arXiv subjects

Loïc Joubert-Doriol

Publications and source records attributed to Loïc Joubert-Doriol.

11 recordsLinked to original sources

arXiv2025

Quantum computer formulation of the FKP-operator eigenvalue problem for probabilistic learning on manifolds

We present a quantum computing formulation to address a challenging problem in the development of probabilistic learning on manifolds (PLoM). It involves solving the spectral problem of the high-dimensional Fokker-Planck (FKP) operator, which remains beyond the reach of classical computing. Our ultimate goal is to develop an efficient approach for practical computations on quantum computers. For now, we focus on an adapted formulation tailored to quantum computing. The methodological aspects covered in this work include the construction of the FKP equation, where the invariant probability measure is derived from a training dataset, and the formulation of the eigenvalue problem for the FKP operator. The eigen equation is transformed into a Schrödinger equation with a potential V, a non-algebraic function that is neither simple nor a polynomial representation. To address this, we propose a methodology for constructing a multivariate polynomial approximation of V, leveraging polynomial chaos expansion within the Gaussian Sobolev space. This approach preserves the algebraic properties of the potential and adapts it for quantum algorithms. The quantum computing formulation employs a finite basis representation, incorporating second quantization with creation and annihilation operators. Explicit formulas for the Laplacian and potential are derived and mapped onto qubits using Pauli matrix expressions. Additionally, we outline the design of quantum circuits and the implementation of measurements to construct and observe specific quantum states. Information is extracted through quantum measurements, with eigenstates constructed and overlap measurements evaluated using universal quantum gates.

quant-ph↗

arXiv2024

Optimizing Conical Intersections Without Explicit Use of Non-Adiabatic Couplings

We present two alternative methods for optimizing minimum energy conical intersection (MECI) molecular geometries without knowledge of the derivative coupling (DC). These methods are based on the utilization of Lagrange multipliers: i) one method uses an approximate calculation of the DC, while the other ii) do not require the DC. Both methods use the fact that information of the DC is contained in the Hessian of the squared energy difference. Tests done on a set of small molecular systems, in comparison with other methods, show the ability of the proposed methods to optimize MECIs. Finally, we apply the methods to the furimamide molecule, to optimize and characterize its S$_1$ /S$_2$ MECI, and to optimizing the S$_0$ /S$_1$ MECI of the silver trimer.

physics.chem-ph↗

arXiv2022

Quantum Kinetic Rates within the Nonequilibrium Steady State

The nonequilibrium steady state (NESS) of a quantum network is central to a host of physical and biological scenarios. Examples include natural processes such as vision and photosynthesis, as well as technical devices such as photocells, both activated by incoherent light (e.g. sunlight) and leading to quantum transport. Here, a completely general approach to defining components of a quantum network in the NESS, and obtaining rates of processes between these components is provided. Quantum effects are explicitly included throughout, both in (a) defining network components via projection operators, and (b) in determining the role of coherences in rate processes. As examples, the methodology is applied to model cases, two versions of the V-level system, and to the spin-boson model, wherein the role of the environment and of internal system properties in determining the rates is examined. In addition, the role of Markovian vs. non-Markovian contributions is quantified, exposing conditions under which NESS rates can be obtained by perturbing the nonequilibrium steady state.

quant-ph↗

arXiv2022

A variational approach for linearly dependent moving bases in quantum dynamics: application to Gaussian functions

In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on: i) the definition of a linearly independent working space, and ii) a variational construction of the propagator over finite time-steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are done on a quartic double-well potential with Gaussian basis function whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.

quant-ph↗

arXiv2022

The moving crude adiabatic alternative to the adiabatic representation in excited state dynamics

The choice of the electronic representation in on-the-fly quantum dynamics is crucial. The adiabatic representation is appealing since adiabatic states are readily available from quantum chemistry packages. The nuclear wavepackets are then expanded in a basis of Gaussian functions, which follow trajectories to explore the potential energy surfaces and approximate the potential using a local expansion of the adiabatic quantities. Nevertheless, the adiabatic representation is plagued with severe limitations when conical intersections are involved: the diagonal Born-Oppenheimer corrections (DBOCs) are non-integrable, and the geometric phase effect on the nuclear wavepackets cannot be accounted for unless a model is available. To circumvent these difficulties, the moving crude adiabatic (MCA) representation was proposed and successfully tested in low energy dynamics where the wavepacket skirts the conical intersection. We assess the MCA representation in the case of non-adiabatic transitions through conical intersections. First, we show that using a Gaussian basis in the adiabatic representation indeed exhibits the aforementioned difficulties with a special emphasis on the possibility to regularize the DBOC terms. Then, we show that MCA is indeed able to properly model non-adiabatic transitions. Tests are done on linear vibronic coupling models for the bis(methylene) adamantyl cation and the butatriene cation.

physics.chem-ph↗

arXiv2022

Controlling energy conservation in quantum dynamics with independently moving basis functions: Application to Multi-Configuration Ehrenfest

Application of the time-dependent variational principle to a linear combination of frozen-width Gaussians describing the nuclear wavefunction provides a formalism where the total energy is conserved. The computational downside of this formalism is that trajectories of individual Gaussians are solutions of a coupled system of differential equations, limiting implementation to serial propagation algorithms. To allow for parallelization and acceleration of the computation, independent trajectories based on simplified equations of motion were suggested. Unfortunately, within practical realizations involving finite Gaussian bases, this simplification leads to breaking the energy conservation. We offer a solution for this problem by using Lagrange multipliers to ensure the energy and norm conservation regardless of basis function trajectories or basis completeness. We illustrate our approach within the Multi-Configuration Ehrenfest method considering a linear vibronic coupling model.

physics.chem-ph↗

arXiv2018

Nonadiabatic Quantum Dynamics with Frozen-Width Gaussians

We review techniques for simulating fully quantum nonadiabatic dynamics using the frozen-width moving Gaussian basis functions to represent the nuclear wavefunction. A choice of these basis functions is primarily motivated by the idea of the on-the-fly dynamics that will involve electronic structure calculations done locally in the vicinity of each Gaussian center and thus avoiding the "curse of dimensionality" appearing in large systems. For quantum dynamics involving multiple electronic states there are several aspects that need to be addressed. First, the choice of the electronic state representation is one of most defining in terms of formulation of resulting equations of motion. We will discuss pros and cons of the standard adiabatic and diabatic representations as well as the relatively new moving crude adiabatic representation. Second, if the number of electronic states can be fixed throughout the dynamics, the situation is different for the number of Gaussians needed for an accurate expansion of the total wavefunction. The latter increases its complexity along the course of the dynamics and a protocol extending the number of Gaussians is needed. We will consider two common approaches for the extension: 1) spawning and 2) cloning. Third, equations of motion for individual Gaussians can be chosen in different ways, implications for the energy conservation related to these ways will be discussed. Finally, to extend the success of moving basis approaches to quantum dynamics of open systems we will consider the Non-stochastic Open System Schrödinger Equation (NOSSE).

physics.chem-ph↗

arXiv2018

Variational nonadiabatic dynamics in the moving crude adiabatic representation: Further merging of nuclear dynamics and electronic structure

A new methodology of simulating nonadiabatic dynamics using frozen-width Gaussian wavepackets within the moving crude adiabatic representation with the on-the-fly evaluation of electronic structure is presented. The main feature of the new approach is elimination of any global or local model representation of electronic potential energy surfaces, instead, the electron-nuclear interaction is treated explicitly using the Gaussian integration. As a result, the new scheme does not introduce any uncontrolled approximations. The employed variational principle ensures the energy conservation and leaves the number of electronic and nuclear basis functions as the only parameter determining the accuracy. To assess performance of the approach, a model with two electronic and two nuclear spacial degrees of freedom containing conical intersections between potential energy surfaces has been considered. Dynamical features associated with nonadiabatic transitions and nontrivial geometric (or Berry) phases were successfully reproduced within a limited basis expansion.

physics.chem-ph↗

arXiv2017

Geometric phase effects in excited state dynamics through a conical intersection in large molecules: N-dimensional linear vibronic coupling model study

We investigate geometric phase (GP) effects in nonadiabatic transitions through a conical intersection (CI) in an N-dimensional linear vibronic coupling (ND-LVC) model. This model allows for the coordinate transformation encompassing all nonadiabatic effects within a two-dimensional (2D) subsystem while the other N-2 dimensions form a system of uncoupled harmonic oscillators identical for both electronic states and coupled bi-linearly with the subsystem coordinates. The 2D subsystem governs ultra-fast nonadiabatic dynamics through the CI and provides a convenient model for studying GP effects. Parameters of the original ND-LVC model define the Hamiltonian of the transformed 2D subsystem and thus influence GP effects directly. Our analysis reveals what values of ND-LVC parameters can introduce symmetry breaking in the 2D subsystem that diminishes GP effects.

physics.chem-ph↗

arXiv2017

Quantum Nonadiabatic Cloning of Entangled Coherent States

We propose a systematic approach to the basis set extension for nonadiabatic dynamics of entangled combination of nuclear coherent states (CSs) evolving according to the time-dependent variational principle (TDVP). TDVP provides a rigorous framework for fully quantum nonadiabatic dynamics of closed systems, however, quality of results strongly depends on available basis functions. Starting with a single nuclear CS replicated vertically on all electronic states, our approach clones this function when replicas of the CS on different electronic states experience increasingly different forces. Created clones move away from each other (decohere) extending the basis set. To determine a moment for cloning we introduce generalized forces based on derivatives that maximally contribute to a variation of the total quantum action and thus account for entanglement of all basis functions.

physics.chem-ph↗

arXiv2014

Non-stochastic matrix Schrödinger equation for open systems

We propose an extension of the Schrödinger equation for a quantum system interacting with environment. This equation describes dynamics of auxiliary wave-functions $\mathbf{m}$, from which the system density matrix can be reconstructed as $\hatρ = \mathbf{m} \mathbf{m}^\dagger$. We formulate a compatibility condition, which ensures that the reconstructed density satisfies a given quantum master equation for the system density. The resulting non-stochastic evolution equation preserves positive-definiteness of the system density and is applicable to both Markovian and non-Markovian system-bath treatments. Our formalism also resolves a long-standing problem of energy non-conservation in the time-dependent variational principle applied to mixed states of closed systems.

physics.chem-ph↗