area between computational physics and information science, which makes an interdisciplinary study necessary and helpful. \n\n\n\n I devote my research to this new interdisciplinary area by seeking new connections in both models and algorithms. This dissertation consists of two themes. The first theme is about the research of spin glasses, including a study on the universality of a diluted spin-glass model and the development of algorithms by introducing new tools from artificial intelligence and extending the current algorithms for more general use. I demonstrate that to detect the phase transition in a spin-glass model we can train convolutional neural networks with a different spin-glass model. Then, use the convolutional neural networks to precisely predict the phase transition of the model. I also show that by training the convolutional neural networks with poisoned data, the results can be very unpredictable. Next, I introduce research in modifying the isoenergetic cluster moves for highly connected topologies in spin glasses. The numerical results show that isoenergetic cluster moves can only be effective for graphs with low connectivities due to the relatively small percolation thresholds in topologies with high connectivities and the trade-off between the cluster size and acceptance rate forced by the detailed balance condition. To conclude the first theme, I study the universality of the two dimensional Ising model with bond dilution. This project is inspired by a debate in the grey area that is not covered by Harris criterion: the universality of a two-dimensional Ising model with disorder can not be theoretically explained. The simulation results for the two-dimensional Ising model with bond dilution show that the critical exponent of the correction to the correlation length does not change with disorder, which supports the strong universality scenario. The second theme of this dissertation is the study of the Boolean satisfiability problem through the use of Ising spin form. The Boolean satisfiability problem is the problem of determining if there exists an assignment of Boolean variables that satisfies a given Boolean formula. Considering that there are rarely global algorithms for the Boolean satisfiability problem and that the NP-complete Ising spin glass is written in Boolean variables as well, I use Monte Carlo simulations to find the optimal solutions for the Boolean satisfiability problem by implementing the Metropolis algorithm combined with the Parallel tempering algorithm. The advantage of this method is that the ergodicity of the thermodynamic procedure makes this algorithm global, which can help search for optimal solutions through the entire configuration space. The numerical results show that this physical algorithm has an obvious advantage over the current state-of-art algorithms on a derivative problem, the Boolean maximum satisfiability problem. Finally, I introduce my research on the Boolean satisfiability problem-based membership filter. Since the physical approach has the ability of globally seeking solutions it can help find uncorrelated solutions for the Boolean satisfiability problem-based membership filter, which is the key to improving efficiency for this new filter. Furthermore, I improve the efficiency of the filter by turning the Boolean satisfiability instance into a not-all-equal form to guarantee that the solutions found by our algorithm, Borealis, are uncorrelated. The numerical results show that the efficiency of Boolean satisfiability-based filter can be significantly improved while keeping the false positive rate low.
Abstract We present an algorithm for the simulation of three-dimensional spin glasses deep in the low-temperature phase: Parallel Tempering enhanced with Houdayer moves and with an entropic reservoir (PTHR). Although differences with the standard Houdayer algorithm are small, PTHR allows us to equilibrate a large number of samples of L = 16 lattices with Gaussian couplings for temperatures T ⩾ 0.2 . We show that the computational complexity displays better size scaling than standard Parallel Tempering. For finite sizes, our method outperforms other cluster algorithms by a speedup factor of around 64. In close analogy with standard Parallel Tempering, PTHR’s computational complexity strongly relates to temperature chaos.
Finding the ground state of Ising spin glasses is notoriously difficult due to disorder and frustration. Often, this challenge is framed as a combinatorial optimization problem, for which a common strategy employs simulated annealing, a Monte Carlo (MC)-based algorithm that updates spins one at a time. Yet, these localized updates can cause the system to become trapped in local minima. Cluster algorithms (CAs) were developed to address this limitation and have demonstrated considerable success in studying ferromagnetic systems; however, they tend to encounter percolation issues when applied to generic spin glasses. In this work, we introduce a novel CA designed to tackle these challenges by leveraging precomputed two-point correlations, aiming solve combinatorial optimization problems in the form of Max-Cut more efficiently. In our approach, clusters are formed probabilistically based on these correlations. Various classical and quantum algorithms can be employed to generate correlations that embody information about the energy landscape of the problem. By utilizing this information, the algorithm aims to identify groups of spins whose simultaneous flipping induces large transitions in configuration space with high acceptance probability - even at low energy levels - thereby escaping local minima more effectively. Notably, clusters generated using correlations from the Quantum Approximate Optimization Algorithm exhibit high acceptance rates at low temperatures. These acceptance rates often increase with circuit depth, accelerating the algorithm and enabling more efficient exploration of the solution space.
The main subjects of this dissertation are spin glass applications in other disciplines and spin glass algorithms. Spin glasses are magnetic systems with disorder and frustration, and the essential physics of spin glasses lies not in the details of their microscopic interactions but rather in the competition between quenched ferromagnetic and antiferromagnetic interactions. Concepts that arose in the study of spin glasses have led to applications in areas as diverse as computer science, biology, and finance, as well as a variety of others.\n\nIn the first part of this dissertation I study the equilibrium and non-equilibrium properties of Boolean decision problems with competing interactions on scale-free networks in an external bias (a magnetic field). First, I perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and I show that the system has a spin-glass phase in a field, i.e., it exhibits a de Almeida–Thouless line. Then I study avalanche distributions when the system is driven by a field at zero temperature to test whether the system displays self-organized criticality. The numerical results suggest that avalanches (damage) can spread across the entire system with nonzero probability when the decay exponent of the interaction degree is less than or equal to 2, i.e., that Boolean decision problems on scale-free networks with competing interactions can be fragile when the system is not in thermal equilibrium.\n\nIn the second part of this dissertation I discuss the best-case performance of quantum annealers on native spin-glass benchmarks, i.e., how chaos can affect success probabilities. We perform classical parallel-tempering Monte Carlo simulations of the archetypal benchmark problem, an Ising spin glass, on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, I study the best-case resilience, or the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We compute the upper-bound success probabilities for different instance classes based on these simple error models, and I present strategies for developing robust and hard benchmark instances.\n\nIn the third part of this dissertation I present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by several orders of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits (improved thermalization and achievement of more equiprobable sampling of ground states) of the isoenergetic cluster moves in two and three space dimensions, as well as in the nonplanar Chimera topology found in the D-Wave quantum annealing machine.\n\nFinally, I study the thermodynamic properties of the two-dimensional Edwards-Anderson Ising spin-glass model on a square lattice using the tensor renormalization group method based on a higher-order singular-value decomposition. Our estimates of the partition function without a high precision data type lead to negative values at very low temperatures, thus illustrating that the method can not be applied to frustrated magnetic systems.
Large-scale quantum annealing dynamics of Ising spin glasses were recently implemented on D-Wave's Advantage2 system on a range of lattices. After extensive comparison with existing numerical methods, these experiments were claimed to be beyond the reach of classical computation. Here, we simulated these spin-glass models with lattice-specific tensor networks, using belief propagation (BP) to keep up with the entanglement generated during the time evolution and then extracting expectation values with more sophisticated variants of BP. We found that state-of-the-art accuracies could be achieved with modest computational resources. Moreover, our results are scalable in both two and three dimensions, which we leveraged to verify universal Kibble-Zurek physics on systems involving hundreds of qubits.