Spin glasses are, in essence, simply systems of magnetic moments with disordered interactions. Despite this basic premise, spin glasses have become relevant in a wide variety of fields, including condensed matter physics, complexity and optimization theory, error correction, quantum computing, machine learning, and even neuroscience. The pervasiveness of spin glasses stems from the fact that they are paradigmatic examples of how global complexity emerges from disordered interactions between a large number of basic constituents. The inclusion of quantum effects in spin glasses creates an even richer picture, due to the interplay between the energy barriers arising from disorder and quantum fluctuations which allow for tunneling. This thesis initially focuses on quantum glasses and how their initial states explore their Hilbert spaces. Two complementary approaches are used. One is comparison of the spectral statistics, particularly the spectral form factor, with the spectral statistics of random matrix ensembles, a hallmark of quantum chaotic behavior. The other approach uses the Thouless-Anderson-Palmer mean-field equations to determine the number of local minima which appear in the free energy landscape. This thesis then applies methods from spin glass theory to the field of machine learning. Neural networks, like spin glasses, are systems in which remarkable behavior emerges from interactions between basic constituents. By studying the phase diagrams of spin models analogous to snapshots of neural networks throughout their training process, insight is gained into the mechanisms of learning. Finally, this thesis investigates how quantum effects can lead to advantages in machine learning. It is shown, through classical simulation and experiments on both trapped-ion and superconducting quantum computing hardware, that a natural quantization of a classical neural network can demonstrate superior generalization performance due to the injection of quantum uncertainty.
Spin glasses are, in essence, simply systems of magnetic moments with disordered interactions. Despite this basic premise, spin glasses have become relevant in a wide variety of fields, including condensed matter physics, complexity and optimization theory, error correction, quantum computing, machine learning, and even neuroscience. The pervasiveness of spin glasses stems from the fact that they are paradigmatic examples of how global complexity emerges from disordered interactions between a large number of basic constituents. The inclusion of quantum effects in spin glasses creates an even richer picture, due to the interplay between the energy barriers arising from disorder and quantum fluctuations which allow for tunneling. This thesis initially focuses on quantum glasses and how their initial states explore their Hilbert spaces. Two complementary approaches are used. One is comparison of the spectral statistics, particularly the spectral form factor, with the spectral statistics of random matrix ensembles, a hallmark of quantum chaotic behavior. The other approach uses the Thouless-Anderson-Palmer mean-field equations to determine the number of local minima which appear in the free energy landscape. This thesis then applies methods from spin glass theory to the field of machine learning. Neural networks, like spin glasses, are systems in which remarkable behavior emerges from interactions between basic constituents. By studying the phase diagrams of spin models analogous to snapshots of neural networks throughout their training process, insight is gained into the mechanisms of learning. Finally, this thesis investigates how quantum effects can lead to advantages in machine learning. It is shown, through classical simulation and experiments on both trapped-ion and superconducting quantum computing hardware, that a natural quantization of a classical neural network can demonstrate superior generalization performance due to the injection of quantum uncertainty.
Neural network approaches to machine learning problems rely on statistical frameworks that are designed to generalize to many situations, whereas traditional physics-based approaches to quantum problems stem from specific physical laws that govern the behavior of certain classes of systems. While quantum physics and machine learning have been largely disparate fields for most of their history, their interplay has gained considerable attention in recent years, primarily driven by the premise that the most powerful computational models are those whose foundations stem from both physical and statistical principles. Two overarching strategies in this area include building physics-based machine learning models from the ground up, or alternatively, looking for hidden ways that statistical models include aspects of physical systems. Many-body quantum systems turn out to be a useful testing ground for both of these approaches, due to the complex behavior that arises from simple fundamental rules. Here, we present several ways that neural networks can represent many-body spin states, which enables the exploration into the underlying nature of neural networks, as well as gaining insight into how effective different classes of models are for simulating physical systems. The main focus areas of this thesis are (1) connecting convolutional neural networks to maximum-entropy distributions when applied to spin systems, (2) using generative models to gain physical insights into experimental tomographic data, and (3) applying inverse reinforcement learning to probe the nature of agent-based quantum circuit design.
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.
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.
Variational neural network models have achieved remarkable success in solving ground-state problems of quantum many-body systems. However, addressing classical and quantum spin glasses remains challenging, as disorder and energy frustration give rise to an exponentially large number of local energy minima separated by high-energy barriers, hindering the efficiency of conventional Metropolis-based Monte Carlo methods. To bridge this gap, we introduce Deep Boltzmann Quantum States, a class of neural quantum states inspired by deep Boltzmann machines that inherit efficient block Gibbs sampling. We also propose two key advances in the training algorithm. Firstly, we combine natural-gradient updates with state-of-the-art stochastic optimizers. Secondly, we gradually tune the hardness of the problem Hamiltonian by interpolating from an easy to a hard regime, without the need to closely approximate the instantaneous adiabatic state at intermediate times. We match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. We also solve instances of the NP-hard Job Shop Scheduling Problem exceeding the current limitations of quantum annealing hardware. To summarize, deep neural architectures with efficient global update rules and trained within an annealing-like scheme, provide a powerful framework for solving real-world hard combinatorial optimization and for investigating disordered quantum many-body systems.
This dissertation investigates the role of disorder in quantum many-body systems, focusing on strange metals and spin glass systems. For strange metals, we analyze the low-frequency spectra of complex Sachdev-Ye-Kitaev (SYK) models at arbitrary densities, combining analytical and numerical methods. Building on these insights, we study the overdoped random t-J model, demonstrating that strange metal behavior persists across the entire doping range. We also show that the critical metal develops an instability to a low-doping spin-glass phase, and compute the critical doping value. Furthermore, we explore the chaotic properties of a distinct strange metal model—the large N theory of a critical Fermi surface—and establish that it exhibits maximal quantum chaos. For spin glass systems, we examine the low-energy spectrum and Parisi replica-symmetry-breaking function in the quantum Ising model with infinite-range random exchange interactions with the presence of transverse and longitudinal fields. In the spin glass phase, we find that the local spin spectra are gapless and provide a detailed analysis of the phase diagram. Additionally, we investigate the quantum spherical p-rotor model in an external field, showing that the spin glass phase remains robust across a wide range of field strengths. The potential experimental applications of these findings are also discussed.
Spin glasses are disordered magnets with random interactions that are, generally, in conflict with each other. Finding the ground states of spin glasses is not only essential for understanding the nature of disordered magnets and many other physical systems, but also useful to solve a broad array of hard combinatorial optimization problems across multiple disciplines. Despite decades-long efforts, an algorithm with both high accuracy and high efficiency is still lacking. Here we introduce DIRAC - a deep reinforcement learning framework, which can be trained purely on small-scale spin glass instances and then applied to arbitrarily large ones. DIRAC displays better scalability than other methods and can be leveraged to enhance any thermal annealing method. Extensive calculations on 2D, 3D and 4D Edwards-Anderson spin glass instances demonstrate the superior performance of DIRAC over existing methods. The presented framework will help us better understand the nature of the low-temperature spin-glass phase, which is a fundamental challenge in statistical physics. Moreover, the gauge transformation technique adopted in DIRAC builds a deep connection between physics and artificial intelligence. In particular, this opens up a promising avenue for reinforcement learning models to explore in the enormous configuration space, which would be extremely helpful to solve many other hard combinatorial optimization problems.