arXiv subjects

Mads Greisen Højlund

Publications and source records attributed to Mads Greisen Højlund.

7 recordsLinked to original sources

arXiv2026

Don't Get Your Kroneckers in a Twist: Gaussian Processes on High-Dimensional Incomplete Grids

We introduce CUTS-GPR, a new method for performing numerically exact GPR in high-dimensional settings. The key component of CUTS-GPR is an extremely fast kernel matrix-vector product, which exhibits near-linear or even linear scaling with the amount of training data, $N$, and low-order polynomial scaling with dimensionality, $D$. This is obtained by combining an additive kernel with an incomplete grid and exploiting the resulting structure of the kernel matrix. The scalability of the matrix-vector product is verified by benchmarks with billions of data points and thousands of dimensions. We demonstrate the end-to-end scalability of CUTS-GPR by running full GPR calculations, including hyperparameter optimization, on synthetic datasets with up to $N = 4\,494\,001$ and $D = 500$. As a realistic and challenging test, we finally apply CUTS-GPR to a set of ten potential energy surfaces (PESs) with $N = 447\,265$ and $D = 24$. The calculations are completed in a matter of hours, showing that CUTS-GPR enables Bayesian modelling of high-dimensional PESs - a longstanding challenge in computational chemistry.

cs.LG↗

arXiv2024

The time-dependent bivariational principle: Theoretical foundation for real-time propagation methods of coupled-cluster type

Real-time propagation methods for chemistry and physics are invariably formulated using variational techniques. The time-dependent bivariational principle (TD-BIVP) is known to be the proper framework for coupled-cluster type methods, and is here studied from a differential geometric point of view. It is demonstrated how two distinct classical Hamilton's equations of motion arise from considering the real and imaginary parts of the action integral. The latter is new, and can in principle be used to develop novel propagation methods. Conservation laws and Poisson brackets are introduced, completing the analogy with classical mechanics. An overview of established real-time propagation methods is given in the context of our formulation of the TD-BIVP, namely time-dependent traditional coupled-cluster theory, orbital-adaptive coupled-cluster theory, time-dependent orthogonal optimized coupled-cluster theory, and equation-of-motion coupled cluster theory.

physics.chem-ph↗

arXiv2024

Optimizing the number of measurements for vibrational structure on quantum computers: coordinates and measurement schemes

One of the primary challenges prohibiting demonstrations of practical quantum advantages for near-term devices amounts to excessive measurement overheads for estimating relevant physical quantities such as ground state energies. However, with major differences between the electronic and vibrational structure of molecules, the question of how the resource requirements of computing anharmonic, vibrational states can be reduced remains relatively unexplored compared to its electronic counterpart. Importantly, bosonic commutation relations, distinguishable Hilbert spaces and vibrational coordinates allow manipulations of the vibrational system that can be exploited to minimize resource requirements. In this work, we investigate the impact of different coordinate systems and measurement schemes on the number of measurements needed to estimate anharmonic, vibrational states for a variety of three-mode (six-mode) molecules. We demonstrate an average of 3-fold (1.5-fold), with up to 7-fold (2.5-fold), reduction in the number of measurements required by employing appropriate coordinate transformations, based on an automized construction of qubit Hamiltonians from a conventional vibrational structure program.

quant-ph↗

arXiv2024

A bivariational, stable and convergent hierarchy for time-dependent coupled cluster with adaptive basis sets

We propose a new formulation of time-dependent coupled cluster with adaptive basis functions and division of the one-particle space into active and secondary subspaces. The formalism is fully bivariational in the sense of a real-valued time-dependent bivariational principle and converges to the complete-active-space solution, a property that is obtained by the use of biorthogonal basis functions. A key and distinguishing feature of the theory is that the active bra and ket functions span the same space by construction. This ensures numerical stability and is achieved by employing a split unitary/non-unitary basis set transformation: The unitary part changes the active space itself, while the non-unitary part transforms the active basis. The formulation covers vibrational as well as electron dynamics. Detailed equations of motion are derived and implemented in the context of vibrational dynamics, and the numerical behavior is studied and compared to related methods.

physics.chem-ph↗

arXiv2023

Time-dependent coupled cluster with orthogonal adaptive basis functions: General formalism and application to the vibrational problem

We derive equations of motion for bivariational wave functions with orthogonal adaptive basis sets and specialize the formalism to the coupled cluster ansatz. The equations are related to the biorthogonal case in a transparent way, and similarities and differences are analyzed. We show that the amplitude equations are identical in the orthogonal and biorthogonal formalisms, while the linear equations that determined the basis set time evolution differ by symmetrization. Applying the orthogonal framework to the nuclear dynamics problem, we introduce and implement the orthogonal time-dependent modal vibrational coupled cluster (oTDMVCC) method and benchmark it against exact reference results for four triatomic molecules as well as a 5D trans-bithiophene model. We confirm numerically that the biorthogonal TDMVCC hierarchy converges to the exact solution, while oTDMVCC does not. The differences between TDMVCC and oTDMVCC are found to be small for three of the five cases, but we also identify one case where the formal deficiency of the oTDMVCC approach results in clear and visible errors relative to the exact result. For the remaining example, oTDMVCC exhibits rather modest but visible errors.

physics.chem-ph↗

arXiv2023

Efficient time-dependent vibrational coupled cluster computations with time-dependent basis sets at the two-mode coupling level: full and hybrid TDMVCC[2]

The computation of the nuclear quantum dynamics of molecules is challenging, requiring both accuracy and efficiency to be applicable to systems of interest. Recently, theories have been developed for employing time-dependent basis functions (denoted modals) with vibrational coupled cluster theory (TDMVCC). The TDMVCC method was introduced along with a pilot implementation, which illustrated good accuracy in benchmark computations. In this paper we report an efficient implementation of TDMVCC covering the case where the wave function and Hamiltonian contain up to two-mode couplings. After a careful regrouping of terms, the wave function can be propagated with a cubic computational scaling with respect to the number of degrees of freedom. We discuss the use of a restricted set of active one-mode basis functions for each mode, as well as two interesting limits: i) the use of a full active basis where the variational modal determination amounts essentially to the variational determination of a time-dependent reference state for the cluster expansion; and ii) the use of a single function as active basis for some degrees of freedom. The latter case defines a hybrid TDMVCC/TDH approach which can obtain an even lower computational scaling. The resulting computational scaling for hybrid and full TDMVCC[2] are illustrated for polyaromatic hydrocarbons (PAHs) with up to 264 modes. Finally, computations on the internal vibrational redistribution on benzoic acid (39 modes) are used to show the faster convergence of TDMVCC/TDH hybrid computations towards TDMVCC compared to simple neglect of some degrees of freedom.

physics.chem-ph↗

arXiv2023

General exponential basis set parametrization: Application to time-dependent bivariational wave functions

We present equations of motion (EOMs) for general time-dependent wave functions with exponentially parametrized biorthogonal basis sets. The equations are fully bivariational in the sense of the time-dependent bivariational principle (TDBVP) and offer an alternative, constraint free formulation of adaptive basis sets for bivariational wave functions. We simplify the highly non-linear basis set equations using Lie algebraic techniques and show that the computationally intensive parts of the theory are in fact identical to those that arise with linearly parametrized basis sets. Our approach thus offers easy implementation on top of existing code with minimal computational overhead in the context of both nuclear dynamics and time-dependent electronic structure. Computationally tractable working equations are provided for single and double exponential parametrizations of the basis set evolution. The EOMs are generally applicable for any value of the basis set parameters, unlike the approach of transforming the parameters to zero at each evaluation of the EOMs. We show that the basis set equations contain a well-defined set of singularities, which are identified and removed by a simple scheme. The exponential basis set equations are implemented in conjunction with time-dependent vibrational coupled cluster with time-dependent modals (TDMVCC) and we investigate the propagation properties in terms of the average integrator step size. For the systems we test, the exponentially parametrized basis sets yield slightly larger step sizes compared to linearly parametrized basis set.

physics.chem-ph↗