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Classical nonlinear dynamical systems are often characterized by their steady-state probability distribution functions (PDFs). Typically, PDFs are accumulated from numerical simulations that involve solving the underlying dynamical equations of motion using integration techniques. An alternative procedure, direct statistical simulation (DSS), solves for the statistics directly. One approach to DSS is the Fokker-Planck Equation (FPE), which can be used to find the PDF of classical dynamical systems. Here, we investigate the utility of Noisy Intermediate-Scale Quantum (NISQ) computers to find steady-state solutions to the FPE. We employ the Quantum Phase Estimation (QPE) and the Variational Quantum Eigensolver (VQE) algorithms to find the zero-mode of the FPE for one-dimensional Ornstein-Uhlenbeck problems enabling comparison with exact solutions. The quantum computed steady-state probability distribution functions (PDFs) are demonstrated to be in reasonable agreement with the classically computed PDFs. We conclude with a discussion of potential extensions to higher-dimensional dynamical systems., Comment: 14 pages, 6 figures

The ability to perform ab initio molecular dynamics simulations using potential energies calculated on quantum computers would allow virtually exact dynamics for chemical and biochemical systems, with substantial impacts on the fields of catalysis and biophysics. However, noisy hardware, the costs of computing gradients, and the number of qubits required to simulate large systems present major challenges to realizing the potential of dynamical simulations using quantum hardware. Here, we demonstrate that some of these issues can be mitigated by recent advances in machine learning. By combining transfer learning with techniques for building machine-learned potential energy surfaces, we propose a new path forward for molecular dynamics simulations on quantum hardware. We use transfer learning to reduce the number of energy evaluations that use quantum hardware by first training models on larger, less accurate classical datasets and then refining them on smaller, more accurate quantum datasets. We demonstrate this approach by training machine learning models to predict a molecule's potential energy using Behler-Parrinello neural networks. When successfully trained, the model enables energy gradient predictions necessary for dynamics simulations that cannot be readily obtained directly from quantum hardware. To reduce the quantum resources needed, the model is initially trained with data derived from low-cost techniques, such as Density Functional Theory, and subsequently refined with a smaller dataset obtained from the optimization of the Unitary Coupled Cluster ansatz. We show that this approach significantly reduces the size of the quantum training dataset while capturing the high accuracies needed for quantum chemistry simulations., Comment: 1- pages, 12 figures

The intrinsic dynamics of most proteins are central to their function. Protein tyrosine kinases such as Abl1 undergo significant conformational changes that modulate their activity in response to different stimuli. These conformational changes constitute a conserved mechanism for self-regulation that dramatically impacts kinases' affinities for inhibitors. Few studies have attempted to extensively sample the pathways and elucidate the mechanisms that underlie kinase inactivation. Seeking to bridge this knowledge gap, we present a thorough analysis of the ``DFG flip'' inactivation pathway in Abl1 kinase. By leveraging the power of the Weighted Ensemble methodology, which accelerates sampling without the use of biasing forces, we have comprehensively simulated DFG flip events in Abl1 and its inhibitor-resistant variants, revealing a rugged landscape punctuated by potentially druggable intermediate states. Through our strategy, we successfully simulated dozens of uncorrelated DFG flip events distributed along two principal pathways, identified the molecular mechanisms that govern them, and measured their relative probabilities. Further, we show that the compound Glu255Lys/Val Thr315Ile Abl1 variants owe their inhibitor resistance phenotype to an increase in the free energy barrier associated with completing the DFG flip. This barrier stabilizes Abl1 variants in conformations that can lead to loss of binding for Type-II inhibitors such as Imatinib or Ponatinib. Finally, we contrast our Abl1 observations with the relative state distributions and propensity for undergoing a DFG flip of evolutionarily-related protein tyrosine kinases with diverging Type-II inhibitor binding affinities. Altogether, we expect that our work will be of significant importance for protein tyrosine kinase inhibitor discovery., Comment: 34 pages, 10 figures

The Chemical Master Equation (CME) provides a highly accurate, yet extremely resource-intensive representation of a stochastic chemical reaction network and its kinetics due to the exponential scaling of its possible states with the number of reacting species. In this work, we demonstrate how quantum algorithms and hardware can be employed to model stochastic chemical kinetics as described by the CME using the Schl\"ogl Model of a trimolecular reaction network as an illustrative example. To ground our study of the performance of our quantum algorithms, we first determine a range of suitable parameters for constructing the stochastic Schl\"ogl operator in the mono- and bistable regimes of the model using a classical computer and then discuss the appropriateness of our parameter choices for modeling approximate kinetics on a quantum computer. We then apply the Variational Quantum Deflation (VQD) algorithm to evaluate the smallest-magnitude eigenvalues, $\lambda_0$ and $\lambda_1$, which describe the transition rates of both the mono- and bi-stable systems, and the Quantum Phase Estimation (QPE) algorithm combined with the Variational Quantum Singular Value Decomposition (VQSVD) algorithm to estimate the zeromode (ground state) of the bistable case. Our quantum computed results from both noisy and noiseless quantum simulations agree within a few percent with the classically computed eigenvalues and zeromode. Altogether, our work outlines a practical path toward the quantum solution of exponentially complex stochastic chemical kinetics problems and other related stochastic differential equations.

Descriptors are physically-inspired schemes for representing atomistic systems that play a central role in the construction of models of potential energy surfaces. Although physical intuition can be flexibly encoded into descriptor schemes, they are generally ultimately guided only by the spatial or topological arrangement of atoms in the system. Here, we propose a novel approach for the optimization of descriptors based on encoding information about geodesic distances along potential energy manifolds into the hyperparameters of commonly used descriptor schemes. To accomplish this, we combine two ideas: (1) a differential-geometric approach for the fast estimation of approximate geodesic distances; and (2) an information-theoretic evaluation metric - information imbalance - for measuring the shared information between two distance measures. Using the MD22 datasets of ethanol, malonaldehyde, and aspirin, we first show that Euclidean (in Cartesian coordinates) and geodesic distances are inequivalent distance measures, indicating the need for updated ground-truth distance measures that go beyond the Euclidean distance. We then utilize a Bayesian optimization framework to show that descriptors (in this case, atom-centered symmetry functions) can be optimized to maximally express a certain type of distance information, such as Euclidean or geodesic information. We also show that modifying the Bayesian optimization algorithm to minimize a combined Euclidean+geodesic objective function can yield descriptors that not only express both Euclidean and geodesic distance information simultaneously, but in fact resolve substantial disagreements between descriptors optimized to encode only one type of distance measure. We discuss the relevance of our approach to the design of more physically rich and informative descriptors that can encode useful, alternative information about molecular systems.

Stochastic electronic structure theories, e.g., Quantum Monte Carlo methods, enable highly accurate total energy calculations which in principle can be used to construct highly accurate potential energy surfaces. However, their stochastic nature poses a challenge to the computation and use of forces and Hessians, which are typically required in algorithms for minimum-energy pathway (MEP) and transition state (TS) identification, such as the nudged-elastic band (NEB) algorithm and its climbing image formulation. Here, we present strategies that utilize the surrogate Hessian line-search method - previously developed for QMC structural optimization - to efficiently identify MEP and TS structures without requiring force calculations at the level of the stochastic electronic structure theory. By modifying the surrogate Hessian algorithm to operate in path-orthogonal subspaces and on saddle points, we show that it is possible to identify MEPs and TSs using a force-free QMC approach. We demonstrate these strategies via two examples, the inversion of the ammonia molecule and an SN2 reaction. We validate our results using Density Functional Theory- and coupled cluster-based NEB calculations. We then introduce a hybrid DFT-QMC approach to compute thermodynamic and kinetic quantities - free energy differences, rate constants, and equilibrium constants - that incorporates stochastically-optimized structures and their energies, and show that this scheme improves upon DFT accuracy. Our methods generalize straightforwardly to other systems and other high-accuracy theories that similarly face challenges computing energy gradients, paving the way for highly accurate PES mapping, transition state determination, and thermodynamic and kinetic calculations, at significantly reduced computational expense.

This paper presents a novel approach for predicting the relative populations of protein conformations using AlphaFold 2, an AI-powered method that has revolutionized biology by enabling the accurate prediction of protein structures. While AlphaFold 2 has shown exceptional accuracy and speed, it is designed to predict proteins' single ground state conformations and is limited in its ability to predict fold switching and the effects of mutations on conformational landscapes. Here, we demonstrate how AlphaFold 2 can directly predict the relative populations of different conformations of proteins and even accurately predict changes in those populations induced by mutations by subsampling multiple sequence alignments. We tested our method against NMR experiments on two proteins with drastically different amounts of available sequence data, Abl1 kinase and the granulocyte-macrophage colony-stimulating factor, and predicted changes in their relative state populations with accuracies in excess of 80%. Our method offers a fast and cost-effective way to predict protein conformations and their relative populations at even single point mutation resolution, making it a useful tool for pharmacology, analyzing NMR data, and studying the effects of evolution., Comment: 26 pages, 9 figures

Diffusion Monte Carlo (DMC) is one of the most accurate techniques available for calculating the electronic properties of molecules and materials, yet it often remains a challenge to economically compute forces using this technique. As a result, ab initio molecular dynamics simulations and geometry optimizations that employ Diffusion Monte Carlo forces are often out of reach. One potential approach for accelerating the computation of "DMC forces" is to machine learn these forces from DMC energy calculations. In this work, we employ Behler-Parrinello Neural Networks to learn DMC forces from DMC energy calculations for geometry optimization and molecular dynamics simulations of small molecules. We illustrate the unique challenges that stem from learning forces without explicit force data and from noisy energy data by making rigorous comparisons of potential energy surface, dynamics, and optimization predictions among ab initio Density Functional Theory (DFT) simulations and machine learning models trained on DFT energies with forces, DFT energies without forces, and DMC energies without forces. We show for three small molecules - C2, H2O, and CH3Cl - that machine learned DMC dynamics can reproduce average bond lengths and angles within a few percent of known experimental results at a 100th of the typical cost. Our work describes a much-needed means of performing dynamics simulations on high-accuracy, DMC PESs and for generating DMC-quality molecular geometries given current algorithmic constraints.

Diffusion Monte Carlo (DMC) is being recognized as a higher-accuracy, albeit more computationally expensive, alternative to Density Functional Theory (DFT) for energy predictions of catalytic systems. A major computational bottleneck in the use of DMC for catalysis is the need to perform finite-size extrapolations by simulating increasingly large periodic cells (supercells) to eliminate many-body finite-size effects and obtain energies in the thermodynamic limit. Here, we show that this computational cost can be significantly reduced by leveraging the cancellation of many-body finite-size errors that accompanies the evaluation of energy differences when calculating quantities like binding energies and mapping potential energy surfaces. We test the error cancellation and convergence in two well-known adsorbate/slab systems, H2O/LiH(001) and CO/Pt(111). Based on this, we identify strategies for obtaining binding energies in the thermodynamic limit that optimize error cancellation to balance accuracy and computational efficiency. We then predict the correct order of adsorption site preference on CO/Pt(111), a challenging problem for DFT. Our accurate, inexpensive DMC calculations recover the top > bridge > hollow site order, in agreement with experimental observations. We proceed to map the potential energy surface of CO hopping between Pt(111) adsorption sites. This reveals the existence of an L-shaped top-bridge-hollow diffusion trajectory characterized by energy barriers that provide an additional kinetic justification for experimental observations of CO/Pt(111) adsorption. Overall, this work demonstrates that it is routinely possible to achieve order-of-magnitude speedups and memory savings in DMC calculations by taking advantage of error cancellation in the calculation of energy differences that are ubiquitous in heterogeneous catalysis and surface chemistry more broadly.

Ion traps are promising architectures for implementing scalable quantum computing, but they suffer from excessive "anomalous" heating that prevents their full potential from being realized. This heating, which is orders of magnitude larger than that expected from Johnson-Nyquist noise, results in ion motion that leads to decoherence and reduced fidelity in quantum logic gates. The exact origin of anomalous heating is an open question, but experiments point to adsorbates on trap electrodes as a likely source. Many different models of anomalous heating have been proposed, but these models have yet to pinpoint the atomistic origin of the experimentally-observed $1/\omega$ electric field noise scaling observed in ion traps at frequencies between 0.1-10 MHz. In this work, we perform the first computational study of the ion trap electric field noise produced by the motions of multiple monolayers of adsorbates described by first principles potentials. In so doing, we show that correlated adsorbate motions play a definitive role in producing $1/\omega$ noise and identify candidate collective adsorbate motions, including translational and rotational motions of adsorbate patches and multilayer exchanges, that give rise to $1/\omega$ scaling at the MHz frequencies typically employed in ion traps. These results demonstrate that multi-adsorbate systems, even simple ones, can give rise to a set of activated motions that can produce the $1/\omega$ noise observed in ion traps and that collective, rather than individual, adsorbate motions are much more likely to give rise to low-frequency heating.

We present first principles calculations of the electrostatic properties of Ba$_2$NaOsO$_6$ (BNOO), a 5$d^1$ Mott insulator with strong spin orbit coupling (SOC) in its low temperature quantum phases. In light of recent NMR experiments showing that BNOO develops a local octahedral distortion that is accompanied by the emergence of an electric field gradient (EFG) and precedes the formation of long range magnetic order [Lu et al., Nature Comm. 8, 14407 (2017), Liu et al., Phys. Rev. B. 97, 224103 (2018), Liu et al., Physica B. 536, 863 (2018)], we calculated BNOO's EFG tensor for several different model distortions. The local orthorhombic distortion that we identified as mostly strongly agreeing with experiment corresponds to a Q2 distortion mode of the Na-O octahedra, in agreement with conclusions given in [Liu et al., Phys. Rev. B. 97, 224103 (2018)]. Furthermore, we found that the EFG is insensitive to the type of underlying magnetic order. By combining NMR results with first principles modeling, we have thus forged a more complete understanding of BNOO's structural and magnetic properties, which could not be achieved based upon experiment or theory alone., Comment: 11 pages, Submitted to Physical Review B

Molecular data systems have the potential to store information at dramatically higher density than existing electronic media. Some of the first experimental demonstrations of this idea have used DNA, but nature also uses a wide diversity of smaller non-polymeric molecules to preserve, process, and transmit information. In this paper, we present a general framework for quantifying chemical memory, which is not limited to polymers and extends to mixtures of molecules of all types. We show that the theoretical limit for molecular information is two orders of magnitude denser by mass than DNA, although this comes with different practical constraints on total capacity. We experimentally demonstrate kilobyte-scale information storage in mixtures of small synthetic molecules, and we consider some of the new perspectives that will be necessary to harness the information capacity available from the vast non-genomic chemical space.

In the study of strongly-correlated, many-electron systems, the Hubbard Kanamori (HK) model has emerged as one of the prototypes for transition metal oxide physics. The model is multi-band in nature and contains Hund's coupling terms, which have pronounced effects on metal-insulator transitions, high-temperature superconductivity, and other physical properties. In the following, we present a complete theoretical framework for treating the HK model using the ground state Auxiliary Field Quantum Monte Carlo (AFQMC) method and analyze its performance on few-band models whose parameters approximate those observed in ruthenate, rhodates, and other materials exhibiting Hund's physics. Unlike previous studies, the constrained path and phaseless approximations are used to respectively control the sign and phase problems, which enables high accuracy modeling of the HK model's ground state properties within parameter regimes of experimental interest. We demonstrate that, after careful consideration of the Hubbard-Stratonovich transformations and trial wave functions employed, relative errors in the energy of less than 1% can routinely be achieved for moderate to large values of the Hund's coupling constant. Crucially, our methodology also accurately predicts magnetic ordering and phase transitions. The results presented open the door to more predictive modeling of Hund's physics within a wide range of strongly-correlated materials using AFQMC., Comment: 14 pages, 7 figures, 2 tables

In order to realize the full potential of ion trap quantum computers, an improved understanding is required of the motional heating that trapped ions experience. Experimental studies of the temperature-, frequency-, and ion--electrode distance-dependence of the electric field noise responsible for motional heating, as well as the noise before and after ion bombardment cleaning of trap electrodes, suggest that fluctuations of adsorbate dipoles are a likely source of so-called `anomalous heating,' or motional heating of the trapped ions at a rate much higher than the Johnson noise limit. Previous computational studies have investigated how the fluctuation of model adsorbate dipoles affects anomalous heating. However, the way in which specific adsorbates affect the electric field noise has not yet been examined, and an electric dipole model employed in previous studies is only accurate for a small subset of possible adsorbates. Here, we analyze the behavior of both in-plane and out-of-plane vibrational modes of fifteen adsorbate--electrode combinations within the independent fluctuating dipole model, utilizing accurate first principles computational methods to determine the surface-induced dipole moments. We find the chemical specificity of the adsorbate can change the electric field noise by seven orders of magnitude and specifically that soft in-plane modes of weakly-adsorbed hydrocarbons produce the greatest noise and ion heating. We discuss the dynamics captured by the fluctuating dipole model, namely the adsorbate dependent turn-on temperature and electric field noise magnitude, and also discuss the model's failure to reproduce the measured 1/$\omega$ noise frequency scaling with a single adsorbate species. We suggest future research directions for improved, quantitatively predictive models based on extensions of the present framework to multiple interacting adsorbates.

In this work, we introduce a new type of linear classifier that is implemented in a chemical form. We propose a novel encoding technique which simultaneously represents multiple datasets in an array of microliter-scale chemical mixtures. Parallel computations on these datasets are performed as robotic liquid handling sequences, whose outputs are analyzed by high-performance liquid chromatography. As a proof of concept, we chemically encode several MNIST images of handwritten digits and demonstrate successful chemical-domain classification of the digits using volumetric perceptrons. We additionally quantify the performance of our method with a larger dataset of binary vectors and compare the experimental measurements against predicted results. Paired with appropriate chemical analysis tools, our approach can work on increasingly parallel datasets. We anticipate that related approaches will be scalable to multilayer neural networks and other more complex algorithms. Much like recent demonstrations of archival data storage in DNA, this work blurs the line between chemical and electrical information systems, and offers early insight into the computational efficiency and massive parallelism which may come with computing in chemical domains., Comment: Accepted to 2018 IEEE International Conference on Rebooting Computing

Neutral molecules with sufficiently large dipole moments can bind electrons in diffuse nonvalence orbitals with most of their charge density far from the nuclei, forming so-called dipole-bound anions. Because long-range correlation effects play an important role in the binding of an excess electron and overall binding energies are often only of the order of 10-100s of wave numbers, predictively modeling dipole-bound anions remains a challenge. Here, we demonstrate that quantum Monte Carlo methods can accurately characterize molecular dipole-bound anions with near threshold dipole moments. We also show that correlated sampling Auxiliary Field Quantum Monte Carlo is particularly well-suited for resolving the fine energy differences between the neutral and anionic species. These results shed light on the fundamental limitations of quantum Monte Carlo methods and pave the way toward using them for the study of weakly-bound species that are too large to model using traditional electron structure methods.

QMCPACK is an open source quantum Monte Carlo package for ab-initio electronic structure calculations. It supports calculations of metallic and insulating solids, molecules, atoms, and some model Hamiltonians. Implemented real space quantum Monte Carlo algorithms include variational, diffusion, and reptation Monte Carlo. QMCPACK uses Slater-Jastrow type trial wave functions in conjunction with a sophisticated optimizer capable of optimizing tens of thousands of parameters. The orbital space auxiliary field quantum Monte Carlo method is also implemented, enabling cross validation between different highly accurate methods. The code is specifically optimized for calculations with large numbers of electrons on the latest high performance computing architectures, including multicore central processing unit (CPU) and graphical processing unit (GPU) systems. We detail the program's capabilities, outline its structure, and give examples of its use in current research calculations. The package is available at http://www.qmcpack.org .

Fractional derivatives are nonlocal differential operators of real order that often appear in models of anomalous diffusion and a variety of nonlocal phenomena. Recently, a version of the Schr\"odinger Equation containing a fractional Laplacian has been proposed. In this work, we develop a Fractional Path Integral Monte Carlo algorithm that can be used to study the finite temperature behavior of the time-independent Fractional Schr\"odinger Equation for a variety of potentials. In so doing, we derive an analytic form for the finite temperature fractional free particle density matrix and demonstrate how it can be sampled to acquire new sets of particle positions. We employ this algorithm to simulate both the free particle and $^{4}$He (Aziz) Hamiltonians. We find that the fractional Laplacian strongly encourages particle delocalization, even in the presence of interactions, suggesting that fractional Hamiltonians may manifest atypical forms of condensation. Our work opens the door to studying fractional Hamiltonians with arbitrarily complex potentials that escape analytical solutions., Comment: 18 pages, 14 figures

Quantum Monte Carlo (QMC) algorithms have long relied on Jastrow factors to incorporate dynamic correlation into trial wave functions. While Jastrow-type wave functions have been widely employed in real-space algorithms, they have seen limited use in second-quantized QMC methods, particularly in projection methods that involve a stochastic evolution of the wave function in imaginary time. Here we propose a scheme for generating Jastrow-type correlated trial wave functions for auxiliary-field QMC methods. The method is based on decoupling the two-body Jastrow into one-body projectors coupled to auxiliary fields, which then operate on a single determinant to produce a multi-determinant trial wave function. We demonstrate that intelligent sampling of the most significant determinants in this expansion can produce compact trial wave functions that reduce errors in the calculated energies. Our technique may be readily generalized to accommodate a wide range of two-body Jastrow factors and applied to a variety of model and chemical systems., Comment: Revised version with 7 pages, 6 figures, 2 tables

We present a quantum Monte Carlo (QMC) technique for calculating the exact finite-temperature properties of Bose-Fermi mixtures. The Bose-Fermi Auxiliary-Field Quantum Monte Carlo (BF-AFQMC) algorithm combines two methods, a finite-temperature AFQMC algorithm for bosons and a variant of the standard AFQMC algorithm for fermions, into one algorithm for mixtures. We demonstrate the accuracy of our method by comparing its results for the Bose-Hubbard and Bose-Fermi-Hubbard models against those produced using exact diagonalization for small systems. Comparisons are also made with mean-field theory and the worm algorithm for larger systems. As is the case with most fermion Hamiltonians, a sign or phase problem is present in BF-AFQMC. We discuss the nature of these problems in this framework and describe how they can be controlled with well-studied approximations to expand BF-AFQMC's reach. The new algorithm can serve as an essential tool for answering many unresolved questions about many-body physics in mixed Bose-Fermi systems., Comment: 19 pages, 6 figures

The extent of coupling between the folding of a protein and its binding to a substrate varies from protein to protein. Some proteins have highly structured native states in solution, while others are natively disordered and only fold fully upon binding. In this Letter, we use Monte Carlo simulations to investigate how disordered polymer chains grafted around a binding site affect the folding and binding of three model proteins. The protein that approaches the substrate fully folded is more hindered during the binding process than those whose folding and binding are cooperative. The polymer chains act as localized crowding agents and can select correctly folded and bound configurations in favor of non-specifically adsorbed states. The free energy change for forming all intra-protein and protein-substrate contacts can depend non-monotonically on the polymer length., Comment: 5 pages, 5 figures, 1 table