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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Quantum algorithm to simulate Lindblad master equations

We present a quantum algorithm for simulating a family of Markovian master equations that can be realized through a probabilistic application of unitary channels and state preparation. Our approach employs a second-order product formula for the Lindblad master equation, achieved by decomposing the dynamics into dissipative and Hamiltonian components and replacing the dissipative segments with randomly compiled, easily implementable elements. The sampling approach eliminates the need for ancillary qubits to simulate the dissipation process and reduces the gate complexity in terms of the number of jump operators. We provide a rigorous performance analysis of the algorithm. We also extend the algorithm to time-dependent Lindblad equations, generalize the family of Markovian master equations it can be applied to, and explore applications beyond the Markovian noise model. A new error bound, in terms of the diamond norm, for second-order product formulas for time-dependent Liouvillians is provided that might be of independent interest. Published by the American Physical Society 2025

Borras, Evan (ORCID:000900017709037X)↗
DOE OSTI2025

A neural master equation framework for multiscale modeling of molecular processes: application to atomic-scale plasma processes

Plasma-surface interactions (PSI) play a crucial role in microelectronics fabrication; however, their multiscale nature and array of complex, often unknown interactions make computational modeling of PSIs extremely difficult. To this end, we propose a general neural master equation (NME) framework that uses master equations to describe the dynamics of a molecular process, wherein neural networks learned from atomistic simulations represent unknown transitions between different system states. By leveraging the physics-based structure of master equations and data-driven state transitions, the NME framework promotes generalizability and physics interpretability, and can bridge disparate length and time scales. The framework is demonstrated for multiscale modeling of Si atomic layer etching and reactive ion etching, where the learned NME-based surface kinetic models exhibit good predictive and extrapolative capabilities for predicting experimentally relevant observables as a function of process parameters. The NME-based surface kinetic models obey physical constraints, which are violated in models based on neural ordinary differential equations. The proposed NME framework for multiscale modeling of molecular processes can pave the way for the discovery of new chemistries and materials in atomic-scale plasma processes.

Chemical engineering↗
DOE OSTI2025

On the discretization error of the discrete generalized quantum master equation

The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 112 , 110401 (2014)] can be considered a discrete-time formulation of the Nakajima–Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 21 , 5037 (2025)] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time t = 0. Here, this work presents a detailed analysis of the discretization structure of the TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel K N and the continuous-time NZ-QME kernel $\mathscr{K}$( N Δ t ). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as Δ t → 0. While the TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri’s work. The relative performance of various schemes for either computing accurate $\mathscr{K}$( N Δ t ) from exact dynamics or obtaining accurate dynamics from exact $\mathscr{K}$( N Δ t ) warrants further investigation.

Density-matrix↗
DOE OSTI2026

Generalized master equation for particle transport in binary random media with renewal statistics

Particle transport in binary stochastic mixtures is classically modeled assuming Markovian or exponential mixing statistics but in many applications material memory invalidates the Markov assumption. For non-Markovian mixing characterized by alternating renewal processes, a transport-theoretic framework is presented that provides an exact description of transport in nonscattering random binary media with general non-exponential statistics. Our approach is to Markovianize the problem by augmenting the {material type, particle flux} state space with the age or distance from the last interface. A Chapman-Kolmogorov equation is formulated for the joint probability density of the material type, particle flux, and age, and subsequently reduced to a generalized Master equation (GME) in differential form. This constitutes the primary result of this work. A state-updating Monte Carlo algorithm consistent with the GME is developed and benchmarked against analytical solutions for multiple chord-length laws. For purely absorbing renewal statistical media, the GME reproduces analytical benchmarks for the equilibrium age distribution, interior mean/variance of material-conditioned fluxes, and boundary transmittance. Simulations further demonstrate that a Markov (exponential) approximation of non-exponential statistics can introduce large errors in transmittance and interior flux profiles. Lastly, the reintroduction of memory due to scattering is briefly addressed through heuristic considerations.

Fluctuations & noise↗
DOE OSTI2025

Simulating quantum-classical interfaces via the Lindblad master equation

In hybrid quantum systems, the interface between quantum and classical domains is essential for the generation, control, and measurement of quantum states. Quantum-classical interfaces (QCIs) are ubiquitous in devices such as optical modulators, quantum sensors, and signal processors, where classical signals influence quantum dynamics. In this paper, we employ the Lindblad master equation to simulate the evolution of a quantum system interacting with a classical control system. Our model captures both linear and nonlinear interactions by incorporating first- and second-order susceptibilities, and it quantifies the influence of externally applied control parameters on decoherence and state evolution. As an illustrative example, we analyze an optical modulator and demonstrate how variations in material response and drive conditions affect photon statistics, coherence, and phase-space distributions. In conclusion, the findings offer a path to an all-encompassing model for understanding and optimizing QCIs, with wide-ranging implications for the performance, design, and robustness of next-generation quantum devices.

Quantum engineering↗
DOE OSTI2024

Generalized quantum master equations can improve the accuracy of semiclassical predictions of multitime correlation functions

Multitime quantum correlation functions are central objects in physical science, offering a direct link between the experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control can now measure such quantities, the accurate simulation of such responses remains computationally expensive and sometimes impossible, depending on the system’s complexity. A natural tool to employ is the generalized quantum master equation (GQME), which can offer computational savings by extending reference dynamics at a comparatively trivial cost. However, dynamical methods that can tackle chemical systems with atomistic resolution, such as those in the semiclassical hierarchy, often suffer from poor accuracy, limiting the credence one might lend to their results. By combining work on the accuracy-boosting formulation of semiclassical memory kernels with recent work on the multitime GQME, here we show for the first time that one can exploit a multitime semiclassical GQME to dramatically improve both the accuracy of coarse mean-field Ehrenfest dynamics and obtain orders of magnitude efficiency gains.

Chemistry↗
DOE OSTI2025

The Forward Master Equation for the Joint Neutron-Photon Number Probability Distribution

The model is similar to the Binary Fission Model (BFM) in chapter three of the Stochastic Neutronics Primer Volume I, however, we add photons as a product of induced fission events (IFEs). We will find that tracking the population of an additional particle adds an additional layer of complexity because we are now looking for a joint probability distribution.

42 ENGINEERING↗
DOE OSTI2026

Learning thermodynamic master equations for open quantum systems

The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.

Mathematics and Computing↗
DOE OSTI2025

Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments

In this work, we introduce an efficient method, TTN-HEOM, for exactly calculating the open quantum dynamics for driven quantum systems interacting with highly structured bosonic baths by combining the tree tensor network (TTN) decomposition scheme with the bexcitonic generalization of the numerically exact hierarchical equations of motion (HEOM). The method yields a series of quantum master equations for all core tensors in the TTN that efficiently and accurately capture the open quantum dynamics for non-Markovian environments to all orders in the system–bath interaction. These master equations are constructed based on the time-dependent Dirac–Frenkel variational principle, which isolates the optimal dynamics for the core tensors given the TTN ansatz. The dynamics converges to the HEOM when increasing the rank of the core tensors, a limit in which the TTN ansatz becomes exact. We introduce TENSO, tensor equations for non-Markovian structured open systems, as a general-purpose Python code to propagate the TTN-HEOM dynamics. We implement three general propagators for the coupled master equations: two fixed-rank methods that require a constant memory footprint during the dynamics and one adaptive-rank method with a variable memory footprint controlled by the target level of computational error. We exemplify the utility of these methods by simulating a two-level system coupled to a structured bath containing one Drude–Lorentz component and eight Brownian oscillators, which is beyond what can presently be computed using the standard HEOM. Our results show that the TTN-HEOM is capable of simulating both dephasing and relaxation dynamics of driven quantum systems interacting with structured baths, even those of chemical complexity, with an affordable computational cost.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Lectures on statistical mechanics

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

plasma dynamics↗
DOE OSTI2025

QEpsilon v0.1.0

QEpsilon is a Python package designed to minimize the effort required to build a data-driven quantum master equation of an open quantum system and to perform time evolution of the master equation. Applications of QEpsilon span from quantum computing to condensed matter systems.

Xie, Pinchen [Lawrence Berkeley National Laborator↗
DOE OSTI2026

Pressure-dependent kinetics of cyclopentene consumption

Recent experiments have uncovered deficiencies in the ability of current kinetic models for cyclopentane combustion to predict species profiles as a function of O2 concentration. One likely source of these discrepancies flows from the limited availability of relevant kinetic data for key functionalized intermediates, requiring the use of distant analogies to estimate their rates. This work fills this gap for cyclopentene, the most prominent C5 intermediate in low-temperature cyclopentane combustion. Ab initio transition-state-theory based master equation methods are used to calculate pressure-dependent rate constants across four potential energy surfaces (C5H8 + OH, C5H8 + HO2, C5H7, and C5H7 + O2). Potential energy surfaces are characterized at the CCSD(T)-F12/cc-pVDZ-F12//revDSD-PBEP86-D3BJ/def2-TZVPP level of theory. Pressure-dependent rate constants and branching fractions are determined via 1D master equation calculations. The results are used to analyze the competition for radicals between cyclopentene and cyclopentane in the intermediate temperature region, and to identify the dominant pathways flowing from these initiation channels. On the OH initiation surface, we find good agreement with recently published experimental rate constants at high temperature, and we elucidate the complex temperature dependence below 800 K, where 𝜋𝜋 addition becomes prominent. On the HO2 initiation surface, we find an interesting alternative route to bicyclic ether formation via well-skipping over the HO2 adduct well. On the cyclopentenyl radical surface, the major product is cyclopentadiene plus H. On the cyclopentenyl + O2 surface, we find that resonance stabilization largely prevents the allylic radical from undergoing oxidation, whereas the alkylic radical has a somewhat elevated propensity for oxidation pathways relative to cyclopentyl.

ab initio calculations↗
DOE OSTI2024

Association Kinetics for Perfluorinated n -Alkyl Radicals

Radical-radical reaction channels are important in the pyrolysis and oxidation chemistry of perfluoroalkyl substances (PFAS). In particular, unimolecular dissociation reactions within unbranched n-perfluoroalkyl chains, and their corresponding reverse barrierless association reactions, are expected to be significant contributors to the gas-phase thermal decomposition of families of species such as perfluorinated carboxylic acids and perfluorinated sulfonic acids. Unfortunately, experimental data for these reactions are scarce and uncertain. Furthermore, obtaining reliable theoretical predictions for such reactions is a laborious and computationally intensive task. Here, in this work, the chemical kinetics of the various association/decomposition reactions producing/decomposing the C 2 -C 4 series of unbranched n-perfluoroalkanes (C 2 F 6 , C 3 F 8 , and C 4 F 10 ) are examined using state-of-the-art ab initio transition-state-theory-based master-equation calculations. The variable-reaction-coordinate transition-state theory (VRC-TST) formalism is employed in computing the microcanonical and canonical rates for the association reactions. Reaction thermochemistry is obtained via composite quantum chemistry calculations and the laddering of error-canceling reaction schemes via a connectivity-based hierarchy approach employing ANL1/ANL0-style reference energies. Lennard-Jones collision model parameters for the considered systems were estimated by a direct dynamics approach, and collisional energy transfer parameters were obtained from analogies to systems of similar size and heavy-atom connectivity. A one-dimensional master equation approach was used to convert the microcanonical rate coefficients from the VRC-TST analysis into temperature- and pressure-dependent rate constants for the association reactions and the reverse dissociation reactions. The data are reported in standardized formats for usage in comprehensive chemical kinetic models for PFAS thermal destruction.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Implementation of new mixture rules has a substantial impact on combustion predictions for H 2 and NH 3

Complex-forming reactions comprise a substantial fraction of all important combustion reactions and are central to combustion behavior. Despite being often called “pressure-dependent” reactions, their rate constants depend on not only the pressure but also the composition. While modern combustion codes allow arbitrarily high accuracy in treating pressure dependence, recent work has consistently demonstrated dramatic failures of essentially all available treatments of mixture dependence. In situations where mixture dependence is treated at all, it is inevitably treated through specification of pressure-dependent rate constants for a set of pure bath gases, which are then combined to estimate the rate constant in a mixture via a “mixture rule.” While there had been a generally unquestioning confidence in these mixture rules, they had, in reality, been scarcely tested until the last decade, when comparisons against master equation calculations revealed order-of-magnitude errors for important pressure-dependent reactions. New mixture rules, based on the reduced pressure, have recently been proposed and shown to reproduce master equation calculations for broad classes of complex-forming reactions very accurately. Here, in this work, we present an implementation of one such new mixture rule (“LMR-R”) in Cantera and then use it to enable simulations that use new high-accuracy ab initio data for individual bath gases (for the first time, since codes previously could not accommodate the complex bath gas dependence). Demonstrations focus on combustion of H 2 and NH 3 , where (1) high-accuracy ab initio data are available and (2) the impact is expected to be large due to the high fractions of efficient colliders (e.g., H 2 O and NH 3 ) in the burned and unburned gases. Indeed, we find the impact of this treatment to be substantial and may explain previous modeling difficulties for these important carbon-free fuels, particularly for NH 3 , whose extraordinarily high third-body efficiency (~20) is often omitted from kinetic models.

Ammonia↗
DOE OSTI2026

Kinetic Model of HoxEFU reduction by NADH [SWR-26-087]

This repository is used to release code generated for manuscripts on the Photosynthetic Energy Transduction core program. This code simulates the reduction of HoxEFU by NADH. The electro transfer rate constants for the simulation are specified in the .csv files. The two .csv files correspond tot he two models described in Dawson et al. Cell. Rep. Phys. Sci. 2026. The code utilizes a chemical master equation, a set of differential equations, defining the time evolution of the oxidation and reduction kinetics of NAD+, NADH, a FMN flavin, and a set of iron sulfur clusters. The kinetics of HoxEFU reduction by NADH are evaluated by numerical integration of the chemical master equation using a variable-time-step Runge-Kutta algorithm.

Dahl, Peter [National Laboratory of the Rockies (N↗
DOE OSTI2024

Real-time dynamics of the Schwinger model as an open quantum system with Neural Density Operators

Ab-initio simulations of multiple heavy quarks propagating in a Quark-Gluon Plasma are computationally difficult to perform due to the large dimension of the space of density matrices. This work develops machine learning algorithms to overcome this difficulty by approximating exact quantum states with neural network parametrisations, specifically Neural Density Operators. As a proof of principle demonstration in a QCD-like theory, the approach is applied to solve the Lindblad master equation in the 1 + 1d lattice Schwinger Model as an open quantum system. Neural Density Operators enable the study of in-medium dynamics on large lattice volumes, where multiple-string interactions and their effects on string-breaking and recombination phenomena can be studied. Thermal properties of the system at equilibrium can also be probed with these methods by variationally constructing the steady state of the Lindblad master equation. Scaling of this approach with system size is studied, and numerical demonstrations on up to 32 spatial lattice sites and with up to 3 interacting strings are performed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

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↗