Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries
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Zhu, Guanyu (ORCID:0000000233754445)↗
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Results for “Computational topology”
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PC-CUBE: A Personal Computer Based Hypercube
PC-CUBE is an ensemble of IBM PCs or close compatibles connected in the hypercube topology with ordinary computer cables. Communication occurs at the rate of 115.2 K-band via the RS-232 serial links. Available for PC-CUBE is the Crystalline Operating System III (CrOS III), Mercury Operating System, CUBIX and PLOTIX which are parallel I/O and graphics libraries. A CrOS performance monitor was developed to facilitate the measurement of communication and computation time of a program and their effects on performance. Also available are CXLISP, a parallel version of the XLISP interpreter; GRAFIX, some graphics routines for the EGA and CGA; and a general execution profiler for determining execution time spent by program subroutines. PC-CUBE provides a programming environment similar to all hypercube systems running CrOS III, Mercury and CUBIX. In addition, every node (personal computer) has its own graphics display monitor and storage devices. These allow data to be displayed or stored at every processor, which has much instructional value and enables easier debugging of applications. Some application programs which are taken from the book Solving Problems on Concurrent Processors (Fox 88) were implemented with graphics enhancement on PC-CUBE. The applications range from solving the Mandelbrot set, Laplace equation, wave equation, long range force interaction, to WaTor, an ecological simulation.
Ho, Alex↗
Fusion dynamics of Majorana zero modes
Braiding and fusion of Majorana zero modes are key elements of any future topological Majorana-based quantum computer. In this article, we investigate the fusion dynamics of Majorana zero modes in the spinless Kitaev model, as well as in a spinful model describing magnet-superconductor hybrid structures. We consider various scenarios allowing us to reproduce the fusion rules of the Ising anyon model. Particular emphasis is given to the charge of the fermion obtained after fusing two Majorana zero modes: as long as it remains on the superconductor, charge quantization is absent. When moving the fermion to a nonsuperconducting region, such as a quantum dot, nearly quantized charge can be measured. Our findings confirm for both platforms that fusion dynamics of Majorana zero modes can indeed be used for the readout of Majorana qubits.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Mapping unstructured grid problems to the connection machine
We present a highly parallel graph mapping technique that enables one to solve unstructured grid problems on massively parallel computers. Many implicit and explicit methods for solving discretizated partial differential equations require each point in the discretization to exchange data with its neighboring points every time step or iteration. The time spent communicating can limit the high performance promised by massively parallel computing. To eliminate this bottleneck, we map the graph of the irregular problem to the graph representing the interconnection topology of the computer such that the sum of the distances that the messages travel is minimized. We show that, in comparison to a naive assignment of processors, our heuristic mapping algorithm significantly reduces the communication time on the Connection Machine, CM-2.
Hammond, Steven W.↗
Solving unstructured grid problems on massively parallel computers
A highly parallel graph mapping technique that enables one to efficiently solve unstructured grid problems on massively parallel computers is presented. Many implicit and explicit methods for solving discretized partial differential equations require each point in the discretization to exchange data with its neighboring points every time step or iteration. The cost of this communication can negate the high performance promised by massively parallel computing. To eliminate this bottleneck, the graph of the irregular problem is mapped into the graph representing the interconnection topology of the computer such that the sum of the distances that the messages travel is minimized. It is shown that using the heuristic mapping algorithm significantly reduces the communication time compared to a naive assignment of processes to processors.
Hammond, Steven W.↗
Experimental Observations of the Topology of Convolutional Neural Network Activations
Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of parameters associated with a series of transformations defined by the model architecture resulting in high-dimensional, difficult to interpret internal representations of input data. As DNNs become more ubiquitous across multiple sectors of our society, there is increasing recognition that mathematical methods are needed to aid analysts, researchers, and practitioners in understanding and interpreting how these models' internal representations relate to the final classification. In this paper we apply cutting edge techniques from TDA with the goal of gaining insight towards interpretability of convolutional neural networks used for image classification. We use two common TDA approaches to explore several methods for modeling hidden layer activations as high-dimensional point clouds, and provide experimental evidence that these point clouds capture valuable structural information about the model's process. First, we demonstrate that a distance metric based on persistent homology can be used to quantify meaningful differences between layers and discuss these distances in the broader context of existing representational similarity metrics for neural network interpretability. Second, we show that a mapper graph can provide semantic insight as to how these models organize hierarchical class knowledge at each layer. These observations demonstrate that TDA is a useful tool to help deep learning practitioners unlock the hidden structures of their models.
topological data analysis, deep learning↗
Parallel processing for scientific computations
The main contribution of the effort in the last two years is the introduction of the MOPPS system. After doing extensive literature search, we introduced the system which is described next. MOPPS employs a new solution to the problem of managing programs which solve scientific and engineering applications on a distributed processing environment. Autonomous computers cooperate efficiently in solving large scientific problems with this solution. MOPPS has the advantage of not assuming the presence of any particular network topology or configuration, computer architecture, or operating system. It imposes little overhead on network and processor resources while efficiently managing programs concurrently. The core of MOPPS is an intelligent program manager that builds a knowledge base of the execution performance of the parallel programs it is managing under various conditions. The manager applies this knowledge to improve the performance of future runs. The program manager learns from experience.
Alkhatib, Hasan S.↗
Correlated oxide Dirac semimetal in the extreme quantum limit
Quantum materials (QMs) with strong correlation and nontrivial topology are indispensable to next-generation information and computing technologies. Exploitation of topological band structure is an ideal starting point to realize correlated topological QMs. Here, we report that strain-induced symmetry modification in correlated oxide SrNbO 3 thin films creates an emerging topological band structure. Dirac electrons in strained SrNbO 3 films reveal ultrahigh mobility (μ max ≈ 100,000 cm 2 /Vs), exceptionally small effective mass (m* ~ 0.04m e ), and nonzero Berry phase. Strained SrNbO 3 films reach the extreme quantum limit, exhibiting a sign of fractional occupation of Landau levels and giant mass enhancement. Our results suggest that symmetry-modified SrNbO 3 is a rare example of correlated oxide Dirac semimetals, in which strong correlation of Dirac electrons leads to the realization of a novel correlated topological QM.
36 MATERIALS SCIENCE↗
Engineering magnetic topological insulators in Eu 5 M 2 X 6 Zintl compounds
Magnetic topological insulators provide a prominent material platform for quantum anomalous Hall physics and axion electrodynamics. However, the lack of material realizations with cleanly gapped surfaces hinders technological utilization of these exotic quantum phenomena. Here, using the Zintl concept and the properties of nonsymmorphic space groups, we computationally engineer magnetic topological insulators. Specifically, we explore Eu 5 M 2 X 6 (M=metal, X=pnictide) Zintl compounds and find that Eu 5 Ga 2 Sb 6 , Eu 5 Tl 2 Sb 6 , and Eu 5 In 2 Bi 6 form stable structures with nontrivial Z 2 indices. We also show that epitaxial and uniaxial strain can be used to control the Z 2 index and the bulk energy gap. Lastly, we discuss experimental progress towards the synthesis of the proposed candidates and provide insights that can be used in the search for robust magnetic topological insulators in Zintl compounds.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Constraint Embedding Technique for Multibody System Dynamics
Multibody dynamics play a critical role in simulation testbeds for space missions. There has been a considerable interest in the development of efficient computational algorithms for solving the dynamics of multibody systems. Mass matrix factorization and inversion techniques and the O(N) class of forward dynamics algorithms developed using a spatial operator algebra stand out as important breakthrough on this front. Techniques such as these provide the efficient algorithms and methods for the application and implementation of such multibody dynamics models. However, these methods are limited only to tree-topology multibody systems. Closed-chain topology systems require different techniques that are not as efficient or as broad as those for tree-topology systems. The closed-chain forward dynamics approach consists of treating the closed-chain topology as a tree-topology system subject to additional closure constraints. The resulting forward dynamics solution consists of: (a) ignoring the closure constraints and using the O(N) algorithm to solve for the free unconstrained accelerations for the system; (b) using the tree-topology solution to compute a correction force to enforce the closure constraints; and (c) correcting the unconstrained accelerations with correction accelerations resulting from the correction forces. This constraint-embedding technique shows how to use direct embedding to eliminate local closure-loops in the system and effectively convert the system back to a tree-topology system. At this point, standard tree-topology techniques can be brought to bear on the problem. The approach uses a spatial operator algebra approach to formulating the equations of motion. The operators are block-partitioned around the local body subgroups to convert them into aggregate bodies. Mass matrix operator factorization and inversion techniques are applied to the reformulated tree-topology system. Thus in essence, the new technique allows conversion of a system with closure-constraints into an equivalent tree-topology system, and thus allows one to take advantage of the host of techniques available to the latter class of systems. This technology is highly suitable for the class of multibody systems where the closure-constraints are local, i.e., where they are confined to small groupings of bodies within the system. Important examples of such local closure-constraints are constraints associated with four-bar linkages, geared motors, differential suspensions, etc. One can eliminate these closure-constraints and convert the system into a tree-topology system by embedding the constraints directly into the system dynamics and effectively replacing the body groupings with virtual aggregate bodies. Once eliminated, one can apply the well-known results and algorithms for tree-topology systems to solve the dynamics of such closed-chain system.
Woo, Simon S.↗
Band Energy Dependence of Defect Formation in the Topological Semimetal Cd3As2
Cadmium Arsenide (Cd3As2) is a prototypical Dirac semimetal that manifests topological properties in a 3D bulk material. In defect-free Cd3As2, the Fermi level lies at a minimum in the density of states at the Dirac point, but experimentally it forms with excess electron carriers and an elevated EF, thereby masking the topological features. To computationally study the self-doping of Cd3As2, we combine density functional theory (DFT) calculations for defect formation energies with quasi-particle self-consistent GW (QSGW) electronic structure calculations. We demonstrate an innate dependence of the point defect formation energies on carrier concentrations and use the QSGW calculated density of states to extrapolate formation energies to arbitrary electron concentrations. This approach allows the quantitative modeling of thermodynamic defect equilibria in topological semimetals and is used to predict how Cd3As2 growth conditions affect the position of EF relative to the Dirac point.
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS↗
Generalized symmetries of the graviton
We find the set of generalized symmetries associated with the free graviton theory in four dimensions. These are generated by gauge invariant topological operators that violate Haag duality in ring-like regions. As expected from general QFT grounds, we find a set of “electric” and a dual set of “magnetic” topological operators and compute their algebra. To do so, we describe the theory using phase space gauge-invariant electric and magnetic dual variables constructed out of the curvature tensor. Electric and magnetic fields satisfy a set of constraints equivalent to the ones of a stress tensor of a 3d CFT. The constraints give place to a group $\mathbb{R}^{20}$ of topological operators that are charged under space-time symmetries. Finally, we discuss similarities and differences between linearized gravity and tensor gauge theories that have been introduced recently in the context of fractonic systems in condensed matter physics.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Electric Field Control of Magnetic Skyrmion Helicity in a Centrosymmetric 2D van der Waals Magnet
Two-dimensional van der Waals magnets hosting topological magnetic textures, such as skyrmions, show promise for spintronics and quantum computing. Electrical control of these topological spin textures is crucial for enhancing operational performance and functionality. Here, using electron microscopy combined with in situ electric and magnetic biasing, we show that the skyrmion helicity in insulating Cr 2 Ge 2 Te 6 can be controlled by the direction of the external electric field applied during the field cooling process. The electric field-tuned helicity remains stable at low temperature, even amid variations in magnetic and electric fields. Our theoretical investigation reveals that nonzero Dzyaloshinskii-Moriya interactions between nearest neighbors, induced by the electric field, change their sign upon reversing the electric field direction, thereby facilitating helicity selection. The electrical control of magnetic helicity demonstrated in this study can be extended to other centrosymmetric skyrmion-hosting magnets, paving the way for future device designs in topological spintronics and quantum computing.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Performance Study of CXL Memory Topology
This paper presents a comprehensive evaluation of the performance impact of various Compute Express Link (CXL) memory topologies, with a particular emphasis on CXL switches, in the context of High- Performance Computing (HPC) and Large Language Model (LLM) inference workloads. Our study unveils significant performance variations across different topologies, demonstrating that certain configurations yield superior performance for specific workloads. These findings underscore the critical importance of tailored topol- ogy selection in optimizing system performance. Additionally, we address the inherent challenges associated with integrating CXL switches, including overhead considerations and routing complex- ities. Our research highlights the necessity for thorough evalua- tion methodologies to fully leverage CXL technology’s potential in contemporary computing environments. These insights provide valuable guidance for system architects and data center operators in designing and optimizing CXL-based infrastructures for diverse workload requirements.
CXL, memory, Artificial Intelligence (AI), HPC↗
Efficient Berry phase calculation via adaptive variational quantum computing approach
We present an adaptive variational quantum algorithm to estimate the Berry phase accumulated by a nondegenerate ground state under cyclic, adiabatic evolution of a time-dependent Hamiltonian. Our method leverages cyclic adiabatic evolution of the Hamiltonian and employs adaptive variational quantum algorithms for state preparation and evolution, optimizing circuit efficiency while maintaining high accuracy. We benchmark our approach on dimerized Fermi–Hubbard chains with four sites, demonstrating precise Berry phase simulations in both noninteracting and interacting regimes. Our results show that circuit depths reach up to 106 layers for noninteracting systems and increase to 279 layers for interacting systems due to added complexity. In addition, we demonstrate the robustness of our scheme across a wide range of parameters governing adiabatic evolution and variational algorithms. These findings highlight the potential of adaptive variational quantum algorithms for advancing quantum simulations of topological materials and computing geometric phases in strongly correlated systems.
Mootz, Martin [Ames Laboratory (AMES), Ames, IA (U↗
A tensor density measure of topological charge in three dimensional nematic phases
A path independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor D, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor D may be nonzero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behavior there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from D.
36 MATERIALS SCIENCE↗
A tensor density measure of topological charge in three dimensional nematic phases
A path-independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three-dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor D, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor D may be non-zero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behaviour there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from D.
36 MATERIALS SCIENCE↗
Asymmetric Fraunhofer spectra in a topological insulator-based Josephson junction
Josephson junctions with topological insulators as their weak link (S–TI–S junctions) are predicted to host Majorana fermions, which are key to creating qubits for topologically protected quantum computing. But the details of the S–TI–S current-phase relation and its interplay with magnetic fields are not well understood. We fabricate a Bi 2 Se 3 junction with NbTi leads and measure the Fraunhofer patterns of the junction with applied in-plane fields. We observe that asymmetric Fraunhofer patterns appear in the resistance maps of B z vs B x,y , with aperiodic node spacings. These asymmetric patterns appear even at zero parallel field and for temperatures up to 1 K. The anomalous features are compared to asymmetric Fraunhofer patterns expected for finite Cooper pair momentum shifts as well as geometric effects. We show that the geometric effects can dominate, independent of in-plane field magnitude. Furthermore, these results are important for differentiating geometrical phase shifts from those caused by Cooper pair momentum shifting, Majorana mode signatures, or other unconventional superconducting behavior.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗