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At least 109 records · Page 6

Sample-efficient verification of continuously-parameterized quantum gates for small quantum processors

Most near-term quantum information processing devices will not be capable of implementing quantum error correction and the associated logical quantum gate set. Instead, quantum circuits will be implemented directly using the physical native gate set of the device. These native gates often have a parameterization (e.g., rotation angles) which provide the ability to perform a continuous range of operations. Verification of the correct operation of these gates across the allowable range of parameters is important for gaining confidence in the reliability of these devices. In this work, we demonstrate a procedure for sample-efficient verification of continuously-parameterized quantum gates for small quantum processors of up to approximately 10 qubits. This procedure involves generating random sequences of randomly-parameterized layers of gates chosen from the native gate set of the device, and then stochastically compiling an approximate inverse to this sequence such that executing the full sequence on the device should leave the system near its initial state. We show that fidelity estimates made via this technique have a lower variance than fidelity estimates made via cross-entropy benchmarking. This provides an experimentally-relevant advantage in sample efficiency when estimating the fidelity loss to some desired precision. We describe the experimental realization of this technique using continuously-parameterized quantum gate sets on a trapped-ion quantum processor from Sandia QSCOUT and a superconducting quantum processor from IBM Q, and we demonstrate the sample efficiency advantage of this technique both numerically and experimentally.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Lastly, potential applications and caveats to the usage of LD are included as a summary.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Optical spin hall effect in exciton–polariton condensates in lead halide perovskite microcavities

An exciton–polariton condensate is a hybrid light–matter state in the quantum fluid phase. The photonic component endows it with characters of spin, as represented by circular polarization. Spin-polarization can form stochastically for quasi-equilibrium exciton–polariton condensates at parallel momentum vector k|| ∼ 0 from bifurcation or deterministically for propagating condensates at k|| > 0 from the optical spin-Hall effect (OSHE). Here, we report deterministic spin-polarization in exciton–polariton condensates at k|| ∼ 0 in microcavities containing methylammonium lead bromide perovskite (CH3NH3PbBr3) single crystals under non-resonant and linearly polarized excitation. We observe two energetically split condensates with opposite circular polarizations and attribute this observation to the presence of strong birefringence, which introduces a large OSHE at k|| ∼ 0 and pins the condensates in a particular spin state. Such spin-polarized exciton–polariton condensates may serve not only as circularly polarized laser sources but also as effective alternatives to ultracold atom Bose–Einstein condensates in quantum simulators of many-body spin–orbit coupling processes.

Chemistry↗

High-Performance Electron Sources: Numerical Methods and Beam Dynamics at the Precision Frontier

Electron sources have a wide range of applications and there are many stakeholders that express continuing need for improvements and performance enhancements. Whether we consider ultra-cold, high-brightness, high-charge or high-average current source needs, there are some common themes from the point of view of the beam dynamics involved. These are the following ones: ability to model accurately the emission processes, including the presence of often complicated cathode and other boundary surfaces with a wide range of spatial scales; ability to deal accurately and efficiently with a large number of particles interacting pair-wise, including the stochastic part of these interactions with a wide range of spatial scales; and ability to propagate the particle distributions in time, including collisions with a wide range of temporal scales. These tasks require high precision and accuracy since the goal is usually generation, transport and preservation of very high-quality beams. This grant addressed one of the rem

43 PARTICLE ACCELERATORS↗

Benchmarking Optimizers for Qumode State Preparation with Variational Quantum Algorithms

Quantum state preparation involves preparing a target state from an initial system, a process integral to applications such as quantum machine learning and solving systems of linear equations. Recently, there has been a growing interest in qumodes due to advancements in the field and their potential applications. However there is a notable gap in the literature specifically addressing this area. This paper aims to bridge this gap by providing performance benchmarks of various optimizers used in state preparation with Variational Quantum Algorithms. We conducted extensive testing across multiple scenarios, including different target states, both ideal and sampling simulations, and varying numbers of basis gate layers. Our evaluations offer insights into the complexity of learning each type of target state and demonstrate that some optimizers perform better than others in this context. Notably, the Powell optimizer was found to be exceptionally robust against sampling errors, making it a preferred choice in scenarios prone to such inaccuracies. Additionally, the Simultaneous Perturbation Stochastic Approximation optimizer was distinguished for its efficiency and ability to handle increased parameter dimensionality effectively.

Kan, Shuwen [Fordham University]↗

Stochastic Modeling in a Multimaterial Continuum Mixture Shock Physics Code

Stochastic modelling approaches are presented to capture random effects at multiple time and length scales. Random processes that occur at the microscale produce nondeterministic effects at the macroscale. Here we present three stochastic modeling approaches that describe random processes at microscopic length scales and map these processes to the macroscopic length scale. The first stochastic modeling approach is based upon a particle based numerical technique to solve a Stochastic Differential Equation (SDE) using an arbitrary diffusion process to capture random processes at the microstructural level. The second approach prescribes a Probability Density Function (PDF) for the drift and diffusion of the random variable derived using the forward and backward Kolmogorov equations. This method requires mean and drift evolution PDF transport equations. The third approach is the coupling of multiple random variables which are dependent on each other. The relationship of the PDFs and a coupling function, known as a copula, produces a Joint Probability Density Function (JPDF). These stochastic modeling approaches are implemented into a Multiple Component (MC) shock physics computational code and used to model statistical fracture and reactive flow applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

On the Virasoro six-point identity block and chaos

We study six-point correlation functions in two dimensional conformal field theory, where the six operators are grouped in pairs with equal conformal dimension. Assuming large central charge $c$ and a sparse spectrum, the leading contribution to this correlation function is the six-point Virasoro identity block - corresponding to each distinct pair of operators fusing into the identity and its descendants. We call this the star channel. One particular term in the star channel identity block is the stress tensor $SL(2,\mathbb{R})$ (global) block, for which we derive an explicit expression. In the holographic context, this object corresponds to a direct measure of nonlinear effects in pure gravity. We calculate additional terms in the star channel identity block that contribute at the same order at large $c$ as the global block using the novel theory of reparametrizations, which extends the shadow operator formalism in a natural way. We investigate these blocks' relevance to quantum chaos in the form of six-point scrambling in an out-of time ordered correlator. Interestingly, the global block does not contribute to the scrambling mode of this correlator, implying that, to leading order, six-point scrambling is insensitive to the three-point graviton coupling in the bulk dual. Finally, we compare our findings with a different OPE channel, called the comb channel, and find the same result for the chaos exponent in this decomposition.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Once-in-a-lifetime encounter models for neutrino media. II. Quasisteady states and miscidynamic flavor evolution

We extended the once-in-a-lifetime encounter (OILE) model to stochastic interactions among neutrinos. As in the original OILE model, the new model reproduces the mean-field behavior of a dense neutrino gas for time 𝑡 ≲ (𝜇⁢𝛾) −1 , where 𝜇 measures the strength of the mean-field neutrino self-interaction potential and is proportional to the neutrino density, and the dimensionless “impact parameter” 𝛾 is a measure of the change in the flavor quantum state of a neutrino during interaction with another neutrino when the wave packets of the two neutrinos overlap. As in the mean-field case, the OILE model with random neutrino velocities experiences kinetic flavor decoherence as the flavor quantum states of the neutrinos diverge from each other. Unlike the mean-field case, however, the OILE model has a “collision term” due to the quantum entanglement among neutrinos. For 𝛾 ≪ 1, this incoherent effect can drive the neutrinos into a quasisteady state that is similar to the collective precession mode in a homogeneous and isotropic neutrino gas in the mean-field approximation. Subsequently, the collision term drives the neutrino gas adiabatically through different quasisteady states and eventually to flavor equilibration. This process is an example of miscidynamic flavor evolution, with the mixing equilibria being the quasisteady precession states.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

LigninGraphs: lignin structure determination with multiscale graph modeling

Lignin is an aromatic biopolymer found in ubiquitous sources of woody biomass. Designing and optimizing lignin valorization processes requires a fundamental understanding of lignin structures. Experimental characterization techniques, such as 2D-heteronuclear single quantum coherence (HSQC) nuclear magnetic resonance (NMR) spectra, could elucidate the global properties of the polymer molecules. Computer models could extend the resolution of experiments by representing structures at the molecular and atomistic scales. We introduce a graph-based multiscale modeling framework for lignin structure generation and visualization. The framework employs accelerated rejection-free polymerization and hierarchical Metropolis Monte Carlo optimization algorithms. We obtain structure libraries for various lignin feedstocks based on literature and new experimental NMR data for poplar wood, pinewood, and herbaceous lignin. The framework could guide researchers towards feasible lignin structures, efficient space exploration, and future kinetics modeling. Its software implementation in Python, LigninGraphs, is open-source and available on GitHub.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

TurboRVB: A many-body toolkit for ab initio electronic simulations by quantum Monte Carlo

TurboRVB is a computational package for ab initio Quantum Monte Carlo (QMC) simulations of both molecular and bulk electronic systems. The code implements two types of well established QMC algorithms: Variational Monte Carlo (VMC) and diffusion Monte Carlo in its robust and efficient lattice regularized variant. A key feature of the code is the possibility of using strongly correlated many-body wave functions (WFs), capable of describing several materials with very high accuracy, even when standard mean-field approaches [e.g., density functional theory (DFT)] fail. The electronic WF is obtained by applying a Jastrow factor, which takes into account dynamical correlations, to the most general mean-field ground state, written either as an antisymmetrized geminal power with spin-singlet pairing or as a Pfaffian, including both singlet and triplet correlations. This WF can be viewed as an efficient implementation of the so-called resonating valence bond (RVB) Ansatz, first proposed by Pauling and Anderson in quantum chemistry [L. Pauling, The Nature of the Chemical Bond (Cornell University Press, 1960)] and condensed matter physics [P.W. Anderson, Mat. Res. Bull 8, 153 (1973)], respectively. The RVB Ansatz implemented in TurboRVB has a large variational freedom, including the Jastrow correlated Slater determinant as its simplest, but nontrivial case. Moreover, it has the remarkable advantage of remaining with an affordable computational cost, proportional to the one spent for the evaluation of a single Slater determinant. Therefore, its application to large systems is computationally feasible. The WF is expanded in a localized basis set. Several basis set functions are implemented, such as Gaussian, Slater, and mixed types, with no restriction on the choice of their contraction. The code implements the adjoint algorithmic differentiation that enables a very efficient evaluation of energy derivatives, comprising the ionic forces. Thus, one can perform structural optimizations and molecular dynamics in the canonical NVT ensemble at the VMC level. For the electronic part, a full WF optimization (Jastrow and antisymmetric parts together) is made possible, thanks to state-of-the-art stochastic algorithms for energy minimization. In the optimization procedure, the first guess can be obtained at the mean-field level by a built-in DFT driver. The code was efficiently parallelized by using a hybrid MPI-OpenMP protocol, which is also an ideal environment for exploiting the computational power of modern Graphics Processing Unit accelerators.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

47 OTHER INSTRUMENTATION↗

A light weight regularization for wave function parameter gradients in quantum Monte Carlo

The parameter derivative of the expectation value of the energy, ∂E/∂p, is a key ingredient in variational Monte Carlo (VMC) wave function optimization methods. In some cases, a naïve estimate of this derivative suffers from an infinite variance, which inhibits the efficiency of optimization methods that rely on a stable estimate of the derivative. In this work, we derive a simple regularization of the naïve estimator, which is trivial to implement in existing VMC codes, has finite variance, and a negligible bias, which can be extrapolated to zero bias with no extra cost. We use this estimator to construct an unbiased, finite variance estimation of ∂E/∂p for a multi-Slater–Jastrow trial wave function on the LiH molecule and in the optimization of a multi-Slater–Jastrow trial wave function on the CuO molecule. This regularized estimator is a simple and efficient estimator of ∂E/∂p for VMC optimization techniques.

36 MATERIALS SCIENCE↗

Randomized Algorithms for Scientific Computing (RASC)

Randomized algorithms have propelled advances in artificial intelligence (AI) and represent a foundational research area in advancing AI for Science. Future advancements in DOE Office of Science priority areas such as climate science, astrophysics, fusion, advanced materials, combustion, and quantum computing all require randomized algorithms for surmounting challenges of complexity, robustness, and scalability. Advances in data collection and numerical simulation have changed the dynamics of scientific research and motivate the need for randomized algorithms. For instance, advances in imaging technologies such as X-ray ptychography, electron microscopy, electron energy loss spectroscopy, or adaptive optics lattice light-sheet microscopy collect hyperspectral imaging and scattering data in terabytes, at breakneck speed enabled by state-of-the-art detectors. The data collection is exceptionally fast compared with its analysis. Likewise, advances in high-performance architectures have made exascale computing a reality and changed the economies of scientific computing in the process. Floating-point operations that create data are essentially free in comparison with data movement. Thus far, most approaches have focused on creating faster hardware. Ironically, this faster hardware has exacerbated the problem by making data still easier to create. Under such an onslaught, scientists often resort to heuristic deterministic sampling schemes (e.g., low-precision arithmetic, sampling every nth element) and sacrifice potentially valuable accuracy. Dramatically better results can be achieved via randomized algorithms, reducing the data size as much as or more than naive deterministic subsampling can achieve, while retaining the high accuracy of computing on the full data set. By randomized algorithms we mean those algorithms that employ some form of randomness in internal algorithmic decisions to accelerate time to solution, increase scalability, or improve reliability. Examples include matrix sketching for solving large-scale least-squares problems (see Figure 1) and stochastic gradient descent for training machine learning models. We are not recommending heuristic methods but rather randomized algorithms that have certificates of correctness and probabilistic guarantees of optimality and near-optimality. Such approaches can be useful beyond acceleration, for example, in understanding how to avoid measure zero worst-case scenarios that plague methods such as QR matrix factorization.

97 MATHEMATICS AND COMPUTING↗

Uncontrolled Learning: Codesign of Neuromorphic Hardware Topology for Neuromorphic Algorithms

Neuromorphic computing has the potential to revolutionize future technologies and our understanding of intelligence, yet it remains challenging to realize in practice. The learning-from-mistakes algorithm, inspired by the brain's simple learning rules of inhibition and pruning, is one of the few brain-like training methods. This algorithm is implemented in neuromorphic memristive hardware through a codesign process that evaluates essential hardware trade-offs. While the algorithm effectively trains small networks as binary classifiers and perceptrons, performance declines significantly with increasing network size unless the hardware is tailored to the algorithm. This work investigates the trade-offs between depth, controllability, and capacity—the number of learnable patterns—in neuromorphic hardware. This highlights the importance of topology and governing equations, providing theoretical tools to evaluate a device's computational capacity based on its measurements and circuit structure. The findings show that breaking neural network symmetry enhances both controllability and capacity. Additionally, by pruning the circuit, neuromorphic algorithms in all-memristive circuits can utilize stochastic resources to create local contrasts in network weights. Through combined experimental and simulation efforts, the parameters are identified that enable networks to exhibit emergent intelligence from simple rules, advancing the potential of neuromorphic computing.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Efficient prediction of attosecond two-colour pulses from an X-ray free-electron laser with machine learning

Abstract X-ray free-electron lasers are sources of coherent, high-intensity X-rays with numerous applications in ultra-fast measurements and dynamic structural imaging. Due to the stochastic nature of the self-amplified spontaneous emission process and the difficulty in controlling injection of electrons, output pulses exhibit significant noise and limited temporal coherence. Standard measurement techniques used for characterizing two-coloured X-ray pulses are challenging, as they are either invasive or diagnostically expensive. In this work, we employ machine learning methods such as neural networks and decision trees to predict the central photon energies of pairs of attosecond fundamental and second harmonic pulses using parameters that are easily recorded at the high-repetition rate of a single shot. Using real experimental data, we apply a detailed feature analysis on the input parameters while optimizing the training time of the machine learning methods. Our predictive models are able to make predictions of central photon energy for one of the pulses without measuring the other pulse, thereby leveraging the use of the spectrometer without having to extend its detection window. We anticipate applications in X-ray spectroscopy using XFELs, such as in time-resolved X-ray absorption and photoemission spectroscopy, where improved measurement of input spectra will lead to better experimental outcomes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Quantal diffusion approach for multinucleon transfer processes in the 58,64 Ni + 208 Pb reactions: Toward the production of unknown neutron-rich nuclei

In recent years, substantial efforts have been made for the study of multinucleon transfer reactions at energies around the Coulomb barrier both experimentally and theoretically, aiming at the production of unknown neutron-rich heavy nuclei. It is crucial to provide reliable theoretical predictions based on microscopic theories with sufficient predictive power. Purpose: This article aims to clarify the applicability of the quantal diffusion approach based on the stochastic mean-field (SMF) theory for multinucleon transfer processes. Isotope production cross sections are evaluated for the reactions of 64 Ni + 208 Pb at E c.m. = 268 MeV and 58 Ni + 208 Pb at E c.m. = 270 MeV and are compared with available experimental data. Methods: Three-dimensional time-dependent Hartree-Fock (TDHF) calculations are carried out for a range of initial orbital angular momenta with Skyrme SLy4d functional. Quantal diffusion equations, derived based on the SMF theory, for variances and covariance of neutron and proton numbers of reaction products are solved, with microscopic drift and diffusion coefficients obtained from time evolution of occupied single-particle orbitals in TDHF. Secondary de-excitation processes, both particle evaporation and fission, are simulated by a statistical compound-nucleus de-excitation model, gemini++. Dynamics of a fast isospin equilibration process followed by a slow drift toward the mass symmetry are commonly observed, as expected. Various reaction outcomes are evaluated, including average mass and charge numbers of reaction products, total kinetic energy loss (TKEL), scattering angle, contact time, and production cross sections for primary and secondary products. By comparing with the experimental data, we find that SMF and TDHF quantitatively reproduce experimental data for few-nucleon-transfer channels around the average values. In contrast, for many-nucleon-transfer channels, we find that the SMF approach provides much better description of the experimentally measured isotopic distributions. The results underline the importance of beyond-mean-field effects, especially one-body (mean-field) fluctuations and correlations, in describing multinucleon transfer processes. Moreover, through a combined analysis of SMF with a statistical model, gemini++, we find a significant contribution of transfer-induced fission, which is consistent with the experimental observation. In some cases, the SMF approach overestimates the isotopic width, requiring further improvements of the theoretical description. Possible ways to improve the description are discussed. The SMF approach is designed to describe the quantum many-body problem according to an ensemble of mean-field trajectories, taking into account part of many-body correlations in the description. As it requires feasible computational costs comparable to the ordinary TDHF approach, together with further model improvements, it will be a promising tool in the search for optimal reaction conditions to produce yet-unknown neutron-rich heavy nuclei through the multinucleon transfer reaction.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Quantum bath augmented stochastic nonequilibrium atomistic simulations for molecular heat conduction

Classical molecular dynamics (MD) has been shown to be effective in simulating heat conduction in certain molecular junctions since it inherently takes into account some essential methodological components which are lacking in the quantum Landauer-type transport model, such as many-body full force-field interactions, anharmonicity effects and nonlinear responses for large temperature biases. However, the classical MD reaches its limit in the environments where the quantum effects are significant (e.g. with low-temperatures substrates, presence of extremely high frequency molecular modes). Here, we present an atomistic simulation methodology for molecular heat conduction that incorporates the quantum Bose–Einstein statistics into an “effective temperature” in the form of a modified Langevin equation. We show that the results from such a quasi-classical effective temperature MD method deviates drastically when the baths temperature approaches zero from classical MD simulations and the results converge to the classical ones when the bath approaches the high-temperature limit, which makes the method suitable for full temperature range. In addition, we show that our quasi-classical thermal transport method can be used to model the conducting substrate layout and molecular composition (e.g. anharmonicities, high-frequency modes). Anharmonic models are explicitly simulated via the Morse potential and compared to pure harmonic interactions to show the effects of anharmonicities under quantum colored bath setups. Finally, the chain length dependence of heat conduction is examined for one-dimensional polymer chains placed in between quantum augmented baths.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗