General Classification of Qubit Encodings in Ultracold Diatomic Molecules.

Owing to their rich internal structure and significant long-range interactions, ultracold molecules have been widely explored as carriers of quantum information. Several different schemes for encoding qubits into molecular states, both bare and field-dressed, have been proposed. At the same time, the rich internal structure of molecules leaves many unexplored possibilities for qubit encodings. We show that all molecular qubit encodings can be classified into four classes by the type of the effective interaction between the qubits. In the case of polar molecules, the four classes are determined by the relative magnitudes of matrix elements of the dipole moment operator in the single-molecule basis. We exemplify our classification scheme by considering the encoding of the effective spin-1/2 system into nonadjacent rotational states (e.g., N = 0 and 2) of polar and nonpolar molecules with the same nuclear spin projection. Our classification scheme is designed to inform the optimal choice of molecular qubit encoding for quantum information storage and processing applications, as well as for dynamical generation of many-body entangled states and for quantum annealing.

Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines.

Machine learning is considered to be one of the most promising applications of quantum computing. Therefore, the search for quantum advantage of the quantum analogues of machine learning models is a key research goal. Here, we show that variational quantum classifiers and support vector machines with quantum kernels can solve a classification problem based on the k-FORRELATION problem, which is known to be PROMISEBQP-complete. Because the PROMISEBQP complexity class includes all Bounded-Error Quantum Polynomial-Time (BQP) decision problems, our results imply that there exists a feature map and a quantum kernel that make variational quantum classifiers and quantum kernel support vector machines efficient solvers for any BQP problem. Hence, this work implies that their feature map and quantum kernel, respectively, can be designed to have a quantum advantage for any classification problem that cannot be classically solved in polynomial time but contrariwise by a quantum computer.

Anderson Localization of Composite Particles.

We investigate the effect of coupling between translational and internal degrees of freedom of composite quantum particles on their localization in a random potential. We show that entanglement between the two degrees of freedom weakens localization due to the upper bound imposed on the inverse participation ratio by purity of a quantum state. We perform numerical calculations for a two-particle system bound by a harmonic force in a 1D disordered lattice and a rigid rotor in a 2D disordered lattice. We illustrate that the coupling has a dramatic effect on localization properties, even with a small number of internal states participating in quantum dynamics.

Universal Probability Distributions of Scattering Observables in Ultracold Molecular Collisions.

Currently, quantum dynamics theory cannot be used for quantitative predictions of molecular scattering observables at low temperatures because of two problems. The first problem is the extreme sensitivity of the low-temperature observables to details of potential energy surfaces (PESs) parametrizing the nuclear Schrödinger equation. The second problem is the large size of the basis sets required for the numerical integration of the Schrödinger equation for strongly interacting molecules in the presence of fields, which precludes the application of rigorous quantum theory to all but a few atom-molecule systems. Here, we show that, if the scattering problem is formulated as a probabilistic prediction, quantum theory can provide reliable results with exponentially reduced numerical effort. Specifically, we show that the probability distributions that an observable is in a certain range of values can be obtained by averaging the results of scattering calculations with much smaller basis sets than required for calculations of individual scattering cross sections. Moreover, we show that such distributions do not rely on the precise knowledge of the PES. This opens the possibility of making probabilistic predictions of experimentally relevant observables for a wide variety of molecular systems, currently considered out of reach of quantum dynamics theory. We demonstrate the approach by computing the probability for elastic scattering of CaH and SrOH molecules by Li atoms and SrF molecules by Rb atoms.

Six-dimensional potential energy surface for NaK-NaK collisions: Gaussian process representation with correct asymptotic form.

Constructing accurate global potential energy surfaces (PESs) describing chemically reactive molecule-molecule collisions of alkali metal dimers presents a major challenge. To be suitable for quantum scattering calculations, such PESs must represent accurately three- and four-body interactions, describe conical intersections, and have a proper asymptotic form at the long range. Here, we demonstrate that such global potentials can be obtained by Gaussian Process (GP) regression merged with the analytic asymptotic expansions at the long range. We propose an efficient sampling technique, which allows us to construct an accurate global PES accounting for different chemical arrangements with <2500 ab initio calculations. We apply this method to (NaK)2 and obtain the first global PES for a system of four alkali metal atoms. The resulting surface exhibits a complex landscape including a pair and a quartet of symmetrically equivalent local minima and a seam of conical intersections. The dissociation energy found from our ab initio calculations is 4534 cm-1. This result is reproduced by the GP models with an error of less than 3%. The GP models of the PES allow us to analyze the features of the global PES, representative of general alkali metal four-atom interactions. Understanding these interactions is of key importance in the field of ultracold chemistry.

Nonadditivity of Optical and Casimir-Polder Potentials.

An atom irradiated by an off-resonant laser field near a surface is expected to experience the sum of two fundamental potentials, the optical potential of the laser field and the Casimir-Polder potential of the surface. Here, we report a new nonadditive potential, namely, the laser-induced Casimir-Polder potential, which arises from a correlated coupling of the atom with both the laser and the quantum vacuum. We apply this result to an experimentally realizable scenario of an atomic mirror with an evanescent laser beam leaking out of a surface. We show that the nonadditive term is significant for realistic experimental parameters, transforming potential barriers into potential wells, which can be used to trap atoms near surfaces.

Neural networks vs Gaussian process regression for representing potential energy surfaces: A comparative study of fit quality and vibrational spectrum accuracy.

For molecules with more than three atoms, it is difficult to fit or interpolate a potential energy surface (PES) from a small number of (usually ab initio) energies at points. Many methods have been proposed in recent decades, each claiming a set of advantages. Unfortunately, there are few comparative studies. In this paper, we compare neural networks (NNs) with Gaussian process (GP) regression. We re-fit an accurate PES of formaldehyde and compare PES errors on the entire point set used to solve the vibrational Schrödinger equation, i.e., the only error that matters in quantum dynamics calculations. We also compare the vibrational spectra computed on the underlying reference PES and the NN and GP potential surfaces. The NN and GP surfaces are constructed with exactly the same points, and the corresponding spectra are computed with the same points and the same basis. The GP fitting error is lower, and the GP spectrum is more accurate. The best NN fits to 625/1250/2500 symmetry unique potential energy points have global PES root mean square errors (RMSEs) of 6.53/2.54/0.86 cm-1, whereas the best GP surfaces have RMSE values of 3.87/1.13/0.62 cm-1, respectively. When fitting 625 symmetry unique points, the error in the first 100 vibrational levels is only 0.06 cm-1 with the best GP fit, whereas the spectrum on the best NN PES has an error of 0.22 cm-1, with respect to the spectrum computed on the reference PES. This error is reduced to about 0.01 cm-1 when fitting 2500 points with either the NN or GP. We also find that the GP surface produces a relatively accurate spectrum when obtained based on as few as 313 points.

Time-Dependent Wave Packet Dynamics Calculations of Cross Sections for Ultracold Scattering of Molecules.

Because the de Broglie wavelength of ultracold molecules is very large, the cross sections for collisions of molecules at ultracold temperatures are always computed by the time-independent quantum scattering approach. Here, we report the first accurate time-dependent wave packet dynamics calculation for reactive scattering of ultracold molecules. Wave packet dynamics calculations can be applied to molecular systems with more dimensions and provide real-time information on the process of bond rearrangement and/or energy exchange in molecular collisions. Our work thus makes possible the extension of rigorous quantum calculations of ultracold reaction properties to polyatomic molecules and adds a new powerful tool for the study of ultracold chemistry.

Light Bipolarons Stabilized by Peierls Electron-Phonon Coupling.

It is widely accepted that phonon-mediated high-temperature superconductivity is impossible at ambient pressure, because of the very large effective masses of polarons or bipolarons at strong electron-phonon coupling. Here we challenge this belief by showing that strongly bound yet very light bipolarons appear for strong Peierls coupling. These bipolarons also exhibit many other unconventional properties; e.g., at strong coupling there are two low-energy bipolaron bands that are stable against strong Coulomb repulsion. Using numerical simulations and analytical arguments, we show that these properties result from the specific form of the phonon-mediated interaction, which is of "pair hopping" instead of regular density-density type. This unusual effective interaction is bound to have nontrivial consequences for the superconducting state expected to arise at finite carrier concentrations and should favor a large critical temperature.

Extrapolating Quantum Observables with Machine Learning: Inferring Multiple Phase Transitions from Properties of a Single Phase.

We present a machine-learning method for predicting sharp transitions in a Hamiltonian phase diagram by extrapolating the properties of quantum systems. The method is based on Gaussian process regression with a combination of kernels chosen through an iterative procedure maximizing the predicting power of the kernels. The method is capable of extrapolating across the transition lines. The calculations within a given phase can be used to predict not only the closest sharp transition but also a transition removed from the available data by a separate phase. This makes the present method particularly valuable for searching phase transitions in the parts of the parameter space that cannot be probed experimentally or theoretically.

A general method for controlling and resolving rotational orientation of molecules in molecule-surface collisions.

The outcome of molecule-surface collisions can be modified by pre-aligning the molecule; however, experiments accomplishing this are rare because of the difficulty of preparing molecules in aligned quantum states. Here we present a general solution to this problem based on magnetic manipulation of the rotational magnetic moment of the incident molecule. We apply the technique to the scattering of H2 from flat and stepped copper surfaces. We demonstrate control of the molecule's initial quantum state, allowing a direct comparison of differences in the stereodynamic scattering from the two surfaces. Our results show that a stepped surface exhibits a much larger dependence of the corrugation of the interaction on the alignment of the molecule than the low-index surface. We also demonstrate an extension of the technique that transforms the set-up into an interferometer, which is sensitive to molecular quantum states both before and after the scattering event.

Gaussian process model for extrapolation of scattering observables for complex molecules: From benzene to benzonitrile.

We consider a problem of extrapolating the collision properties of a large polyatomic molecule A-H to make predictions of the dynamical properties for another molecule related to A-H by the substitution of the H atom with a small molecular group X, without explicitly computing the potential energy surface for A-X. We assume that the effect of the -H →-X substitution is embodied in a multidimensional function with unknown parameters characterizing the change of the potential energy surface. We propose to apply the Gaussian Process model to determine the dependence of the dynamical observables on the unknown parameters. This can be used to produce an interval of the observable values which corresponds to physical variations of the potential parameters. We show that the Gaussian Process model combined with classical trajectory calculations can be used to obtain the dependence of the cross sections for collisions of C6H5CN with He on the unknown parameters describing the interaction of the He atom with the CN fragment of the molecule. The unknown parameters are then varied within physically reasonable ranges to produce a prediction uncertainty of the cross sections. The results are normalized to the cross sections for He - C6H6 collisions obtained from quantum scattering calculations in order to provide a prediction interval of the thermally averaged cross sections for collisions of C6H5CN with He.

Gaussian Process Model for Collision Dynamics of Complex Molecules.

We show that a Gaussian process model can be combined with a small number (of order 100) of scattering calculations to provide a multidimensional dependence of scattering observables on the experimentally controllable parameters (such as the collision energy or temperature) as well as the potential energy surface (PES) parameters. For the case of Ar-C_{6}H_{6} collisions, we show that 200 classical trajectory calculations are sufficient to provide a ten-dimensional hypersurface, giving the dependence of the collision lifetimes on the collision energy, internal temperature, and eight PES parameters. This can be used for solving the inverse scattering problem, for the efficient calculation of thermally averaged observables, for reducing the error of the molecular dynamics calculations by averaging over the PES variations, and for the analysis of the sensitivity of the observables to individual parameters determining the PES. Trained by a combination of classical and quantum calculations, the model provides an accurate description of the quantum scattering cross sections, even near scattering resonances.

Tuning Bimolecular Chemical Reactions by Electric Fields.

We develop a theoretical method for solving the quantum mechanical reactive scattering problem in the presence of external fields based on a hyperspherical coordinate description of the reaction complex combined with the total angular momentum representation for collisions in external fields. The method allows us to obtain converged results for the chemical reaction LiF+H→Li+HF in an electric field. Our calculations demonstrate that, by inducing couplings between states of different total angular momenta, electric fields with magnitudes <150 kV/cm give rise to resonant scattering and a significant modification of the total reaction probabilities, product state distributions, and the branching ratios for reactive versus inelastic scattering.

Collision lifetimes of polyatomic molecules at low temperatures: benzene-benzene vs benzene-rare gas atom collisions.

We use classical trajectory calculations to study the effects of the interaction strength and the geometry of rigid polyatomic molecules on the formation of long-lived collision complexes at low collision energies. We first compare the results of the calculations for collisions of benzene molecules with rare gas atoms He, Ne, Ar, Kr, and Xe. The comparison illustrates that the mean lifetimes of the collision complexes increase monotonically with the strength of the atom-molecule interaction. We then compare the results of the atom-benzene calculations with those for benzene-benzene collisions. The comparison illustrates that the mean lifetimes of the benzene-benzene collision complexes are significantly reduced due to non-ergodic effects prohibiting the molecules from sampling the entire configuration space. We find that the thermally averaged lifetimes of the benzene-benzene collisions are much shorter than those for Xe with benzene and similar to those for Ne with benzene.

Collision dynamics of polyatomic molecules containing carbon rings at low temperatures.

We explore the collision dynamics of complex hydrocarbon molecules (benzene, coronene, adamantane, and anthracene) containing carbon rings in a cold buffer gas of (3)He. For benzene, we present a comparative analysis of the fully classical and fully quantum calculations of elastic and inelastic scattering cross sections at collision energies between 1 and 10 cm(-1). The quantum calculations are performed using the time-independent coupled channel approach and the coupled-states approximation. We show that the coupled-states approximation is accurate at collision energies between 1 and 20 cm(-1). For the classical dynamics calculations, we develop an approach exploiting the rigidity of the carbon rings and including low-energy vibrational modes without holonomic constraints. Our results illustrate the effect of the molecular shape and the vibrational degrees of freedom on the formation of long-lived resonance states that lead to low-temperature clustering.

Investigating polaron transitions with polar molecules.

We determine the phase diagram of a polaron model with mixed breathing-mode and Su-Schrieffer-Heeger couplings and show that it has two sharp transitions, in contrast to pure models which exhibit one (for Su-Schrieffer-Heeger coupling) or no (for breathing-mode coupling) transition. We then show that ultracold molecules trapped in optical lattices can be used as a quantum simulator to study precisely this mixed Hamiltonian, and that the relative contributions of the two couplings can be tuned with external electric fields. The parameters of current experiments place them in the region where one of the transitions occurs. We also propose a scheme to measure the polaron dispersion using stimulated Raman spectroscopy.

Nonadiabatic preparation of spin crystals with ultracold polar molecules.

We study the growth dynamics of ordered structures of strongly interacting polar molecules in optical lattices. Using a dipole blockade of microwave excitations, we map the system onto an interacting spin-1/2 model possessing ground states with crystalline order, and describe a way to prepare these states by nonadiabatically driving the transitions between molecular rotational levels. The proposed technique bypasses the need to cross a phase transition and allows for the creation of ordered domains of considerably larger size compared to approaches relying on adiabatic preparation.

Rotational predissociation of extremely weakly bound atom-molecule complexes produced by Feshbach resonance association.

We study the rotational predissociation of atom-molecule complexes with very small binding energy. Such complexes can be produced by Feshbach resonance association of ultracold molecules with ultracold atoms. Numerical calculations of the predissociation lifetimes based on the computation of the energy dependence of the scattering matrix elements become inaccurate when the binding energy is smaller than the energy width of the predissociating state. We derive expressions that represent accurately the predissociation lifetimes in terms of the real and imaginary parts of the scattering length and effective range for molecules in an excited rotational state. Our results show that the predissociation lifetimes are the longest when the binding energy is positive, i.e., when the predissociating state is just above the excited state threshold.

Efficient numerical method for locating Feshbach resonances of ultracold molecules in external fields.

Collision properties of atoms and molecules in low temperature gases can be controlled by applying an external magnetic or electric field. The external field shifts the energy levels of the colliding particles, which gives rise to Feshbach resonances modifying the scattering cross sections. The resonances occur at particular magnitudes of the external field, where a bound state of the collision complex is degenerate with a scattering state. The positions of the resonances in the external field are usually identified by computing either the scattering cross sections or the bound states of the collision complex as functions of the external field magnitude. We propose a more efficient method for locating Feshbach resonances that requires neither of these computations. In particular, we show that the positions of Feshbach resonances can be identified by computing the log-derivative of the total wave function in a classically allowed region as a function of the external field strength. This procedure is particularly useful for locating narrow Feshbach resonances that may be hard to identify with the other methods.