We release a set of GPU programs for the study of the Quantum (S=1/2) Spin Glass on a square lattice, with binary couplings. The library contains two main codes: MCQSG (that carries out Monte Carlo simulations using both the Metropolis and the Parallel Tempering algorithms, for the problem formulated in the Trotter-Suzuki approximation), and EDQSG (that obtains the extremal eigenvalues of the Transfer Matrix using the Lanczos algorithm). EDQSG has allowed us to diagonalize transfer matrices with size up to 236×236. From its side, MCQSG running on four NVIDIA A100 cards delivers a sub-picosecond time per spin-update, a performance that is competitive with dedicated hardware. We include as well in our library GPU programs for the analysis of the spin configurations generated by MCQSG. Finally, we provide two auxiliary codes: the first generates the lookup tables employed by the random number generator of MCQSG; the second one simplifies the execution of multiple runs using different input data. Program Title: QISG Suite CPC Library link to program files: https://doi.org/10.17632/g97sn2t8z2.1 Licensing provisions: MIT Programming language: CUDA-C Nature of problem: The critical properties of quantum disordered systems are known only in a few, simple, cases whereas there is a growing interest in gaining a better understanding of their behaviour due to the potential application of quantum annealing techniques for solving optimization problems. In this context, we provide a suite of codes, that we have recently developed, to the purpose of studying the 2D Quantum Ising Spin Glass. Solution method: We provide a highly tuned multi-GPU code for the Montecarlo simulation of the 2D QISG based on a combination of Metropolis and Parallel Tempering algorithms. Moreover, we provide a code for the evaluation of the eigenvalues of the transfer matrix of the 2D QISG for size up to L=6. The eigenvalues are computed by using the classic Lanczos algorithm that, however, relies on a custom multi-GPU-CPU matrix-vector product that speeds-up dramatically the execution of the algorithm.
We release a set of GPU programs for the study of the Quantum (S=1/2) Spin Glass on a square lattice, with binary couplings. The library contains two main codes: MCQSG (that carries out Monte Carlo simulations using both the Metropolis and the Parallel Tempering algorithms, for the problem formulated in the Trotter-Suzuki approximation), and EDQSG (that obtains the extremal eigenvalues of the Transfer Matrix using the Lanczos algorithm). EDQSG has allowed us to diagonalize transfer matrices with size up to 236×236. From its side, MCQSG running on four NVIDIA A100 cards delivers a sub-picosecond time per spin-update, a performance that is competitive with dedicated hardware. We include as well in our library GPU programs for the analysis of the spin configurations generated by MCQSG. Finally, we provide two auxiliary codes: the first generates the lookup tables employed by the random number generator of MCQSG; the second one simplifies the execution of multiple runs using different input data. Program Title: QISG Suite CPC Library link to program files: https://doi.org/10.17632/g97sn2t8z2.1 Licensing provisions: MIT Programming language: CUDA-C Nature of problem: The critical properties of quantum disordered systems are known only in a few, simple, cases whereas there is a growing interest in gaining a better understanding of their behaviour due to the potential application of quantum annealing techniques for solving optimization problems. In this context, we provide a suite of codes, that we have recently developed, to the purpose of studying the 2D Quantum Ising Spin Glass. Solution method: We provide a highly tuned multi-GPU code for the Montecarlo simulation of the 2D QISG based on a combination of Metropolis and Parallel Tempering algorithms. Moreover, we provide a code for the evaluation of the eigenvalues of the transfer matrix of the 2D QISG for size up to L=6. The eigenvalues are computed by using the classic Lanczos algorithm that, however, relies on a custom multi-GPU-CPU matrix-vector product that speeds-up dramatically the execution of the algorithm.
We present an open-source software package, VAN-DAMME (Versatile Approaches to Numerically Design, Accelerate, and Manipulate Magnetic Excitations), for massively-parallelized quantum optimal control (QOC) calculations of multi-qubit systems. To enable large QOC calculations, the VAN-DAMME software package utilizes symmetry-based techniques with custom GPU-enhanced algorithms. This combined approach allows for the simultaneous computation of hundreds of matrix exponential propagators that efficiently leverage the intra-GPU parallelism found in high-performance GPUs. In addition, to maximize the computational efficiency of the VAN-DAMME code, we carried out several extensive tests on data layout, computational complexity, memory requirements, and performance. These extensive analyses allowed us to develop computationally efficient approaches for evaluating complex-valued matrix exponential propagators based on Padé approximants. To assess the computational performance of our GPU-accelerated VAN-DAMME code, we carried out QOC calculations of systems containing 10 - 15 qubits, which showed that our GPU implementation is 18.4× faster than the corresponding CPU implementation. Our GPU-accelerated enhancements allow efficient calculations of multi-qubit systems, which can be used for the efficient implementation of QOC applications across multiple domains. Program summary Program Title: VAN-DAMME CPC Library link to program files:: https://doi.org/10.17632/zcgw2n5bjf.1 Licensing provisions: GNU General Public License 3 Programming language: C++ and CUDA Nature of problem: The VAN-DAMME software package utilizes GPU-accelerated routines and new algorithmic improvements to compute optimized time-dependent magnetic fields that can drive a system from a known initial qubit configuration to a specified target state with a large (≈1) transition probability . Solution method: Quantum control , GPU acceleration, analytic gradients, matrix exponential, and gradient ascent optimization.
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.
We present a GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians that targets workstation hardware rather than distributed-memory clusters. The Hamiltonian action is evaluated matrix-free in a row-wise gather formulation. We introduce and compare two state-to-index strategies: a compressed lookup table (CLT), which reduces lookup memory by a factor of 16 relative to a full table while retaining a fixed, branch-light access pattern, and a GPU-adapted combinatorial-ranking scheme that removes the lookup table altogether. Numerical tests against FP64 CPU references show that FP32 GPU arithmetic changes heat capacities and magnetic susceptibilities by amounts several orders of magnitude below the stochastic uncertainty of the FTLM trace estimator at typical sample sizes. Benchmarks show speedups of up to about one order of magnitude over optimized multicore CPU calculations and enable Hilbert-space sectors of dimension ~10^8 on a single workstation GPU. The MATLAB/CUDA implementation, including example input files and benchmark scripts, is openly available at https://github.com/ghasdeke/ftlm-gpu (archived at DOI: 10.5281/zenodo.20378647) under the Apache-2.0 license.
The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap Δ encountered at quantum phase transitions. We investigate the scaling of Δ with system size N for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as N increases. This indicates that the unfavorable super-algebraic scaling of Δ, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to Δ∝N−1/3. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.
This thesis focuses on the development, application, and high-performance implementation of numerical methods for simulating quantum many-body systems. In the first chapter, I provide an overview of several most successful conventional approaches for simulating quantum many-body systems, along with their current limitations. Then, the basic properties of the quantum spin model, which will be the subject of study in the following chapters, and its ground state are briefly summarized. Additionally, I discuss the background and the motivations of the research projects carried out during my PhD research. Chapter 2 introduces two numerical algorithms that are extensively used in this thesis, which are exact diagonalization (ED) and variational Monte Carlo. I outline the process of constructing a Hamiltonian matrix and employing iterative methods for exact diagonalization. Subsequently, I introduce the mechanism of the four important components in the ``standard" variational Monte Carlo that are widely used in the most recent research. In Chapter 3, we propose a symmetry-projected variational ansatz in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics,and achieve state-of-the-art accuracy in the representation of the ground state. In Chapter 4, we present a variational approach based on a Chebyshev expansion of the spectral function, and each Chebyshev polynomial is represented by a neural network quantum state. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave functions." We present results for the one-dimensional and two-dimensional Heisenberg models on the square lattice and compare them with those obtained by other methods in the literature. In Chapter 5, we describe a method to carry out thermodynamic simulations of quantum many-body models using typicality and variational representations of quantum states. Minimally entangled typical thermal states (METTS) are a construction that allows one to solve for the imaginary time evolution of quantum many-body systems. By using wave functions that are weakly entangled, one can take advantage of efficient representations in the form of matrixproduct states. We generalize these ideas to arbitrary variational wave functions and we focus, as an illustration, on the particular case of restricted Boltzmann machines. We demonstrate the application of the method to quantum spin systems in one and two spatial dimensions. In Chapter 6, we study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band. In the final chapter, Chapter 7, we discuss the implementation of architecture-aware optimizations for spin dynamics simulations on x86 CPUs and Nvidia GPUs. The time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. We propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel to reduce memory latency. On Nvidia GPU, we perform the on-the-fly calculation in the epilogue of the GEMM subroutine to avoid saving intermediate results on global memory, which significantly reduces memory usage and achieves better performance than cuBLAS.--Author's abstract
We present jQMC, a Python-based computational package for {\it ab initio} Quantum Monte Carlo (QMC) simulations, designed for modern GPU-accelerated computing environments. jQMC implements two well-established QMC algorithms: Variational Monte Carlo (VMC) and the lattice-regularized variant of Diffusion Monte Carlo (LRDMC). The employed wave function is a Jastrow factor combined with the antisymmetrized geminal power with spin-singlet and spin-triplet pairings, which contains the single Slater determinant as its special lowest-rank case. The wave function can be initialized from external Hartree-Fock/Density Functional Theory calculations through the TREX-IO library (a common wave-function format across electronic-structure packages) and optimized by stochastic reconfiguration and linear-method energy minimization. One of the prominent features of jQMC is its use of JAX, which enables automatic differentiation for wave function optimization and atomic force calculations, and allows the main QMC algorithms to be Just-In-Time (JIT) compiled and portable across CPU and GPU. jQMC is vectorized over walkers at the top level of the QMC algorithms, providing efficient intra-GPU~(CPU) vectorization. The multi-GPU~(CPU) parallelization is also supported through MPI and JAX sharding. To assess the practical performance of this implementation, we benchmarked jQMC performance on NVIDIA GPUs (A100 and H100) and analyzed CUDA kernels. For the test cases analyzed here, with system sizes up to 160 electrons, the current version of jQMC is faster than TurboRVB, a Fortran90 code implementing the same algorithms and wave functions, once jQMC is run on GPUs. In terms of wall-time, the gain can reach an order of magnitude for VMC, while it is more moderate for LRDMC.