Chebyshev recursion methods: Kernel polynomials and maximum entropy

Two Chebyshev recursion methods are presented for calculations with very large sparse Hamiltonians, the kernel polynomial method (KPM) and the maximum entropy method (MEM). They are applicable to physical properties involving large numbers of eigenstates such as densities of states, spectral functions, thermodynamics, total energies for Monte Carlo simulations and forces for tight binding molecular dynamics. this paper emphasizes efficient algorithms.

Kernel Polynomial Approximations for Densities of States and Spectral Functions

The simulation of lattice model systems for quantum materials is one of the most important approaches to understand quantum properties of matter in condensed matter physics. The main task in the simulation is to diagonalize a Hamiltonian matrix for the system and evaluate the electronic density of energy states. Kernel polynomial method (KPM) is one of the promising simulation methods. Because KPM contains a fine-grain recursive part in the algorithm, it is hard to parallelize it under the thread level parallelism such as on a supercomputer or a cluster computer. This paper focuses on methods to parallelize KPM on a massively parallel environment of GPU, aiming to achieve high parallelism for more speedups than the recent CPUs. This paper proposes two implementation methods called the full map and the sliding window methods, and evaluates the performances in the recent GPU platform. To enlarge available simulation sizes and at the same time to enhance the performance, this paper also describes additional optimization techniques depending on the GPU architecture.

The modeling of complex service systems entails capturing many sub-components of the system, and the dependencies that exist among them in the form of a joint probability distribution. Two common methods for constructing joint probability distributions from experts using partial information include maximum entropy methods and copula methods. In this paper we explore the performance of these methods in capturing the dependence between random variables using correlation coefficients and lower-order pairwise assessments. We focus on the case of discrete random variables, and compare the performance of these methods using a Monte Carlo simulation when the variables exhibit both independence and non-linear dependence structures. We show that the maximum entropy method with correlation coefficients and the Gaussian copula method perform similarly, while the maximum entropy method with pairwise assessments performs better particularly when the variables exhibit non-linear dependence.

In this paper, we investigated the possibility of using the magnetic Laplacian to characterize directed networks. We address the problem of characterization of network models and perform the inference of the parameters used to generate these networks under analysis. Many interesting results are obtained, including the finding that the community structure is related to rotational symmetry in the spectral measurements for a type of stochastic block model. Due the hermiticity property of the magnetic Laplacian we show here how to scale our approach to larger networks containing hundreds of thousands of nodes using the Kernel Polynomial Method (KPM), a method commonly used in condensed matter physics. Using a combination of KPM with the Wasserstein metric, we show how we can measure distances between networks, even when these networks are directed, large, and have different sizes, a hard problem that cannot be tackled by previous methods presented in the literature.

We investigate spectral properties of polaronic excitations within the framework of an analog quantum simulator based on inductively coupled superconducting transmon qubits and microwave resonators. This system emulates a lattice model that describes a nonlocal coupling of an itinerant spinless-fermion excitation to dispersionless (Einstein-type) phonons through the Peierls and breathing-mode interaction mechanisms. The model is characterized by a sharp, level-crossing transition at a critical value of the effective excitation-phonon coupling strength; above the transition point, the ground state of this model corresponds to a heavily dressed (small-polaron) excitation. Using the kernel-polynomial method, we evaluate the momentum-frequency resolved spectral function of this system for a broad range of parameters. In particular, we underscore the ramifications of the fact that the zero-quasimomentum Bloch state of a bare excitation represents the exact eigenstate of the Hamiltonian of this system for an arbitrary excitation-phonon coupling strength. We also show that -- based on the numerically evaluated spectral function and its well-known relation with the survival probability of the initial, bare-excitation Bloch state (the Loschmidt echo) -- one can make predictions about the system dynamics following an excitation-phonon interaction quench. To make contact with anticipated experimental realizations, we utilize a previously proposed method for extracting dynamical-response functions in systems with local (single-qubit) addressability using the multiqubit (many-body) version of the Ramsey interference protocol.

We present a detailed study of the applications of two stochastic approaches, stochastic optimization method (SOM) and stochastic analytical inference (SAI), to extract spectral functions from Euclidean correlation functions. SOM has the advantage that it does not require prior information. On the other hand, SAI is a more generalized method based on Bayesian inference. Under mean field approximation SAI reduces to the often-used maximum entropy method (MEM), and for a specific choice of the prior SAI becomes equivalent to SOM. To test the applicability of these two stochastic methods to lattice QCD, firstly, we apply these methods to various reasonably chosen model correlation functions, and present detailed comparisons of the reconstructed spectral functions obtained from SOM, SAI and MEM. Next, we present similar studies for charmonia correlation functions obtained from lattice QCD computations using clover-improved Wilson fermions on large, fine, isotropic lattices at $0.75$ and $1.5T_c$, $T_c$ being the deconfinement transition temperature of a pure gluon plasma. We find that SAI and SOM give consistent results to MEM at these two temperatures.

Analysis of NMR Spin-Relaxation Data Using an Inverse Gaussian Distribution Function

Super-moiré materials represent a novel playground to engineer states of matter beyond the possibilities of conventional moiré materials. However, from the computational point of view, understanding correlated matter in these systems requires solving models with several millions of atoms, a formidable task for state-of-the-art methods. Conventional wavefunction methods for correlated matter scale with a cubic power with the number of sites, a major challenge for super-moiré materials. Here, we introduce a methodology capable of solving correlated states in super-moiré materials by combining a kernel polynomial method with a quantics tensor cross interpolation matrix product state algorithm. This strategy leverages a mapping of the super-moiré structure to a many-body Hilbert space, that is efficiently sampled with tensor cross interpolation with matrix product states, where individual evaluations are performed with a Chebyshev kernel polynomial algorithm. We demonstrate this approach with interacting super-moiré systems with up to several millions of atoms, showing its ability to capture correlated states in moiré-of-moiré systems and domain walls between different moiré systems. Our manuscript puts forward a widely applicable methodology to study correlated matter in ultra-long length scales, enabling rationalizing correlated super-moiré phenomena.

21pBT-13 Development of a time-dependent DMRG method using a kernel polynomial method and its application to quantum annealing

Topological invariants allow one to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wave functions under twisted boundary conditions. However, those procedures do not allow one to calculate a topological invariant by evaluating the system locally, and thus require information about the wave functions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a one-dimensional topological superconductor and a two-dimensional quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the kernel polynomial method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.

The emergence of charge order in doped Mott insulators has received considerable attention due to its relevance to a variety of realistic materials and experiments. To investigate the interplay between magnetic and charge order, we have studied the semiclassical Kondo lattice model, which includes both electronic and magnetic degrees of freedom. By combining Langevin dynamical simulations with the kernel polynomial method, our results reveal the presence of charged stripe order, checkerboard order, and non-uniform charge disorder in the near-half-filling regime. Importantly, our simulations show that both the doping level and the strength of the s-d exchange coupling play a crucial role in facilitating charge order formation. These phases give rise to distinct electronic structures as well as excitations in the magnetic dynamics, providing insights into the underlying mechanism of charge ordering phenomena.

Kernel polynomial method on quantum computer

Randomized Matrix-Free Quadrature: Unified and Uniform Bounds for Stochastic Lanczos Quadrature and the Kernel Polynomial Method

Numerous challenges have to be mastered as applications in scientific computing are being developed for post-petascale parallel systems. While ample parallelism is usually available in the numerical problems at hand, the efficient use of supercomputer resources requires not only good scalability but also a verifiably effective use of resources on the core, the processor, and the accelerator level. Furthermore, power dissipation and energy consumption are becoming further optimization targets besides time-to-solution. Performance Engineering (PE) is the pivotal strategy for developing effective parallel code on all levels of modern architectures. In this paper we report on the development and use of low-level parallel building blocks in the GHOST library (“General, Hybrid, and Optimized Sparse Toolkit”). We demonstrate the use of PE in optimizing a density of states computation using the Kernel Polynomial Method, and show that reduction of runtime and reduction of energy are literally the same goal in this case. We also give a brief overview of the capabilities of GHOST and the applications in which it is being used successfully.