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At least 127 records · Page 7

hls4ml: A Flexible, Open-Source Platform for Deep Learning Acceleration on Reconfigurable Hardware

We present hls4ml, a free and open-source platform that translates machine learning (ML) models from modern deep learning frameworks into high-level synthesis (HLS) code that can be integrated into full designs for field-programmable gate arrays (FPGAs) or application-specific integrated circuits (ASICs). With its flexible and modular design, hls4ml supports a large number of deep learning frameworks and can target HLS compilers from several vendors, including Vitis HLS, Intel oneAPI and Catapult HLS. Together with a wider eco-system for software-hardware co-design, hls4ml has enabled the acceleration of ML inference in a wide range of commercial and scientific applications where low latency, resource usage, and power consumption are critical. In this paper, we describe the structure and functionality of the hls4ml platform. The overarching design considerations for the generated HLS code are discussed, together with selected performance results.

FOS: Computer and information sciences↗

hls4ml: A Flexible, Open-Source Platform for Deep Learning Acceleration on Reconfigurable Hardware

We present hls4ml, a free and open-source platform that translates machine learning (ML) models from modern deep learning frameworks into high-level synthesis (HLS) code that can be integrated into full designs for field-programmable gate arrays (FPGAs) or application-specific integrated circuits (ASICs). With its flexible and modular design, hls4ml supports a large number of deep learning frameworks and can target HLS compilers from several vendors, including Vitis HLS, Intel oneAPI and Catapult HLS. Together with a wider eco-system for software-hardware co-design, hls4ml has enabled the acceleration of ML inference in a wide range of commercial and scientific applications where low latency, resource usage, and power consumption are critical. In this paper, we describe the structure and functionality of the hls4ml platform. The overarching design considerations for the generated HLS code are discussed, together with selected performance results.

Schulte, Jan-Frederik [Purdue U.] (ORCID:000000034↗

Resilient Operation of Networked Community Microgrids with High Solar Penetration

This project, funded by the US Department of Energy’s Solar Energy Technologies Office (SETO), focused on the operation of microgrids as a coordinated network. The primary objective, which was successfully achieved, was to develop both control strategies and hardware solutions to support the resilient and efficient operation of networked microgrids with high solar penetration. The work was structured around the following four main tasks: • Development of distributed and scalable optimization algorithms for AC-coupled networked microgrids. • Design and implementation of a novel DC interconnection hardware to enable precise power exchange between microgrids. • Laboratory operational validation of the developed technologies using 480 V testbeds and commercially available hardware. • Field operational validation of the complete solution in Adjuntas, Puerto Rico, interconnecting two kW-scale, split-phase microgrids of Casa Pueblo’s microgrids. This project addressed multiple technical challenges across the domains of optimization, control, hardware interconnection, and protection. One of its key contributions was delivering tangible, real-world solutions for networking microgrids. In contrast to purely theoretical or simulation-based work, this project included full-scale hardware operational validation both in the lab and in the field. The work conducted as part of this project—in collaboration with the University of Puerto Rico; the University of Tennessee, Knoxville; the University of Central Florida; and Casa Pueblo—has advanced the state of the art in networked microgrids. Key contributions include the development of distributed control strategies, practical solutions for real-world implementation challenges, and the introduction of a novel DC interlink approach for microgrid interconnection. The project featured both laboratory and field validation using commercial off-the-shelf components. The field deployment successfully validated that a group of microgrids can operate in a coordinated manner, enabling precise power flow between systems and mutual support during extreme events. This project resulted in 15 journal publications and 15 conference papers; 5 graduate students and 15 undergraduate students were supported. The codes of distributed optimization and forecasting were made open-source through OSTI.gov for distributed optimization and forecasting. All the publications are available in the ORNL-hosted project landing page. The DC interlink with state-of-charge balancing control was operationally validated in Adjuntas by interconnecting two real-world, 240 V split-phase microgrids. To the best knowledge of the team, this represents the first operational validation of AC microgrids interconnected via DC-interlinks. As a culmination of this project, a follow-on grant was awarded to support the technology transfer of the distributed optimization framework to a commercial microgrid controller, Stellar Edge, developed by the California-based company New Sun Road.

14 SOLAR ENERGY↗

Surrogate Neural Architecture Codesign Package (SNAC-Pack)

Neural Architecture Search is a powerful approach for automating model design, but existing methods struggle to accurately optimize for real hardware performance, often relying on proxy metrics such as bit operations. We present Surrogate Neural Architecture Codesign Package (SNAC-Pack), an integrated framework that automates the discovery and optimization of neural networks focusing on FPGA deployment. SNAC-Pack combines Neural Architecture Codesign's multi-stage search capabilities with the Resource Utilization and Latency Estimator, enabling multi-objective optimization across accuracy, FPGA resource utilization, and latency without requiring time-intensive synthesis for each candidate model. We demonstrate SNAC-Pack on a high energy physics jet classification task, achieving 63.84% accuracy with resource estimation. When synthesized on a Xilinx Virtex UltraScale+ VU13P FPGA, the SNAC-Pack model matches baseline accuracy while maintaining comparable resource utilization to models optimized using traditional BOPs metrics. This work demonstrates the potential of hardware-aware neural architecture search for resource-constrained deployments and provides an open-source framework for automating the design of efficient FPGA-accelerated models.

Weitz, Jason [UC, San Diego] (ORCID:00090004631535↗

ARQUIN: Architectures for Multinode Superconducting Quantum Computers

Many proposals to scale quantum technology rely on modular or distributed designs wherein individual quantum processors, called nodes, are linked together to form one large multinode quantum computer (MNQC). One scalable method to construct an MNQC is using superconducting quantum systems with optical interconnects. However, internode gates in these systems may be two to three orders of magnitude noisier and slower than local operations. Surmounting the limitations of internode gates will require improvements in entanglement generation, use of entanglement distillation, and optimized software and compilers. Still, it remains unclear what performance is possible with current hardware and what performance algorithms require. In this article, we employ a systems analysis approach to quantify overall MNQC performance in terms of hardware models of internode links, entanglement distillation, and local architecture. We show how to navigate tradeoffs in entanglement generation and distillation in the context of algorithm performance, lay out how compilers and software should balance between local and internode gates, and discuss when noisy quantum internode links have an advantage over purely classical links. Here, we find that a factor of 10–100× better link performance is required and introduce a research roadmap for the co-design of hardware and software towards the realization of early MNQCs. While we focus on superconducting devices with optical interconnects, our approach is general across MNQC implementations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

LEED: A Lightwave Energy-Efficient Datacenter

The Lightwave Energy-Efficient Datacenter (LEED) program is a disruptive “green-field” approach that provides a quantum leap in the energy efficiency of datacenters. LEED’s fundamental value proposition is that a novel and re-architected optical network—RotorNet— can deliver “more bandwidth per buck” as well as unique system-level attributes that significantly improve overall datacenter energy efficiency and performance. LEED has developed three system-level testbeds. The first testbed uses calibrated hardware and software power measurements to determine server energy efficiency as a function of network bandwidth and workload. These measurements have shown that increasing network communications bandwidth dramatically increases server energy efficiency providing a realistic path to the overall ENLITENED program goal of doubling the number of transactions per joule. The second testbed demonstrates key hardware: a prototype low-loss, high-port count optical “selector switch”. This switch was fabricated, racked, and tested. Measured switch characteristics include loss, bandwidth, crosstalk, switch time, system-level switch time (including the transceivers), and bit error rate. The third testbed demonstrates a fully working and manufactured pinwheel design which dramatically lowers the cost of design, while delivering high switch radix and low reconfiguration times. The LEED project has tied these three novel photonic switch prototypes together with production servers and software through the development of a novel FPGA-based NIC platform called Corundum. Corundum ensures that the packet-switched protocols supported by commodity operating systems and devices can interface with the Rotor switch design. The LEED group has used this combined hardware and software prototype to characterize applications running at a commercially relevant scale. The project has used a combination of enhanced optical modulation amplitude (OMA) modulators, broadband multiplexers and demultiplexers, avalanche photodiodes, and a novel burst-mode receivers to enable the insertion of LEED-developed optical switches without the need for expensive optical amplification. Our modeling has shown that measured LEED-developed device characteristics can achieve link characteristics of 2 pJ/bit including both transceivers and the Rotor switch. In summary, the LEED program has demonstrated a credible and practical path, through novel hardware and software, to realize the program objectives of ENLITENED. The net result will ensure that the United States maintains its strength in the crucial sector of Information Technology, which is vital to both our economic security and our national security.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

An Integrated Framework for Memory-Centric Analysis: From Trace Collection to Co-Design

The memory wall phenomenon—where advances in processor performance significantly outpace those in memory subsystems—poses a fundamental challenge for contemporary computing systems. In memory-bound applications, memory subsystem behavior dominates performance, yet existing analysis approaches present significant limitations: detailed microarchitectural simulators require days to weeks to simulate modest workloads; hardware performance counters provide only aggregate statistics that obscure temporal and spatial access patterns; and scaled simulation approaches face challenges in capturing certain behaviors that emerge at larger scales. These limitations reflect a processor-centric design philosophy increasingly misaligned with memory-bound workloads where detailed understanding of memory access patterns, cache hierarchy interactions, and contention is critical for effective optimization. This paper presents an integrated framework for memory-centric analysis that enables effective hardware-software co-design. We describe practical trace collection techniques, including hardware-assisted processor tracing with minimal overhead and portable software-based instrumentation with statistical sampling. We present multi-perspective analysis methods that examine memory behavior from temporal, sequential, spatial, and relational viewpoints, revealing distinct optimization opportunities invisible in aggregate metrics. We detail an architectural modeling framework that uses sampled traces with temporal interpolation and confidence-based filtering to evaluate cache and memory configurations. Evaluation on representative benchmarks demonstrates that this framework achieves practical accuracy (L2 cache errors of 2.64\%, confidence-filtered L3 errors of 9.92\%, bandwidth errors of 7.33\%) while providing substantial speedup (26.8×) over cycle-accurate simulation, enabling rapid design space exploration. We demonstrate how this integrated framework enables systematic identification of both hardware optimizations (memory controller tuning, bank partitioning, NUMA configuration) and software optimizations (data layout restructuring, prefetching strategies, memory-aware scheduling). Through this comprehensive treatment of the memory-centric analysis pipeline—from trace collection through architectural modeling to co-design application—we provide researchers and practitioners with practical techniques for addressing memory bottlenecks in contemporary computing systems.

Gajaria, Dhruv Mayur↗

Fragme∩t: An Open‐Source Framework for Multiscale Quantum Chemistry Based on Fragmentation

Fragment-based quantum chemistry offers a means to circumvent the nonlinear computational scaling of conventional electronic structure calculations, by partitioning a large calculation into smaller subsystems then considering the many-body interactions between them. Variants of this approach have been used to parameterize classical force fields and machine learning potentials, applications that benefit from interoperability between quantum chemistry codes. However, there is a dearth of software that provides interoperability yet is purpose-built to handle the combinatorial complexity of fragment-based calculations. To fill this void we introduce “Fragme∩t”, an open-source software application that provides a tool for community validation of fragment-based methods, a platform for developing new approximations, and a framework for analyzing many-body interactions. Fragme∩t includes algorithms for automatic fragment generation and structure modification, and for distance- and energy-based screening of the requisite subsystems. Checkpointing, database management, and parallelization are handled internally and results are archived in a portable database. Interfaces to various quantum chemistry engines are easy to write and exist already for Q-Chem, PySCF, xTB, Orca, CP2K, MRCC, Psi4, NWChem, GAMESS, and MOPAC. Applications reported here demonstrate parallel efficiencies around 96% on more than 1000 processors but also showcase that the code can handle large-scale protein fragmentation using only workstation hardware, all with a codebase that is designed to be usable by non-experts. Fragme∩t conforms to modern software engineering best practices and is built upon well established technologies including Python, SQLite, and Ray. The source code is available under the Apache 2.0 license.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

High-power test of a C-band linear accelerating structure with an RFSoC-based LLRF system

Normal conducting linear particle accelerators consist of multiple rf stations with accelerating structure cavities. Low-level rf (LLRF) systems are employed to set the phase and amplitude of the field in the accelerating structure and to compensate for the pulse-to-pulse fluctuation of the rf field in the accelerating structures with a feedback loop. The LLRF systems are typically implemented with analog rf mixers, heterodyne-based architectures, and discrete data converters. There are multiple rf signals from each of the rf stations, so the number of rf channels required increases rapidly with multiple rf stations. With a large number of rf channels, the footprint, component cost, and system complexity of the LLRF hardware will increase significantly. To meet the design goals of being compact and affordable for future accelerators, we have designed the next-generation LLRF (NG-LLRF) with a higher integration level based on RFSoC technology. The NG-LLRF system samples rf signals directly and performs rf mixing digitally. Further, the NG-LLRF has been characterized in loopback mode to evaluate the performance of the system and has also been tested with a standing-wave accelerating structure, a prototype for the Cool Copper Collider (C 3 ) with a peak rf power level up to 16.45 MW. The loopback test demonstrated amplitude fluctuation below 0.15% and phase fluctuation below 0.15°, which are considerably better than the requirements of C 3 . The rf signals from the different stages of the accelerating structure at different power levels are measured by the NG-LLRF, which will be critical references for the control algorithm designs. The NG-LLRF also offers flexibility in waveform modulation, so we have used rf pulses with various modulation schemes, which could be useful for controlling some of the rf stations in accelerators. In this paper, the high-power test results at different stages of the test setup will be summarized, analyzed, and discussed.

47 OTHER INSTRUMENTATION↗

Shape-shifting Elephants: Multi-modal Transport for Integrated Research Infrastructure

Data Acquisition (DAQ) workloads form an important class of scientific network traffic that by its nature (1) flows across different research infrastructure, including remote instruments and supercomputer clusters, (2) has ever-increasing throughput demands, and (3) has ever-increasing integration demands---for example, observations at one instrument could trigger a reconfiguration of another instrument. Today's DAQ transfers rely on UDP and (heavily tuned) TCP, but this is driven by convenience rather than suitability. The mismatch between Internet transport protocols and scientific workloads becomes more stark with the steady increase in link capacities, data generation, and integration across research infrastructure.This position paper argues the importance of developing specialized transport protocols for DAQ workloads. It proposes a new transport feature for this kind of elephant flow: multi-modality involves the network actively configuring the transport protocol to change how DAQ flows are processed across different underlying networks that connect scientific research infrastructure. Multi-modality is a layering violation that is proposed as a pragmatic technique for DAQ transport protocol design. It takes advantage of programmable network hardware that is increasingly being deployed in scientific research infrastructure. The paper presents an initial evaluation through a pilot study that includes a Tofino2 switch and Alveo FPGA cards, and using data from a particle detector.

97 MATHEMATICS AND COMPUTING↗

Next Generation Sensor Development (CRADA Final Report)

B. Project Scope This was a collaborative effort between Lawrence Livermore National Security, LLC (LLNS), as manager and operator of Lawrence Livermore National Laboratory (LLNL) and The Federal Reserve Bank of San Francisco ("FRBSF" or "Participant"), to research and develop advanced processing techniques to measure the fitness and authenticity of U.S. currency. The FRBSF previously partnered with LLNL to evaluate end-of-life predictions, review sensor requirements, develop communications protocols, and perform thermal analysis. These Strategic Partnership Projects (SPP) included SPP No. L20960 - Federal Reserve Currency Technology Office Studies, SPP No. L15900 - Development of a Common Detector Interface (CDI 2.0) Specification and Hardware Simulators, and SPP No. L15610 - Limited Design Review of a Second-Generation E-Material Authentication Sensor (EMAS2).

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

Powering the Woods Hole X-Spar Buoy with Ocean Wave Energy—A Control Co-Design Feasibility Study

Despite its success in measuring air–sea exchange, the Woods Hole Oceanographic Institution’s (WHOI) X-Spar Buoy faces operational limitations due to energy constraints, motivating the integration of an energy harvesting apparatus to improve its deployment duration and capabilities. This work explores the feasibility of an augmented, self-powered system in two parts. Part 1 presents the collaborative design between X-Spar developers and wave energy researchers translating user needs into specific functional requirements. Based on requirements like desired power levels, deployability, survivability, and minimal interference with environmental data collection, unsuitable concepts are pre-eliminated from further feasibility study consideration. In part 2, we focus on one of the promising concepts: an internal rigid body wave energy converter. We apply control co-design methods to consider commercial of the shelf hardware components in the dynamic models and investigate the concept’s power conversion capabilities using linear 2-port wave-to-wire models with concurrently optimized control algorithms that are distinct for every considered hardware configuration. During this feasibility study we utilize two different control algorithms, the numerically optimal (but acausal) benchmark and the optimized damping feedback. We assess the sensitivity of average power to variations in drive-train friction, a parameter with high uncertainty, and analyze stroke limitations to ensure operational constraints are met. Our results indicate that a well-designed power take-off (PTO) system could significantly extend the WEC-Spar’s mission by providing additional electrical power without compromising data quality.

autonomous systems↗

Milestone 49 Report: Batched Sparse LA Phase 5 Implementation

Batched sparse linear algebra operations in general, and solvers in particular, have become the major algorithmic development activity and foremost performance engineering effort in the numerical software libraries work on modern hardware with accelerators such as GPUs. Many applications, ECP and non-ECP alike, require simultaneous solutions of many small linear systems of equations that are structurally sparse in one form or another. In order to move towards high hardware utilization levels, it is important to provide these applications with appropriate interface designs to be both functionally efficient and performance portable and give full access to the appropriate batched sparse solvers running on modern hardware accelerators prevalent across DOE supercomputing sites since the inception of ECP. To this end, we present here a summary of recent advances on the interface designs in use by HPC software libraries supporting batched sparse linear algebra and the development of sparse batched kernel codes for solvers and preconditioners. We also address the potential interoperability opportunities to keep the corresponding software portable between the major hardware accelerators from AMD, Intel, and NVIDIA, while maintaining the appropriate disclosure levels conforming to the active NDA agreements. The presented interface specifications include a mix of batched band, sparse iterative, and sparse direct solvers with their accompanying functionality that is already required by the application codes or we anticipated to be needed in the near future. This report summarizes progress in Kokkos Kernels and the xSDK libraries MAGMA, Ginkgo, hypre, PETSc, and SuperLU.

97 MATHEMATICS AND COMPUTING↗

Comparing Control Performance Between Simulation and Experiment using the Microreactor Automated Control System Testbed

In the advanced reactor domain, a flexible and scalable software/hardware infrastructure is crucial for integrating and validating various control technologies. This study used the Microreactor Automated Control System (MACS) hardware platform as a testbed. MACS was originally designed to mirror Idaho National Laboratory (INL)'s Microreactor Applications Research Validation and Evaluation (MARVEL), a 85-kW thermal fission microreactor. It features control drums for simulated reactivity control; lights that function as a surrogate reactor core, with the brightness being proportional to the reactor power; and light sensors that emulate neutron detectors. To transform MACS into a physical twin of MARVEL for evaluating control methods, the Control and Optimization Modular Modeling Application for Nuclear Deployment (COMMAND) software was employed. This software integrated the hardware with two models of the MARVEL core, based on Reactor Excursion and Leak Analysis Program (RELAP5-3D) and Monte Carlo N-Particle (MCNP) models. The study aimed to demonstrate the gap between control theory and actual practice—a gap that often necessitates empirical adjustments such as control gain retuning, filters, time discretization, and integrator anti-windup measures. Controllers were developed based on increasingly complex simulations without hardware, starting from the base MARVEL model and then introducing actuator saturation constraints and sensor noise. The final control strategy was then tested using MACS, and a comparative performance analysis was conducted.

46 - INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AN↗

A Cost-Effective Wave Energy Harvesting System with Maximum Power Point Tracking Capability

In this paper, we propose a low-cost bidirectional four-switch inverter that transfers three-phase energy between a permanent magnet synchronous generator (PMSG) driven by a wave energy collector and the power grid or a local ac load. To achieve high control performance, the machine-side inverter uses a high-gain observer for rotor position estimation and a Lyapunov-based approach for torque control. For the gridside inverter control, we estimate the filter capacitor voltage with a model reference adaptive system and control the dc-link voltage with a Lyapunov-based energy function. We also track the maximum power point from the sea waves with an integral control law that actively matches the generator impedance with that of the wave collection device. The proposed system supports both grid-following/forming modes. We validate our control design with simulation results on a 3-kW hardware setup.

wave energy, power systems integration, control de↗

An end-to-end workflow for executing a classically bootstrapped variational quantum algorithm on an academic quantum computer

Academic quantum computing platforms often face unique challenges in executing quantum workloads due to fragmented software environments and limited engineering support. Unlike commercial ecosystems, academic devices typically evolve without full-stack integration in mind, making it difficult to run complex applications—such as variational quantum algorithms (VQA)—reliably and efficiently. Issues such as incompatible software layers and lack of automated job management significantly increase the overhead of theory-experiment collaboration. To address these challenges, we develop a modular, end-to-end workflow that decouples application-layer code from low-level hardware control, automates circuit submission and result collection, and supports fine-grained circuit-level job scheduling and recovery. The architecture employs a dual-end application programming interface (API) design, enabling robust operation across unstable or resource-constrained hardware backends. For practical use, the framework is lightweight and user-friendly, allowing rapid prototyping of full-stack workflows using basic Python tools. We validate this workflow on a high-fidelity trapped-ion quantum computer by demonstrating a variational quantum eigensolver (VQE) experiment with a classically bootstrapped ansatz initialization technique. The system successfully executed over 60,000 circuits across multiple molecular test cases with minimal human intervention, highlighting the framework’s effectiveness in enabling reproducible, resilient quantum experimentation in academic settings.

Clifford↗

Design, Analysis, and Experimental Testing of Hydrogen Lean Direct Injection Nozzles at Elevated Pressure

Abstract There are many challenges of commissioning a hydrogen combustor into future gas turbine engines; especially regarding achieving emissions goals. Previously, Escudero et al. and Tran et al. conducted a study to adapt the liquid fuel Lean Direct Injection (LDI) concept from Jet-A to gaseous natural gas-hydrogen blends and pure hydrogen [1], [2]. Experimental data was collected at atmospheric conditions using a Box Behnken design of experiments. The design of experiments suggested that biasing the air split in favor of the inner air circuit and increasing the swirl strength of this inner air passage resulted in improved NOx emissions, while the inverse was true for stability, which was quantified by studying the lean blowoff point (LBO) [1], [2]. The trends revealed by the original experiment [1], [2] provided a design direction for further iterations of the experimental hardware. The study presented herein describes the further investigation of such LDI injectors through experimental methods and computational fluid dynamic (CFD) simulations at atmospheric conditions, which were used to identify potential flow behaviors driving enhanced emissions performance. Further evaluation of select injectors from both studies was then conducted at elevated pressures up to 6 atmospheres. The results from both experiments are presented in this study, which include flame observations, emissions measurements, and operational challenges. NOx emissions results are reported on a volume basis in ppmvd corrected to 15% O2 and corrected for fuel. A predictive model for relating NOx emissions to test conditions at atmospheric conditions show high significance to adiabatic flame temperature while little to no significance to fuel composition for the best performing configurations. The results illustrate the connection between atmospheric testing and testing elevated pressures. The design direction indicated by the initial tests and CFD results in promising configurations for implementation into a Multi-point LDI array.

08 HYDROGEN↗

Identifiability and characterization of transmon qutrits through Bayesian experimental design

Robust control of a quantum system is essential to utilize the current noisy quantum hardware to its full potential, such as quantum algorithms. To achieve such a goal, a systematic search for an optimal control for any given experiment is essential. The design of optimal control pulses requires accurate numerical models and, therefore, accurate characterization of the system parameters. We present an online Bayesian approach for quantum characterization of qutrit systems, which automatically and systematically identifies optimal experiments that provide maximum information on the system parameters, thereby greatly reducing the number of experiments that need to be performed on the quantum testbed. Unlike most characterization protocols that provide point-estimates of the parameters, the proposed approach is able to estimate their probability distribution. The applicability of the Bayesian experimental design technique was demonstrated on test problems, where each experiment was defined by a parameterized control pulse. In addition to this, we also present an approach for iterative pulse extension, which is robust under uncertainties in transition frequencies and coherence times, and shot noise, despite being initialized with wide uninformative priors. Furthermore, we provide a mathematical proof of the theoretical identifiability of the model parameters and present conditions on the quantum state under which the parameters are identifiable. The proof and conditions for identifiability are presented for both closed and open quantum systems using the Schrödinger equation and the Lindblad master equation, respectively.

97 MATHEMATICS AND COMPUTING↗