Algorithms for quantum molecular dynamics: from matrix product states to path integrals

This thesis describes several novel approaches in quantum molecular dynamics for obtaining properties of molecular systems in different regimes. We investigate ground state properties of chains of linear rotors with dipole–dipole interactions via the density matrix renormalization group (DMRG), by deriving the appropriate form of the interaction operator and implementing it in ITensor. This provides us with further evidence of a quantum phase transition in this system. We also improve the sampling of Gaussian mixture distributions for finite temperature path integral Monte Carlo (PIMC) of vibronic Hamiltonians. To do this, we replace random sampling by quasi-random sampling, and improve sampling distributions by optimizing their parameters. Finally, we introduce estimators and integrators for constrained free energy simulations in path integral molecular dynamics (PIMD). This method is applied to the study of a water dimer, for which we obtain a quantum potential of mean force

Variations on PIGS: Non-standard approaches for imaginary-time path integrals

The second Rényi entropy has been used as a measure of entanglement in various model systems, including those on a lattice and in the continuum. The present work focuses on extending the existing ideas to measurement of entanglement in physically relevant systems, such as molecular clusters. We show that using the simple estimator with the regular Path Integral Ground State (PIGS) distribution is not effective, but a superior estimator exists so long as one has access to other configuration sectors. To this end, we implement the ability to explore different sectors in the Molecular Modelling Toolkit (MMTK) and use it to obtain the entanglement entropy for a test system of coupled harmonic oscillators. The Semiclassical Initial Value Representation (SC-IVR) method for real-time dynamics using the Herman-Kluk propagator is known to be an effective semiclassical method. In the present work, we combine this approximate real-time propagator with exact and approximate ground state wavefunctions in order to find ground state survival amplitudes. The necessary integrals are first performed numerically on a grid (which is feasible for only low-dimensional systems) and then stochastically using MMTK (which has applicability to high-dimensional systems). The stochastic approach is used to compare two estimators, and it is again demonstrated that better results are obtained in a specialized configuration sector

Ground states of linear rotor chains via the density matrix renormalization group

In recent years, experimental techniques have enabled the creation of endofullerene peapod nanomolecular assemblies. It was previously suggested that the rotor model resulting from the placement of dipolar linear rotors in one-dimensional lattices at low temperature has a transition between ordered and disordered phases. We use the density matrix renormalization group (DMRG) to compute ground states of chains of up to 50 rotors and provide further evidence of the phase transition in the form of a diverging entanglement entropy. We also propose two methods and present some first steps towards rotational spectra of such nanomolecular assemblies using DMRG. The present work showcases the power of DMRG in this new context of interacting molecular rotors and opens the door to the study of fundamental questions regarding criticality in systems with continuous degrees of freedom.Comment: 5 pages, 4 figure

Quantifying entanglement of rotor chains using basis truncation: Application to dipolar endofullerene peapods

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in Halverson, T., Iouchtchenko, D., & Roy, P.-N. (2018). Quantifying entanglement of rotor chains using basis truncation: Application to dipolar endofullerene peapods. Journal of Chemical Physics, 148(7), 074112 and may be found at https://doi.org/10.1063/1.5011769We propose a variational approach for the calculation of the quantum entanglement entropy of assemblies of rotating dipolar molecules. A basis truncation scheme based on the total angular momentum quantum number is proposed. The method is tested on hydrogen fluoride (HF) molecules confined in C60 fullerene cages themselves trapped in a nanotube to form a carbon peapod. The rotational degrees of freedom of the HF molecules and dipolar interactions between neighboring molecules are considered in our model Hamiltonian. Both screened and unscreened dipoles are simulated and results are obtained for the ground state and one excited state that is expected to be accessible via a far-infrared collective excitation. The effect of basis truncation on energetic and entanglement properties is examined and discussed in terms of size extensivity. It is empirically found that for unscreened dipoles, a total angular momentum cutoff that increases linearly with the number of rotors is required in order to obtain proper system size scaling of the chemical potential and entanglement entropy. Recent experiments [A. Krachmalnicoff et al., Nat. Chem. 8, 953 (2016)] suggest substantial screening of the HF dipole moment, so much smaller basis sets are required to obtain converged results in this realistic case. Static correlation functions are also computed and are shown to decay much quicker in the case of screened dipoles. Our variational results are also used to test the accuracy of perturbative and pairwise ansatz treatments.Natural Sciences and Engineering Research Council Ontario Ministry of Research and Innovation Canada Research Chair program Canada Foundation for Innovation Canada First Research Excellence Fun

A path integral ground state replica trick approach for the computation of entanglement entropy of dipolar linear rotors

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Chem. Phys. 152, 184113 (2020) and may be found at https://doi.org/10.1063/5.0004602.We calculate the second Rényi entanglement entropy for systems of interacting linear rotors in their ground state as a measure of entanglement for continuous rotational degrees of freedom. The entropy is defined in relation to the purity of a subsystem in a bipartite quantum system, and to compute it, we compare two sampling ensembles based on the path integral ground state (PIGS) formalism. This scheme centers on the replica trick and is aided by the ratio trick, both developed in this context by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)]. We study a system composed of linear quantum rotors on a lattice in one dimension, interacting via an anisotropic dipole–dipole potential. The ground state second Rényi entropies estimated by PIGS are benchmarked against those from the density matrix renormalization group for various interaction strengths and system sizes. We find that the entropy grows with an increase in interaction strength, and for large enough systems, it appears to plateau near log(2). We posit that the limiting case of many strongly interacting rotors behaves akin to a lattice of two-level particles in a cat state, in which one naturally finds an entanglement entropy of log(2).Natural Sciences and Engineering Research Council (NSERC), Grant RGPIN-2016-04403 || Ontario Ministry of Research and Innovation (MRI) || Canada Research Chair program, Grant 950-231024 || Canada Foundation for Innovation (CFI), Grant 35232 || Compute Canada || Canada First Research Excellence Fund (CFREF

Reconstructing quantum molecular rotor ground states

Nanomolecular assemblies of C$_{60}$ can be synthesized to enclose dipolar molecules. The low-temperature states of such endofullerenes are described by quantum mechanical rotors, which are candidates for quantum information devices with higher-dimensional local Hilbert spaces. The experimental exploration of endofullerene arrays comes at a time when machine learning techniques are rapidly being adopted to characterize, verify, and reconstruct quantum states from measurement data. In this paper, we develop a strategy for reconstructing the ground state of chains of dipolar rotors using restricted Boltzmann machines (RBMs) adapted to train on data from higher-dimensional Hilbert spaces. We demonstrate accurate generation of energy expectation values from an RBM trained on data in the free-rotor eigenstate basis, and explore the learning resources required for various chain lengths and dipolar interaction strengths. Finally, we show evidence for fundamental limitations in the accuracy achievable by RBMs due to the difficulty in imposing symmetries in the sampling procedure. We discuss possible avenues to overcome this limitation in the future, including the further development of autoregressive models such as recurrent neural networks for the purposes of quantum state reconstruction.Comment: 11 pages, 7 figure

Comparison of the multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) for ground state properties of linear rotor chains

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Mainali, S., Gatti, F., Iouchtchenko, D., Roy, P.-N., & Meyer, H.-D. (2021). Comparison of the multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) for ground state properties of linear rotor chains. The Journal of Chemical Physics, 154(17), 174106. https://doi.org/10.1063/5.0047090 and may be found at https://aip.scitation.org/doi/10.1063/5.0047090We demonstrate the applicability of the Multi-Layer Multi-Configuration Time-Dependent Hartree (ML-MCTDH) method to the problem of computing ground states of one-dimensional chains of linear rotors with dipolar interactions. Specifically, we successfully obtain energies, entanglement entropies, and orientational correlations that are in agreement with the Density Matrix Renormalization Group (DMRG), which has been previously used for this system. We find that the entropies calculated by ML-MCTDH for larger system sizes contain nonmonotonicity, as expected in the vicinity of a second-order quantum phase transition between ordered and disordered rotor states. We observe that this effect remains when all couplings besides nearest-neighbor are omitted from the Hamiltonian, which suggests that it is not sensitive to the rate of decay of the interactions. In contrast to DMRG, which is tailored to the one-dimensional case, ML-MCTDH (as implemented in the Heidelberg MCTDH package) requires more computational time and memory, although the requirements are still within reach of commodity hardware. The numerical convergence and computational demand of two practical implementations of ML-MCTDH and DMRG are presented in detail for various combinations of system parameters.Natural Sciences and Engineering Research Council (NSERC), Grant RGPIN-2016-04403 || Ontario Ministry of Research and Innovation (MRI) || Canada Research Chair program, Grant 950-231024 || Canada Foundation for Innovation (CFI), Grant 35232 || Canada First Research Excellence Fund (CFREF

Entangling qubit registers via many-body states of ultracold atoms

© 2016. American Physical Society, https://doi.org/10.1103/PhysRevA.93.042336Inspired by the experimental measurement of the Rényi entanglement entropy in a lattice of ultracold atoms by Islam et al. [Nature (London) 528, 77 (2015)], we propose a method to entangle two spatially separated qubits using the quantum many-body state as a resource. Through local operations accessible in an experiment, entanglement is transferred to a qubit register from atoms at the ends of a one-dimensional chain. We compute the operational entanglement, which bounds the entanglement physically transferable from the many-body resource to the register, and discuss a protocol for its experimental measurement. Finally, we explore measures for the amount of entanglement available in the register after transfer, suitable for use in quantum information applications.Natural Sciences and Engineering Research Council Canada Research Chair Program Perimeter Institute for Theoretical Physics National Science Foundation || NSF PHY11-2591

A path integral methodology for obtaining thermodynamic properties of nonadiabatic systems using Gaussian mixture distributions

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in Raymond, N., Iouchtchenko, D., Roy, P.-N., & Nooijen, M. (2018). A path integral methodology for obtaining thermodynamic properties of nonadiabatic systems using Gaussian mixture distributions. The Journal of Chemical Physics, 148(19), 194110 and may be found at https://doi.org/10.1063/1.5025058We introduce a new path integral Monte Carlo method for investigating nonadiabatic systems in thermal equilibrium and demonstrate an approach to reducing stochastic error. We derive a general path integral expression for the partition function in a product basis of continuous nuclear and discrete electronic degrees of freedom without the use of any mapping schemes. We separate our Hamiltonian into a harmonic portion and a coupling portion; the partition function can then be calculated as the product of a Monte Carlo estimator (of the coupling contribution to the partition function) and a normalization factor (that is evaluated analytically). A Gaussian mixture model is used to evaluate the Monte Carlo estimator in a computationally efficient manner. Using two model systems, we demonstrate our approach to reduce the stochastic error associated with the Monte Carlo estimator. We show that the selection of the harmonic oscillators comprising the sampling distribution directly affects the efficiency of the method. Our results demonstrate that our path integral Monte Carlo method’s deviation from exact Trotter calculations is dominated by the choice of the sampling distribution. By improving the sampling distribution, we can drastically reduce the stochastic error leading to lower computational cost.Natural Sciences and Engineering Research Council of Canada (NSERC) Ontario Ministry of Research and Innovation (MRI) Canada Foundation for Innovation (CFI) Canada Research Chair progra